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+arXiv:2301.13608v1  [quant-ph]  25 Jan 2023
+Orbit quantization in a retarded harmonic oscillator
+Alvaro G. Lopez1
+1Nonlinear Dynamics, Chaos and Complex Systems Group,
+Departamento de F´ısica, Universidad Rey Juan Carlos,
+Tulip´an s/n, 28933 M´ostoles, Madrid, Spain
+(Dated: February 1, 2023)
+Abstract
+We study the dynamics of a damped harmonic oscillator in the presence of a retarded potential
+with state-dependent time-delayed feedback. In the limit of small time-delays, we show that the
+oscillator is equivalent to a Li´enard system. This allows us to analytically predict the value of the
+first Hopf bifurcation, unleashing a self-oscillatory motion. We compute bifurcation diagrams for
+several model parameter values and analyse multistable domains in detail. Using the Lyapunov
+energy function, two well-resolved energy levels represented by two coexisting stable limit cycles
+are discerned. Further exploration of the parameter space reveals the existence of a superposition
+limit cycle, encompassing two degenerate coexisting limit cycles at the fundamental energy level.
+When the system is driven very far from equilibrium, a multiscale strange attractor displaying
+intrinsic and robust intermittency is uncovered.
+1
+
+I.
+INTRODUCTION
+The importance of time-delayed feedback has been extensively confirmed across many
+disciplines in science, ranging from mechanical physical systems [1], to chemical complex
+reactions [2], or complex biological systems, as for example cardiac oscillations [3], the prop-
+agation of impulses through the nervous system [4] or the modelling of the cell cycle [5]. Time
+delays are also inescapable to understand climate phenomena, such as El Ni˜no-Southern Os-
+cillation [6]. In epidemiology and population dynamics retardations can be crucial as well
+[7], just as much as they are in the modelling of economic cycles [8] or transmission lines
+[9]. Indeed, whenever the forces between two physical interacting bodies are mediated by a
+medium or a field, or whenever large causal chains in a network of connections are reduced
+in a model, time-delays must be present. Consequently, the existence of retardations in
+differential equations describing the evolution of dynamical systems should be taken more
+as the rule, than as the exception.
+However, this contrasts with the standard practice, where ordinary differential equations
+are much preferred for their simplicity, both from an analytical and a numerical point of view.
+Furthermore, not much attention has been dedicated to study dynamical systems where the
+the time retardation is state-dependent [10–13], specially in the field of fundamental physics.
+Recent findings in the study of extended electrodynamic bodies using the retarded Li´enard-
+Wiechert potential have shown that these particles can experience nonlinear oscillations due
+to self-forces, with a frequency similar to the “zitterbewegung” frequency [13, 14].
+The
+crucial importance of time-delay in atomic physics had initially been stressed by C. K. Raju
+[15]. Later on, the expression “Atiyah’s hypothesis” was coined, after Sir Michael Atiyah
+claimed the necessity of functional differential equations to faithfully represent the dynamics
+of microscopic bodies, in a lecture entitled “The nature of space”, which was the first annual
+Einstein Public Lecture, delivered in 2005 [16].
+By the same year, Yves Couder and his collaborators demonstrated empirically the po-
+tential of hydrodynamic quantum analogs consisting of silicone oil droplets bouncing on a
+vibrating bath to describe quantum phenomena [17, 18]. These pilot-wave mechanical sys-
+tems present striking similarities with the quantum mechanics of electromagnetically charged
+bodies, such as orbit quantization [19], diffraction and interference phenomena through slits
+[20], tunneling over barriers [21], or the entanglement of particles [22]. In mathematical
+2
+
+models where the fluid is not explicitly represented [23], these features translate into a time-
+delayed feedback arising from the self-affection of the particle through the fluid medium.
+Just as it occurs in electrodynamics, perturbations produced in the past by the particle can
+affect it at the present time, introducing memory effects that can trigger its self-propulsion
+[23].
+In the present work we demonstrate that the phase space orbits of a harmonic oscillator
+with state-dependent time-delayed feedback are quantized and organized conforming a two-
+level system. We also report new compelling dynamical phenomena, as for example the
+superposition of quantized orbits and the existence of robust intermittency. The paper is
+organized as follows. In Sec. 2 we introduce the mathematical model of our oscillator and
+explain its origin.
+Then, in Sec.
+3 we analytically bridge state-dependent time-delayed
+oscillators and Li´enard systems, proving that the periodic motion corresponds to a self-
+oscillation [24]. In the following section we show that there exist quantized orbits, which
+are well-resolved in the energy landscape given by the harmonic external potential. Secs. 4
+and 5 are dedicated to introduce two new phenomena, which might be crucial to understand
+other aspects of microscopic physics, such as the superposition of orbits and the passage
+of particles over external potential barriers. Finally, in the conclusions, we summarize the
+main results of the present work and the future perspectives, as usual.
+II.
+MODEL
+We use an apparently simple model consisting in a harmonic oscillator with linear damp-
+ing, according to Stokes’s law of dissipation [25], an external quadratic potential V (x) repre-
+senting Hooke’s law and another quadratic potential with state-dependent time-delay Q(xτ),
+where xτ ≡ x(t − τ(x)). Following traditional studies in classical electrodynamics, we bor-
+row the concept of retarded potential [26] hereafter to denote this self-excited contribution.
+Therefore, we can write our dynamical equation of motion in the form
+m¨x + µ ˙x + dV
+dx + dQ
+dxτ
+= 0,
+(1)
+where m is the mass of the oscillator and µ represents the rate of dissipation.
+This model is a simplified version of an oscillator recently encountered in the study of the
+dynamics of extended electromagnetic bodies, for which the presence of self-forces produces
+3
+
+self-oscillations through a Hopf bifurcation [13, 14].
+Thus, if desired, we can physically
+interpret to some extent this new time-delay term as the result of some complicated mass
+self-interactions in a mass-spring system, arising from the mass’ structure. A similar, though
+more sophisticated, model has been previously used in the literature to study the effect of
+state-dependence of the delay on the phenomenon of vibrational resonance [11]. In summary,
+we can mathematically express the external potential as V (x) = kx2/2 and the same holds
+for Q(xτ) = αx2
+τ/2, yielding the differential equation
+m¨x(t) + µ ˙x(t) + kx(t) + αx(t − τ(x)) = 0.
+(2)
+For convenience and without loss of generality, we shall consider m = 1, k = 1 and µ = 0.1
+hereafter. We are intending to describe the dynamics of our oscillator in the phase space
+representation, as it is traditionally done in the study of nonlinear dynamical systems,
+specially regarding mechanical and electronic oscillators. However, we notice that, rigorously
+speaking, the true phase space of our dynamical systems is infinite-dimensional, since history
+functions have to be provided to integrate the Eq. (2), instead of mere initial conditions [27].
+In the phase space (x, y) we can write the differential equation as follows
+˙x = y
+(3)
+˙y = −0.1y − x − αxτ.
+(4)
+One of the crucial issues of the present work is the nature of the function τ(x). Some
+constraints on this function to ensure that the system is well-behaved must be provided.
+For example, we want the trajectories to remain bounded in the external well for x → ±∞,
+so that the state-dependent delay decays to zero asymptotically. It is also reasonable to
+demand that the delay function τ(x) remains bounded all over its domain, guaranteeing
+that the feedback coming from the past history of the dynamics does not extend to minus
+infinity. In this manner, we bound the memory of this non-Markovian system to a finite
+domain of its temporal past. Finally, symmetry with respect to spatial reflections (x → −x)
+is also present in the original model [13]. Moreover, as we show ahead, we can exploit the
+degeneracy introduced by this symmetry to obtain intriguing new dynamical phenomena. A
+simple function that has been used in previous works is the Gaussian distribution [11], which
+accounts for these three requirements. In conclusion, we assume that τ(x) = τ0e−x2/2σ2, and
+fix σ = 1/
+√
+2 unless otherwise stated. The parameter τ0 represents the maximum value of
+4
+
+the time-delay feedback, attained at the centre of the potential well. It consitutes one of the
+two key parameters investigated in the present study.
+All things considered, we have a time-delayed nonlinear oscillator with two independent
+parameters α and τ0. Interestingly, we note that the system’s nonlinearity comes entirely
+from the retardation, since both potentials have been assumed harmonic. Even though this
+system has been designed following previous findings in electrodynamics, we would like to
+stress all the simplifications performed. Firstly, the delay in the functional differential equa-
+tion appearing in Ref. [13] depends both on the speed and the acceleration of the particle.
+Secondly, such differential equation is of the advanced type [13], since the acceleration and
+the speed appearing in the Li´enard-Wiechert potentials are retarded themselves. Unfortu-
+nately, numerical schemes to integrate advanced differential equations with state-dependent
+delays are lacking. This has motivated the authors to develop the present approximated
+model. Finally, some speed-dependent nonlinearities appearing in the dissipation term and
+also in the restoring force term have been neglected.
+They are related to the Lorentz’s
+gamma factor, which is required to comply with the principle of relativity in classical elec-
+trodynamics. Consequently, the present model remains somewhat abstract. It is not our
+purpose to rigorously fit it to any specific physical system. We just use it to illustrate some
+physical phenomena that are frequently believed to belong exclusively to the atomic realm
+of physics.
+III.
+RELATED LI´ENARD SYSTEM
+Given the fact that there is dissipation in the system, in the absence of retardation
+(α = 0 or τ0 = 0), it can be immediately proved using the Lyapunov energy function
+E(x, y) = (x2 + y2)/2 that the rest state at the equilibrium x = 0 is the only global stable
+fixed point of the system, which asymptotically attracts all the initial conditions in the phase
+space [28]. We recall that this function only comprises the conservative part of the energetic
+content of our dynamical system. However, when the retarded potential is activated for
+α > 0, as we increase τ0 bellow a critical value, a Hopf bifurcation appears destabilizing
+such an equilibrium point. A fundamental energy level appears with non-zero energy fluctu-
+ations, in which the system performs limit cycle oscillations. Therefore, the orbit becomes
+quantized as a consequence of the time-delayed feedback. The system becomes unstable and
+5
+
+locally active [29], performing a periodic self-oscillatory motion around the minimum of the
+square well potential. Of course, this is only possible at the expense of an energy input in
+the system, which must come from external field sources [13, 24]. Therefore, the present
+dynamical system must be regarded as as a non-equilibrium open physical system, whose
+nonlinear periodic motion can be interpreted as a cyclic thermodynamic engine [30]. Due
+to the existence of energy losses, these dynamical systems are frequently named dissipative
+structures. This contrasts to conservative dynamical systems, which are generally equipped
+with a symplectic structure [31].
+We now prove that the dynamics of the system is a self-oscillation, triggered by the well-
+known Hopf bifurcation. To compute the value of τ0 at the bifurcation point, we approximate
+this system to a Li´enard system. Expanding in Taylor series the retarded potential to second
+order yields
+x(t − τ(x)) = x(t) − τ(x) ˙x(t) + 1
+2τ 2(x)¨x(t) + O(τ 3).
+(5)
+When the time-delay is small, we can neglect the third and higher order terms, substitute
+the two leading order contributions in the Eq. 2, and obtain
+¨x + f(x) ˙x + g(x) = 0,
+(6)
+where the functions f(x) = (µ − ατ(x))/(1 + ατ 2(x)/2) and g(x) = (k + α)/(1 + ατ 2(x)/2)
+have been defined. Thus, as we can see, small retardations have two fundamental physical
+consequences. Firstly, an antidamping correction to the drag force appears to first order.
+Secondly, the inertia of the mass becomes dependent on the dynamical state through its
+evolution along the trajectory. To demonstrate that this Li´enard system with f(x) and
+g(x) as defined, fulfils the conditions required to produce the Hopf bifurcation, we appeal to
+Li´enard’s theorem. Given the importance of this theorem, we first enunciate it, so that the
+reader is aware of all the technical details [32]. For this purpose it is convenient to introduce
+the primitive function F(x) =
+� x
+0 f(s)ds, since it is alluded in the theorem, which reads
+Theorem 1 (Li´enard, 1928) Under the assumptions that the functions F(x), g(x) ∈
+C1(R) are odd, xg(x) > 0 for x ̸= 0, F(0) = 0, F ′(0) < 0 and F has one single root
+for x = a, beyond which it increases monotonically to infinity, it follows that there exists
+only one limit cycle and that it is stable.
+For a proof of the theorem we refer the reader to Ref. [33]. It is immediate to confirm
+that all the requirements for the existence of the Hopf bifurcation are accomplished. Indeed,
+6
+
+the function F fulfils F(0) = 0, has a root at a ∈ R+, and for x ≥ a we find that it
+increases monotonically towards infinity (see Fig. 1(a)). This follows from an analytical
+estimation of F(x), which can be provided by neglecting the postive term τ 2(x) ≪ 2/α in
+the denominator of f(x), yielding the function F(x) = µx −
+√
+2πατ0 erf(x/
+√
+2), with erf(x)
+the error function. It is also confirmed by the numerical solution of the integral, which has
+been computed using the trapezoidal rule. Recall, this rule works by approximating the
+region under the graph of a function as a trapezoid and calculating its area. The condition
+xg(x) > 0 is trivially verified, while the condition F ′(0) > 0 can be used to find out the value
+of the Hopf bifurcation, since F ′(0) = (µ − ατ0)/(1 + ατ 2
+0 /2), which entails that τ0 > µ/α.
+For the parameter values here considered and α = 1/2, we can approximate the value of
+the point where the Hopf bifurcation takes place to τ0 = 1/5. This analytical results holds
+nicely, as depicted in Fig. 1(b).
+In the following section we show that, as we push the
+system even further from thermodynamic equilibrium [30] by increasing the effects of the
+time-delay feedback, the oscillator undergoes further bifurcation phenomena producing a
+second quantized excited orbit, at a higher energy level.
+IV.
+ENERGY LEVELS: MULTISTABILITY
+Once we have demonstrated that the time retardation can destabilize the rest state,
+generating a fundamental energy level with zero-point fluctuations, it is worth asking if by
+posing the system even further from equilibrium, i.e. by increasing the delay feedback, orbits
+with larger amplitude representing excited energy levels can appear. For this purpose we
+have computed the bifurcation diagrams of the related maxima map of the system. As it is
+well-known, this map can be constructed by computing the local maxima of the temporal
+series. Together with its related minima map, this is the simplest general way to discretize
+the dynamics of delayed differential equations.
+We insist again that, strictly speaking,
+the phase space of retarded differential equations is infinite-dimensional.
+An alternative
+possibility is to build an embedding from the temporal series and to construct a Poincar´e
+section out of it. However, this technique it computationally more intensive and does not
+produce better insights into the dynamics of the system.
+To compute the bifurcation diagrams we have to integrate the Eq. (2). This requires
+to consider history functions [27].
+Since in the absence of time retardation the system
+7
+
+Figure 1: Hopf bifurcation. The conditions for Li´enard’s theorem are shown. (a) The function
+F for τ0 = 1 and α = 1/2. It clearly verifies F(0) = 0, F ′(0) < 0, has a single root for a ≈ 10 and
+increases monotonically without bounds thereafter. An analytical approximation using the error
+function. (b) The bifurcation diagram, with the maximum value of x along the orbit resulting
+from the numerical solution of the retarded differential (red curve), showing a Hopf bifurcation for
+the value τ0c = 0.195, very close to the analytical prediction of the related Li´enard system, which
+corresponds to τ0c = 0.2. The rest state becomes unstable beyond the critical bifurcation point,
+entailing the zero-point fluctuations of the fundamental energy level.
+is harmonic, we consider that the most natural choice of history functions are periodic
+solutions, Therefore, we take the functions x(t) = A sin(ωt + ϕ) for t < 0. Moreover, this
+choice can be used later to ascertain relevant dynamical aspects of the system under finite-
+time external periodic drivings, which can be physically interpreted as brief pulses exerted
+on the oscillator. As it is expected, external perturbations acting on the system can produce
+transitions between the energy levels.
+Because our aim is to figure out if there exists multistable parameter regimes, repre-
+sented by two or more coexisting stable limit cycles, we throw ten different initial conditions
+randomly chosen in the range A ∈ [0, 3], ω ∈ [−π, π] and ϕ ∈ [0, π]. We compute the trajec-
+tories in the temporal interval t ∈ [0, 2000] using a residual order integrator implemented in
+MATLAB. Transients as long as seven-tenths or even larger of the whole temporal series are
+8
+
+3
+Numerical
+Stable
+ Analytical approximation
+1
+2
+F
+mac
+Hopf
+0
+0
+Toc
+Unstable
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+10
+20
+To
+(a)
+(b)1.5 F
+1.0
+0.5
+0.0
+0.5
+0
+5
+10
+15
+20
+253
+2.5
+20.4
+0.51.5
+1
+0.5
+0
+-0.5
+0
+0.1
+0.2
+0.3Figure 2: Bifurcation diagrams (α > 0). The bifurcation diagrams of the maxima map of x are
+represented for increasing values of the maximum delay τ0. A total number of ten initial randomly
+chosen histories have been used, and depicted using two different colors to represent the asymptotic
+sets, clearly distinguishing multistable regions (green background). (a) For α = 0.5 several Hopf
+bifurcations (arrows), interrupted by an amplitude death region (AD), end in a quasiperiodic
+route to chaos.
+Multistability (MS) starts at the critical value τ0c = 6.1, when a high energy
+limit cycle (green) is born, coexisting with the fundamental energy level, which involves periodic,
+quasiperiodic or chaotic attractors (blue). (b) For α = 0.9 similar results are observed, except for
+the disappearance of the amplitude death region, and the fact that the mustilstable regions appear
+and disappear through several crises.
+discarded, since time-delayed systems usually display long transient phenomena [34]. Finally,
+we obtain the maxima map and represent these points for 1200 varying parameter values of
+the maximum time-delay in the range τ0 ∈ [0, 11]. Recalling that several conditions can lead
+to the same asymptotic limit cycle, we have coloured the bifurcations diagrams in two colors,
+to clearly distinguish the two energy levels, whenever they exist. As we can see in Fig. 2(a),
+and as detailed in the previous section, for α = 1/2, as the time-delay is increased from
+zero, a first Hopf bifurcation reveals at τ0 = 1/5. Then, if we increase further the maximum
+delay τ0, the fundamental orbit first enlarges reaching a maximum amplitude of xmax = 6.0
+for τ0 close to 2.0, then shrinks again and, finally, it disappears. This is the well-studied
+9
+
+AD
+MS
+6
+MS
+8
+6
+4
+mac
+4
+Cmac
+2
+2
+α= 0.5
+0
+α = 0.9
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+To
+To
+(a)
+(b)10
+8
+6: 2. .:
+8
+104
+2
+0
+2
+0
+2
+4
+6AD
+MS
+6
+Cmac
+2
+0
+4
+6
+8
+10
+2
+To6
+58
+104
+3
+2
+0
+0
+2
+4
+6phenomena of amplitude death (AD), frequently displayed by time-delayed differential equa-
+tions [35]. However, for values beyond τ0 = 5.25 a Hopf bifurcation shows anew, which is
+now followed by a secondary Hopf bifurcation, giving rise to quasiperiodic motion. For an
+approximate critical value of the maximum time-delay τ0c = 6.1, a new periodic limit cycle
+of higher amplitude is born, rendering a multistable (MS) two-level system. The second
+quantized excited state shall persist all along the bifurcation diagram and remains periodic,
+although its amplitude shrinks as the retardation increases. Then, the fundamental energy
+level experiences further bifurcations through a quasiperiodic route to chaos [36], ending in a
+chaotic strange attractor (see Figs. 3(a) and (b)). The chaotic attractor experiences a crisis
+at approximately τ0 = 9.0, yielding two coexisting periodic limit cycles, which are depicted
+in Fig. 3(c). For α = 0.9 similar results are observed in Fig. 2(b), except for the fact that the
+amplitude death region is missing, and also the multistable regions appear and disappear
+intermittently along the bifurcation diagram through several crises. Some new interesting
+dynamical features are also discerned, as for example the coexistence of two quasiperiodic
+attractors for τ0 = 9.5. Naturally, whenever a strange attractor disappears through a cri-
+sis, transient chaos phenomena [37] can be observed, where a trajectory can spend large
+transients in the fundamental level, and then spiral away towards the first excited level.
+We now investigate if the two energy levels are well-resolved across the different energy
+shells. For this purpose, and also for aesthetic purposes, we have used a value of α = 0.9
+and τ0 = 5.87 to illustrate this two-level system. For these parameter values, we can find
+two stable symmetric degenerate coexisting orbits at the fundamental level, as shown in
+Fig. 4(a). This degeneracy is a consequence of the fact that the Eq. (2) is invariant under
+spatial reflections, and the splitting of these two orbits constitutes a typical phenomenon
+of symmetry breaking at the fundamental energy level. We recall that symmetry breaking
+is an ordinary phenomenon frequently observed in nonlinear self-excited systems [29]. In
+Fig. 4(b) we have plotted the harmonic external potential in red. We have used the Lyapunov
+energy function E(x, y) = (x2 + y2)/2 to compute the energy of the particle along the
+limit cycles [33], and numerically integrated its average value along these periodic orbits,
+using the trapezoidal rule once more. The average energy has been plotted in the energy
+diagram in dashed lines, together with the energy fluctuations that the stable quantized
+orbits experience along their periodic motions. As we can clearly appreciate, despite the
+fact that the fluctuations are substantial and the oscillator performs excursions out of shell
+10
+
+Figure 3: Multistability. Three phase space portraits along a quasiperiodic route to chaos in the
+multistable region for α = 0.5 (see first bifurcation diagram in Fig. 2). (a) Two doubly-degenerate
+quasiperiodic attractors coexist with a higher amplitude periodic limit cycle surrounding them.
+(b) The quasiperiodic attractors have merged into a single chaotic attractor as the delay increases,
+while the most exterior limit cycle has enlarged. (c) The chaotic orbit disappears through a crisis
+for even higher time-delays, yielding two coexisting periodic limit cycles of different amplitude.
+with respect the average energy, the two levels are well differentiated and they do not overlap
+in the energy diagram. Consequently, it can be safely stated that the present system displays
+quantized stable orbits at two independent energies, which can be denoted as E1 = 0.4 and
+E2 = 26.
+To conclude this section, we have also studied the basins of attraction of the system for
+this particular situation, to ascertain if there exists sensitivity to external perturbations.
+This is of crucial importance, for if an external perturbation is effected on this system,
+we may wonder which of the possible asymptotic limit cycles is attained in the end. Or,
+equivalently, we may ask about the ultimate energy of the oscillator when it is perturbed from
+the outside. In Fig. 5 we show the basins of attraction in the history subspace of periodic
+functions. We have used a resolution of 300 × 300, fixed an amplitude of A = 0.43, and
+computed trajectories until they get close enough to one of the three attractors. Depending
+on which attractor is approached, each initial history is plotted in the parameter space with
+a different color. As we can see, the basins are fractalized, what introduces unpredictability
+at all the scales of precision [38]. However, this basin does not posses the Wada property
+[27]. In general, unless infinite experimental accuracy is accessible, the best that we can say
+11
+
+4
+3
+22
+3
+41
+0
+-1
+-2
+-3
+-4
+-4
+-3
+-2
+-1
+0
+12
+1.5
+11
+1.5
+20.5
+0
+-0.5
+-1
+-1.5
+-2
+-2
+-1.5
+-1
+-0.5
+0
+0.54
+3
+21
+2
+31
+0
+-1
+-2
+-3
+-3
+-2 
+-1
+0Figure 4: Energy levels. A two-level system for α = 0.9 and τ0 = 5.87. The fundamental level E1
+is doubly degenerate, with two coexisting symmetric (under reflection) limit cycles, E1,+ and E1,−.
+(a) Limit cycles representing the quantization of orbits, with two different average energies, one
+corresponding to the fundamental level, and the other to the first and last excited level. (b) The
+harmonic potential is represented in red, while the average energy of the limit cycles is represented
+with dashed lines. In gray we can see the detour of the orbit through different energy shells. The
+fluctuations are considerable, although the two levels are well resolved.
+is that there exists some probability that the system might end in one of the two energy
+levels.
+This probability can be roughly approximated by merging the two basins of the
+respective orbits at the fundamental level, and by computing the size of the resulting basins
+of attraction in the parameter space. The fraction of volume of each basin in relation to the
+total volume in the parameter space in the region at investigation allows to introduce the
+concept of basin stability [39]. In addition, the asymptotic uncertainty can be further studied
+through the concept of basin entropy, which offers a more concise probabilistic account of
+the hidden structure of the basins [40].
+12
+
+10
+30
+E2
+E2
+6
+25
+2
+20
+E1,
+E1,+
+E
+y
+15
+-2
+10
+-6
+5
+E1
+E1,+
+-10
+0
+-6
+-4
+4
+6
+-5
+0
+5
+(b)
+aFigure 5: Unpredictability. The basins of attraction in the history space of the three stable
+attractive orbits for α = 0.9 and τ0 = 5.87.
+The two energy levels are clearly mixed in the
+phase space of initial histories, rendering the basins their fractal nature. Thus, arbitrarily small
+perturbations in the initial histories can lead to different asymptotic energy levels. (a) The basin
+of attraction for A = 0.43 and varying frequency in the phase space of the periodic histories. (b)
+A blow-up of the basins, evincing the sensitivity of the system to initial conditions, which entails
+unpredictability at all scales of precision.
+V.
+LIMIT CYCLE SUPERPOSITION
+The present section is dedicated to describe a new dynamical phenomenon that we have
+encountered for α < 0, which reminds of phenomena typically appearing in microscopic
+physics. In fact, the Eq. (2) with α < 0 resembles more exactly the electrodynamic self-
+oscillator encountered in previous works [13]. Specifically, we refer to the existence of states
+of superposition of orbits. In the present case, this corresponds to a quasiperiodic limit cycle
+encompassing two smaller symmetric degenerate limit cycles. This phenomenon can only be
+detected when the effects of the retarded potential are comparable to the magnitude of the
+external potential. Here we have selected a value of α = −0.9 to illustrate the phenomenon,
+which in absolute value is rather close to the value k = 1.
+In the first place we plot the bifurcation diagram. It has been computed following exactly
+the same recipe described in the previous section. As the reader can see in Fig. 6, for α < 0
+we cannot find a corresponding Li´enard system that experiences a Hopf bifurcation for
+13
+
+2.2
+1.8
+1.6
+1.4
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+1.2
+2.64.5
+4
+3.5
+3
+2.5
+1.5
+1
+0.5
+.5
+5Figure 6: Bifurcation diagram (α < 0). The bifurcation diagrams of the maxima map of x
+are represented for increasing values of the maximum delay τ0 and α = −0.9. A total number
+of ten initial randomly chosen histories have been used and depicted using two different colors to
+represent the asymptotic sets. A first Hopf bifurcation (arrow) appears now for τ0 = 2.4, followed
+by a Pitchfork bifurcation (green and blue branches). We can distinguish a region where limit cycle
+superposition (LCS) is detected (red background). For τ0 > 9.2 a strange attractor with robust
+intermittency (RI) appears.
+small values of the maximum time-delay τ0. This occurs because the change in the sign of α
+precludes the antidamping effect produced in the first derivative of x appearing in Eq. (6).
+However, as τ0 is further increased, again a Hopf bifurcation reveals at the approximate
+maximum delay critical value τ0c = 2.4. Thus, now, the instability occurs when the system
+is posed quite far from the original equilibrium. It must be the result of high-order terms
+in the Taylor expansion of the delayed potential, involving the jerk, the jounce and other
+derivatives of higher order. Later on, at the critical value τ0c = 2.7, a Pitchfork bifurcation
+ensues, which then transits to the chaotic regime, as we keep increasing the retardation.
+As far as we have computed, a period three orbit coexisting with the two period one orbits
+suddenly appears. As we zoom in the bifurcation diagram, we can see that these period-3
+orbits then experience a period doubling bifurcation. Nevertheless, the cascade cannot be
+clearly distinguished, since it includes very complicated dynamics with truly large chaotic
+14
+
+3
+LCS
+RI
+2
+Cmac
+0
+-1
+0
+2
+4
+6
+8
+10
+To3
+2.5
+28
+101.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+0
+2
+4
+6transients, involving heterogeneous alternating motions.
+For higher values of the maximum time-delay, around τ0 = 8.5, we can find a window
+of parameter values in the bifurcation diagram where a limit cycle superposition can be
+found. We describe this new phenomenon in detail. In this region, we have numerically
+detected at least five different coexisting limit cycles, by scrutinizing the history parameter
+space. Two of them are symmetric and have lower amplitude. They could describe a first
+fundamental level, but this time unresolved from the second, which is the one concerned
+now. For simplicity, we omit them from our analysis. Then, another two limit cycles of
+larger amplitude have also been found, which consist in two complicated period-6 stable
+symmetric degenerate orbits. By varying initial histories in the parameter space (A, ω, ϕ),
+one can find many past histories leading to any these two limit cycles, just as shown in Fig. 5
+for E1,±. But, to our surprise, we have also found a superposition limit cycle travelling along
+both limit cycles (see Figs. 7(a)). This orbit spends some time going close to one of the
+degenerate stable periodic orbits, and then switches to the other one, alternating between
+them in a regular fashion.
+The new limit set corresponds to an apparently quasiperiodic stable attractor, and it can
+also be accessed from many parameter values (A, ω, ϕ) in the parameter space chosen as
+initial histories. Since this superposition limit cycle resembles to its encompassed orbits, it
+can be numerically shown that its average energy is, although slightly below, close to the
+average energy of the other two period-6 orbits. The small difference arises because the
+superposition limit set visits regions of the phase space with lower energy (closer to the
+origin of the square well), which are not covered by the periodic trajectories. Thus, as far as
+we are concerned, we describe here for the first time a stable limit cycle that can be partly
+constructed from two smaller stable orbits, by nearly taking their union in the phase space.
+This would be impossible in a finite-dimensional dynamical system represented by some
+set of ordinary differential equations, as they are frequently used to describe conventional
+mechanical conservative systems: two orbits cannot cross in the phase space of a finite-
+dimensional continuous system.
+Of course, if we interpret the true phase space of our
+retarded oscillator as infinite-dimensional, neither they do cross here.
+To conclude our analysis, because the superposition state takes after the two encom-
+passed smaller cycles, we have computed the power spectra (see Figs. 7(b) and (c)) of the
+temporal series of the quantized periodic orbits and their superposition orbit, to ascertain
+15
+
+Figure 7: Limit cycle superposition. We can see two symmetric degenerate periodic limit cycles
+at the fundamental (red and blue orbits) level for τ0 = 8.5 and α = −0.9. Another limit cycle
+(green orbit) encompassing the previous two orbits can be appreciated. (b) Power spectra of the
+periodic orbits. (c) Power spectra of the superposition limit cycle encompassing the periodic orbits,
+where the lower frequencies (arrows) are different, rendering this attractor its quasiperiodic nature.
+the periodicity of the later. As expected, the power spectra of both orbits take after one
+another, since their average energy is similar. However, we can see that differences appear
+in the lower frequency domain of the spectrum, which render the superposition limit cycle
+quasiperiodic or, in the worst case, of a very high period, as compared to the other orbits.
+Nevertheless, without taking advantage of spectral analysis, it is really striking to see how
+this quasiperiodic orbit resembles to the underlying periodic limit cycles.
+VI.
+ROBUST INTERMITTENCY
+We now investigate an interesting dynamical phenomenon that is encountered in our
+retarded oscillator for α = −0.9 when the maximum time-delay is substantially increased
+(see Fig. 8(a)). This phenomenon consists in a multiscale strange attractor that appears
+to be robust [41] and which also exhibits intrinsic intermittency in a double sense.
+To
+understand it properly, we first show some complicated symmetric degenerate limit cycles
+16
+
+0.45
+0.4
+0.35.4
+0.50.3
+Powe
+0.2
+0.15
+0.1
+0.05
+0.1
+0.2
+0.3
+Frequency (Hz)1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.50.5
+0.40.4
+0.5Power
+0.3
+0.2
+0.1
+0.1
+0.2
+0.3
+Frequency (Hz)Figure 8: Multiscale limit cycles. (a) Two degenerate symmetric limit quiasiperiodic attractors
+for α = −0.9 and τ0 = 3.35. The trajectories are reminiscent of a saddle-focus projected on the
+2D phase space. (b) The magnitude of the continuous wavelet transform is represented (colorbar),
+using the analytic Morse wavelet with the symmetry parameter equal to 3 and a time-bandwidth
+product equal to 60.
+We can already appreciate in the temporal evolution of the spectrum a
+complex on-off periodic oscillatory behaviour. The inset shows the total power spectra, with a
+bimodal distribution displaying a rich frequency content.
+with two intrinsic scales. By intrinsic we mean a property that results from the structure
+of the limit cycles, and not as a consequence of some crises at a bifurcation point.
+As
+shown in Fig. 8(a), for τ0 = 3.35, this attracting orbits spiral out of the rest state and
+then are reinjected back to the limit cycle, drifting slowly towards the equilibrium point
+without oscillating at all.
+They clearly evoke a saddle-focus structure, as appearing in
+Shilnikov’s bifurcation [42], specially when embedded in a higher dimensional subspace of the
+full infinite-dimensional true phase space (see below). Their frequency spectrum is very rich,
+having two maxima and many frequencies at different scales. Interestingly, by implementing
+a continuous wavelet transform method, we can capture dynamical phenomena that is not
+displayed by conventional stationary spectral analysis. As it can be appreciated in Fig. 8(b),
+this time-multiscale method uses several time-windows, showing how the frequency spectrum
+evolves in time, and evincing the alternation in the system between oscillatory dynamics
+and low-speed silent drifts. This dynamics is somewhat reminiscent of relaxation oscillators,
+17
+
+0.20.3
+0.35
+0.40.15
+Power
+0.1
+0.05
+0.05
+0.1
+0.15
+0.2
+0.25
+Frequency (Hz)1.5
+0.50.5
+1
+1.50
+-0.5
+-1
+-1.5
+-1.5
+-1
+-0.5
+0Magnitude Scalogram
+0.7
+8
+4
+0.6
+2
+0.5
+1
+Frequency (Hz)
+0.5
+Magnitude
+0.4
+0.25
+0.125
+0.3
+0.0625
+0.2
+0.03125
+0.015625
+0.1
+0.0078125
+0
+2
+4
+6
+8
+10
+Time (mins)Figure 9: Intrinsic intermittency. (a) The two degenerate symmetric limit quiasiperiodic at-
+tractors have merged into a chaotic strange attractor in the 2D phase space. This attractor possess
+two dynamical and well differentiated scales. (b) The continuous wavelet transform is represented
+(colorbar), using again the analytic Morse wavelet with the symmetry parameter equal to 3 and
+a time-bandwidth product equal to 60. We can newly appreciate in the temporal evolution of
+the spectrum a complex behaviour that switches between two oscillatory motions with different
+amplitude. (c) The time series of x and its derivative y in the phase space. A sequence of bursts
+is clearly appreciated. Note how the trajectories can be reinserted into the attractor through two
+different arms, making the phenomenon doubly intermittent.
+although these limit cycles are way more sophisticated in the present case [43].
+For higher parameter values, as for example for τ0 = 9.5, these two complex limit cycles
+have merged into a strange chaotic attractor, as shown in Fig. 9(a). Now we find that the
+system alternates between two different states of chaotic oscillation, one with low amplitude
+18
+
+Magnitude Scalogram
+2
+1.2
+1
+0.5
+0.25
+(zH)
+0.125
+0.8
+Magnitude
+Frequency (
+0.0625
+0.6
+0.03125
+0.015625
+0.4
+0.0078125
+0.00390625
+0.2
+0.00195312
+0
+10
+20
+30
+40
+Time (mins)1.5
+1
+0.5
+0
+-0.5
+-1
+-1.5
+-2
+2
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2Figure 10: Robustness. (a) The largest Lyapunov exponent has been computed by using embed-
+ding techniques across different values of the time-delay for α = −0.9. A value of 0.05 has been
+chosen as a threshold to determine if the motion is chaotic. We see that its positive value rarely
+goes below the threshold, what entails great robustness of the attractor to parameter perturba-
+tions. (b) The attractor embedded in D = 3 dimensions, reconstructed with an embedding delay
+τ = 10. We see how it unfolds in this higher-dimensional space, so that the saddle-focus hidden
+structure is more clearly appreciated. Its projected shadow manifestly resembles the attractor in
+2D phase space.
+and another with a higher amplitude (Fig. 9(c)).
+In this sense, we can affirm that the
+system displays intermittent behaviour, switching between these two nonperiodic modes of
+oscillation. Comparing this dynamics with the dynamics along the underlying multiscale
+limit cycles previously described, we can say that the low-speed drift towards the original
+equilibrium of the system without retardation, have now become an oscillation of small
+amplitude around it.
+Note also how the system is reinjected into the domain through
+two possible routes: the lower branch and the higher branch of the residual multiscale
+attractors, rendering a second form of intermittency. Importantly, this doubly intermittent
+behavior is intrinsic to the complex heterogeneous nature of the attractor. Simply put, it
+does not require a fine-tuning of the parameter τ0, as opposed to conventional intermittency
+phenomena, which occurs close to bifurcation critical points [44]. Moreover, it can be shown
+that this chaotic attractor does not disappear as we move across the parameter space τ0.
+Thus it is robust under parameter perturbations.
+19
+
+2
+1.5
+0.5
+-0.5
+.1
+-1.5
+-2
+2.5
+-3
+2
+1
+0
+2
+1
+0
+2
+-2Fascinated by this dynamical behavior and by the fact that the attractor seems to be
+robust, in the sense that no periodic windows appear as we zoom in the bifurcation diagram
+around some value of τ0, we have computed the largest Lyapunov exponent (LLE) across a
+continuous interval of parameter values of the maximum time-delay τ0. Since MATLAB’s
+integrator does not allow to compute the LLE dynamically, we have taken advantage of
+embedology and used the entire time series. We follow a method exposed by Rosenstein et
+al. to efficiently compute the LLE from experimental time series [45]. These computations
+have been carried out using an embedding dimension of D = 3, and embedding time-delay
+for the series of τ = 10. The mean period T to compute the LLE considered can be obtained
+from spectral analysis (see Ref. [45]). We have used a value of T = 35, which is an upper
+bound obtained for many parameter values of the attractor. The time of integration has
+been considered t ∈ [0, 3000] and the maximum number of iteration for the algorithm was
+set to 1500, keeping our conservative attitude (see again Ref. [45]). The 3D embedding is
+depicted in Fig. 10(b).
+In Fig. 10(a) we can see the value of the maximum Lyapunov exponent for α = −0.9,
+starting with a periodic orbit at τ0 = 9.1, where the value of the Lyapunov exponents is very
+small or negative, as it should be for a periodic stable motion. When the chaotic attractor
+is born, a sudden jump to positive high values of the exponent is computed. We have set
+a threshold of λmax = 0.05 as the limiting value below which we cannot safely affirm that
+a sensitivity to initial histories occurs. This value is a conservative choice consistent with
+the temporal series of the periodic window, before the chaotic dynamics is triggered. As
+shown in Fig. 10(a), we have performed magnifications at several scales whenever downward
+peak fluctuations in the LLE exponent are present. The threshold limit is rarely exceeded.
+Furthermore, whenever the exponent drops bellow the value of 0.05, we have systematically
+computed bifurcation diagrams to see if the chaotic behavior vanishes. However, we have
+not found any periodic windows, and if periodic orbits exist, they coexist with the chaotic
+attractor. Thus we can conclude that the chaotic attractor is very robust in the present
+dynamical system, even though an analytical proof of robustness can not be easily provided
+in this case, as in previous works [41]. Since the intermittency arises as a consequence of the
+complicated nature of the attractor, which is robust, it is reasonable to say that, in addition
+to being intrinsic, it is robust, as well.
+20
+
+VII.
+CONCLUSIONS
+In the present work we have developed a very simple retarded oscillator with state-
+dependent delays, uncovering crucial dynamical behaviour that is frequently believed to be
+impossible in classical physics. Firstly, we have shown that orbits can be quantized in the
+phase space, producing one or more energy levels. We believe that the fact that these levels
+are produced in a finite number, as compared to having an infinite spectra of energy levels,
+is due to the fact that our delayed differential equations are not of the advanced type, as
+encountered in electrodynamics [46]. Secondly, we have found sensitivity to initial condi-
+tions in the history space, what introduces unpredictability in a simple fashion, making the
+concept of randomness redundant, in principle [47]. Are the apparent random fluctuations
+of fundamental physical systems just a byproduct of the complicated, even heterogeneous
+and high-dimensional [48], chaotic dynamics introduced by the dynamics of fields and the
+subsequent retardation effects in functional differential equations? [46]. Finally, we have
+uncovered a robust intermittency in the absence of multistable external wells, simply caused
+by the inherent multiscale nature of our chaotic system. Of course, this is possible because
+retardation introduces more dimensions in the dynamical system, ultimately approaching
+its center or slow manifold. In this respect, a deep connection between Lorenz-like chaotic
+dynamical systems and walking droplets has been recently proved [49].
+Interestingly, other related phenomena commonly attributed to the microscopic realm,
+such as tunneling through external potential barriers (or in multistable external potentials)
+can be easily demonstrated with our retarded potential by introducing an external Duffing
+potential in replacement of the harmonic well used here [50]. A similar situation occurs when
+studying the flow of electrons through potential barriers, where this paradoxical phenomenon
+becomes explained when interpreted in terms of the quantum potential, which appears in
+the Hamilton-Jacobi equation of the quantum system, and which is frequently disregarded
+when interpreting physical phenomena [51]. For a connection between retarded potentials
+and the quantum potential we refer the reader to previous works [13]. In other words, we
+are suggesting that the switch between different wells leading to an intermittent behavior
+can be interpreted in terms of the robust intermittency phenomenon. This dynamics is due
+to the nonlinear resonances that allow the particle to jump back and forth over the potential
+barrier [50].
+21
+
+Another important phenomena that might be studied with our oscillator is the existence
+of entangled states, which can be explained in terms of synchronization of oscillations [13].
+These states have already been predicted in previous works in classical electrodynamics to
+arise as a consequence of delay-coupling τi(xi, xj) and synchronization between systems of
+self-oscillating bodies. Synchronization phenomena has already found to actually produce
+entanglement in theoretical models of bouncing silicone oil droplets [22], although not with
+dynamical setups closing the locality loophole so far. Synchronization is more complicated
+for fluids, because the dissipation is higher at the scale of macroscopic fluid dynamics.
+Specially when compared to electrodynamic fields, where light travels mostly unimpeded
+when particles communicate through the electrovacuum. This can entail loopholes produced
+by the long-range correlations in the background fields [52].
+Importantly, time-delays are frequently considered constant, so that their dynamical na-
+ture is disregarded.
+Fortunately, thanks to the development of numerical methods and
+computational techniques, an increasing number of works in the literature of dynamical
+systems is being dedicated to the dynamical evolution of time-delays [53]. We have shown
+that the state-dependence of delays can produce very complicated behavior, entailing non-
+linear oscillations through the ubiquitous Hopf bifurcation, and producing counterintuitive
+new complex dynamical chaotic behavior. The connection between state-dependent time-
+delayed differential equations and Li´enard systems had been barely suggested [24]. A much
+deeper exploration has been provided here. It was certainly lacking in the literature, and
+opens forefront possibilities to study new physical nonlinear phenomena.
+In summary, we have provided new evidence in support of Raju-Atiyah’s hypothesis,
+claiming that physical phenomena in the microscopic physical realm can be understood by
+using functional differential equations to study dynamical phenomena produced by time
+retardation in non-Markovian systems. Importantly, we highlight that the dissipation and
+the time-delay, which both constitute genuine radiative phenomena, introduce an arrow of
+time in physical systems [54]. Thus perhaps the time-reversal symmetry of conservative field
+theories might be broken when oscillating and radiating solitons are formed in these fields
+[55]. Partly, the abusive neglect of delayed feedback in physics stems from the tradition of
+Newtonian mechanics, where action at a distance is artificially introduced to simplify forces
+of interaction.
+Certainly, this approximation has rendered many accurate and valuable
+results, allowing a great progress in the knowledge of many macroscopic physical systems,
+22
+
+which would have been impossible otherwise. Quite the opposite, the principle of causality
+in classical field theories produces memory effects that are always present whenever physical
+entities communicate through a background field with themselves, and among each other.
+VIII.
+ACKNOWLEDGMENT
+The author would like to thank Mattia Coccolo for valuable comments on the elaboration
+of the present manuscript, the discussion of some of its ideas and the computation of the
+basins of attraction.
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+26
+

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+Laser Inter-Satellite Link Setup Delay:
+Quantification, Impact, and Tolerable Value
+Dhiraj Bhattacharjee1, Aizaz U. Chaudhry1, Halim Yanikomeroglu1, Peng Hu2, and Guillaume Lamontagne3
+1Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada
+2National Research Council Canada (NRC), 1200 Montreal Road, Ottawa, ON K1A 0R6, Canada
+3MDA, Sainte-Anne-de-Bellevue, QC H9X 3R2, Canada
+1{dhirajbhattacharjee, auhchaud, halim}@sce.carleton.ca, 2peng.hu@nrc-cnrc.gc.ca, 3guillaume.lamontagne@mda.space
+Abstract—Dynamic laser inter-satellite links (LISLs) provide
+the flexibility of connecting a pair of satellites as required (dynam-
+ically) while static LISLs need to be active continuously between
+the energy-constrained satellites. However, due to the LISL estab-
+lishment time (termed herein as LISL setup delay) being in the
+order of seconds, realizing dynamic LISLs is currently unfeasible.
+Towards the realization of dynamic LISLs, we first study the
+quantification of LISL setup delay; then we calculate the end-
+to-end latency of a free-space optical satellite network (FSOSN)
+with the LISL setup delay; subsequently, we analyze the impact
+of LISL setup delay on the end-to-end latency of the FSOSN.
+We also provide design guidelines for the laser communication
+terminal manufacturers in the form of maximum tolerable value
+of LISL setup delay for which the FSOSN based on Starlink’s
+Phase I satellite constellation will be meaningful to use for low-
+latency long-distance inter-continental data communications.
+Index Terms—dynamic laser inter-satellite links, free-space
+optical satellite networks, laser inter-satellite link setup delay,
+Starlink.
+I. INTRODUCTION
+In recent advancements of wireless communication of 6G era,
+satellite networks have been seen as an integral part along with
+terrestrial networks for global broadband coverage specially
+for enabling broadband Internet in rural and remote areas [1],
+low-latency long-distance inter-continental data communications
+[2], and IoT based monitoring and remote surveillance [3]. From
+3GPP definition, satellite payloads could either be transparent
+or regenerative [4]. In transparent scenario, inter-continental
+communication has to go up (ground station to satellite) and
+down (satellite to ground station) frequently to reach from
+source to destination. With regenerative payload, communication
+between satellites over inter-satellite links (ISLs) could be a
+better option in such long-distance communication. Compared
+to RF-based ISLs, laser ISLs (LISLs) have the advantage of
+higher bandwidth, smaller antenna size, higher directivity, less
+power consumption, less chance of interception and interference,
+etc [5]. Exploiting these LISLs in low Earth orbit (LEO) or very
+low Earth orbit (VLEO) satellite mega constellations, free-space
+optical satellite networks (FSOSNs) can be realized in space [6].
+On the basis of an LISL’s active duration, LISLs can be
+classified into two types: static LISLs and dynamic LISLs. Static
+LISLs are those LISLs which are kept active all the time, e.g.,
+SpaceX’s Starlink will have four static LISLs per satellite which
+will be operating all the time [7]. In contrast, dynamic LISLs
+can be established dynamically between satellites (which are
+within the LISL range) at any time on demand depending upon
+data communication requirements. To realize such dynamic
+LISLs instantaneously, we need to have very precise and
+efficient pointing, acquisition, and tracking (PAT) system [8].
+Before two satellites start communicating via LISLs,
+transmitting satellite needs to position its laser beam within
+the field of view of receiver satellite (pointing). Then the
+receiver satellite needs to align itself towards the arriving beam
+(acquisition). Finally, transmitter and receiver continue this
+process as the communication goes on (tracking) [9]. Now, we
+define LISL setup delay as the time taken by the PAT system to
+establish the LISL, i.e., the sum of pointing time and acquisition
+time. This delay will be introduced to the end-to-end latency
+from a source ground station to destination ground station when
+the path over an FSOSN changes. Note that when the path
+changes, it could lead to one or multiple new LISLs. However,
+LISL setup delay will be introduced only once as multiple
+LISLs can be established simultaneously during a time slot.
+Satellites are driven by onboard battery and solar power, and
+satellite battery power is a very precious resource, which should
+be used intelligently. On that regard, static LISLs are always
+active whether they are being used or not. This will drain the
+satellite battery and satellites could be dead more often and they
+need to be de-orbited and new satellites have to be launched.
+This in turn will increase the maintenance expenditure. On the
+other hand, dynamic LISLs will be an energy efficient approach
+where LISLs are only established as required. With dynamic
+LISLs, two neighbour satellites could connect whenever they
+are within LISL range and this will provide more routing
+options. These links between neighbour satellites could be
+inter-orbital plane, crossing orbital plane, inter-shell, and even
+inter-constellation (e.g., between Starlink and OneWeb). Also, as
+the LEO/VLEO satellites are mobile, communications between
+satellites and ground stations will always be through dynamic
+laser links. Furthermore, in an operating satellite constellation, if
+one or many satellites fail, dynamic LISLs will instantaneously
+reroute the traffic by avoiding the dead satellite(s).
+LISL setup delay for current laser communication terminals
+(LCTs) varies from few seconds to tens of seconds [10]. This
+prevents us from realizing dynamic LISLs in next-generation
+FSOSNs (NG-FSOSNs) in late 2020s. In next-next-generation
+FSOSNs (NNG-FSOSNs) (in 2030s), due to advancement in
+satellite PAT technology, LISL setup delay could be reduced to
+the order of a few milliseconds and dynamic LISLs could become
+a reality. In this context, we study the quantification of LISL setup
+delay in the FSOSN based on Starlink’s Phase I constellation [7].
+We calculate the end-to-end latency of this FSOSN using different
+values of the LISL setup delay in different inter-continental
+connection scenarios and different LISL ranges for satellites.
+We investigate the impact of LISL setup delay on overall
+latency and provide design guidelines for LCT manufacturers
+to leverage full potential of NNG-FSOSNs via dynamic LISLs.
+To the best of our knowledge, there exists no study on LISL
+arXiv:2301.05285v1  [cs.NI]  12 Jan 2023
+
+setup delay that examines its quantification, and its impact on
+end-to-end latency along with its maximum tolerable values.
+The authors of [11] state that for terrestrial distances
+larger than 3000 km, FSOSNs could provide a better latency
+performance as compared to the optical fiber terrestrial network
+(OFTN). In high-frequency trading of stocks, even 1 ms
+improvement in latency could generate $100 million of revenue
+per year [12]. Thus, in such long distance-communication,
+FSOSNs could be a better solution compared to the OFTN.
+With this objective, we come up with maximum tolerable values
+of LISL setup delay for which latency performance of the
+FSOSN based on Starlink’s Phase I constellation will be better
+than the OFTN. This maximum value of LISL setup delay
+can be a design guideline for LCT manufacturers to leverage
+advantages of dynamic LISLs in NNG-FSOSNs.
+The paper organization is as follows. We discuss related
+work on network latency of satellite networks and examine
+LISL setup delays of current LCT manufacturers in Section
+II. In Section III, we elaborate on how we quantify LISL setup
+delay, calculate end-to-end latency, and define performance
+metrics. We present our results in Section IV, discuss insights
+and design guidelines in Section V, and conclude our discussion
+with some possible future extensions in Section VI.
+II. RELATED WORK
+Currently, Mynaric’s LCT CONDOR needs 30 seconds to
+establish an LISL between two satellites for the first time. Once
+the orbital parameters are exchanged between satellites, it takes 2
+seconds to setup an LISL for every next time [10]. Tesat [13] and
+General Atomics [14] have LCTs for LEO/VLEO constellations
+which have LISL setup delay in the range of tens of seconds.
+In any communications network, the end-to-end latency from
+source to destination typically has four components: propagation
+delay, transmission delay, queuing delay, and processing delay
+[15]. Based on this latency model, authors of [2] compared the
+latency performance of FSOSNs and OFTNs. As stated earlier
+that latency-wise, FSOSNs can be a better alternative to OFTN for
+longer communication distances [11]. On that regard, authors of
+[12] and [16] have come up with a concept of crossover distance
+to determine that for a certain terrestrial distance, which one will
+provide a better latency performance among FSOSN and OFTN.
+Authors of [17] have suggested ground stations as relays as a
+substitute of ISLs where satellites have transparent payload. With
+this network architecture, they proved that constellations like Star-
+link can provide better latency performance compared to OFTN.
+In addition to that, idle user terminals can also be used which will
+provide further improvement in latency performance. However,
+[18] shows that exploiting ISLs can reduce variation in latency
+performance as well as reduce the effects of weather impairments.
+To analyze network delay, authors of [19] have modeled each
+satellite node as M/M/1 queue in a multihop scenario where
+each satellite can receive packets from ground station as well as
+other satellite node. Authors of [20] highlighted the importance
+of temporary LISLs (defined as LISLs which are established
+temporarily with satellites that are within LISL range) in order
+to achieve better latency performance compared to static LISLs
+in FSOSNs. They showed that with temporary LISLs, there exist
+more number of LISLs which will provide better routing options.
+They also reported that temporary LISLs are more useful at lower
+LISL ranges. Authors of [12] and [20] mentioned about LISL
+setup delay in FSOSNs. With that respect, in FSOSNs, along
+with the other four end-to-end latency components discussed
+earlier, we introduce LISL setup delay to the latency model.
+III. METHODOLOGY
+To quantify LISL setup delay, calculate end-to-end latency
+from source to destination with the LISL setup delay, investigate
+the impact of LISL setup delay on overall latency, and present
+design guidelines for LCT as a form of maximum tolerable value
+of LISL setup delay, we simulate Starlink’s Phase I Version 2 con-
+stellation in AGI’s Systems Tool Kit (STK) platform [21]. This
+constellation has a total of 1584 satellites consisting 24 orbital
+planes with each of them having 66 satellites [7]. The orbits are
+at an inclination of 53° with respect to the equator and satellites
+are at an altitude of 550 km. With these constellation parameters,
+we generate this constellation’s satellites in STK with a certain
+LISL range (i.e., the range over which a satellite in an FSOSN
+can establish an LISL with any other satellite within this range)
+along with ground stations at New York, London, Istanbul, and
+Hanoi. Next we extract the data from STK (e.g., vertices, edges,
+length of edges, etc) at every second for one hour simulation
+period to Python platform. Then we apply Dijkstra’s shortest
+path algorithm [22] to find shortest path at every time slot (equal
+to one second in duration) for the source to destination pairs:
+New York–London, New York–Istanbul, and New York–Hanoi.
+In our investigation, we consider 4 different values of LISL
+range: 1500 km, 1700 km, 2500 km, and 5016 km. The minimum
+range to have communication with nearest neighbor at the imme-
+diate left and right orbital planes is 1500 km in Starlink’s Phase
+I Version 2 constellation. At this range, a satellite can connect to
+two satellites in front and two at rear in the same orbital plane
+making total 6 connections. At 1700 km range, a satellite can
+connect to three immediate neighbors on the left, three immediate
+neighbors on the right, and four intra-orbital plane neighbors
+making a total of 10 possible connections. The maximum possible
+LISL range for Starlink’s Phase 1 constellation can be calculated
+as 5016 km [6]. The 2500 km LISL range is taken as an
+intermediate value between 1700 km and 5016 km.
+A. Quantification of LISL Setup Delay
+We define LISL setup delay indicator (a binary variable)
+as follows: if the shortest paths of (i − 1)th time slot and
+ith time slot are exactly same, no LISL setup delay is to be
+included in the end-to-end latency, and the LISL setup delay
+indicator, αi is 0. If shortest path changes from (i−1)th to
+ith time slot, αi is 1. Considering ηs as LISL setup delay, we
+denote end-to-end latency without and with LISL setup delay
+as ηLE(withoutηs) and ηLE(withηs), respectively. In Table
+I, we show shortest paths (satellite naming convention follows
+[6]) and corresponding values of ηLE(without ηs), αi, and
+ηLE(withηs) for first 6 time slots over the FSOSN for New
+York to Istanbul inter-continental connection at an LISL range
+of 1500 km. From Table I, we can see that shortest path could
+change from time to time. This is due to the fact that as LEO
+satellites are moving at high orbital speeds, either a shortest path
+at one time instance may even not exist in the next time instance
+(due to one or multiple satellites moving out of range) or there
+may become available a new shortest path. ηLE(withoutηs) has
+
+Table I. ηLE(without ηs), αi, ηs, and ηLE(with ηs) of the shortest paths at first 6 time slots
+over the FSOSN for New York–Istanbul inter-continental connection.
+ηLE
+ηLE
+Time
+Shortest
+(without
+αi
+ηs
+(with
+Slot
+Path
+ηs)
+(ms)
+ηs)
+(ms)
+(ms)
+1
+GS at New York, satellite x10919, x11115, x11312, x11509, x11609, x11611, x12166, GS at Istanbul
+38.09
+0
+0
+38.09
+2
+GS at New York, satellite x11503, x11505, x11507, x11509, x11609, x11611, x12166, GS at Istanbul
+38.08
+1
+100
+138.08
+3
+GS at New York, satellite x11503,x11505, x11507, x11509, x11609, x11611, x12166, GS at Istanbul
+38.07
+0
+0
+38.07
+4
+GS at New York, satellite x11503,x11505, x11507, x11509, x11609, x11611, x12166, GS at Istanbul
+38.06
+0
+0
+38.06
+5
+GS at New York, satellite x11503,x11505, x11507, x11509, x11609, x11611, x12166, GS at Istanbul
+38.05
+0
+0
+38.05
+6
+GS at New York, satellite x11503, x11505, x11507, x11508, x11608, x11903, x12166, GS at Istanbul
+38.00
+1
+100
+138.00
+two major components: propagation delay and node delay (sum
+of processing and queuing delay is node delay). We calculate
+propagation delay as sum of lengths of all the laser links in
+the shortest path divided by speed of light in vacuum and we
+consider node delay as 1 ms [23]. From Table I, it is evident that
+shortest paths are not same for 1st and 2nd time slots, so α2 =1.
+Considering ηs =100 ms, ηLE(withηs) will be 38.08+100, i.e.,
+138.08 ms at time slot 2. The shortest path remains unchanged
+from time slot 3 to 5, i.e., α3 =α4 =α5 =0 and corresponding
+ηLE(withηs) values remain same as ηLE(withoutηs). Then
+again at time slot 6, shortest path changes which makes α6 =1.
+B. Path Change Rate
+We simulate for 3600 time slots, one time slot being equal
+to 1 second in duration and we define the path change rate,
+λ as the average number of instances the shortest path from
+source to destination changes (represented in percentage) and
+mathematically it can be calculated as
+λ=
+1
+3600
+3600
+�
+i=1
+αi ×100%.
+(1)
+C. End-to-End Latency
+Averaging ηLE(with ηs) and ηLE(without ηs) over 3600
+time slots, we get average end-to-end latency with and without
+ηs as ηLE(withηs) and ηLE(withoutηs), respectively. They
+are related to λ and ηs as follows:
+ηLE(withηs)=ηLE(withoutηs)+ λ
+100 ηs.
+(2)
+D. Impact of ηs
+To measure the impact of LISL setup delay, ηs on average end-
+to-end latency, we define β as percentage of delay introduced due
+to ηs in average end-to-end latency and calculate it as follows:
+β = ηLE(withηs)−ηLE(withoutηs)
+ηLE(withηs)
+×100%.
+(3)
+E. Tolerable Value of ηs
+For an inter-continental connection, it is meaningful to use
+the FSOSN only when ηLE(withηs) is lesser than end-to-end
+latency of the OFTN, ηLE,OF T N. Using (2), the following can
+be written,
+ηLE(withoutηs)+ λ
+100 ηs ≤ηLE,OF T N.
+(4)
+Now we define the maximum tolerable value of LISL setup
+delay, ηs,max as the maximum value of ηs so that the average
+Figure 1. Path change rate.
+end-to-end latency of the FSOSN is lesser or equal to that of
+the OFTN and calculate it from (4) as follows:
+ηs,max = ηLE,OF T N −ηLE(withoutηs)
+λ/100
+.
+(5)
+To calculate ηLE,OF T N, first we determine the distance
+from the source ground station to the destination ground
+station along the surface of the Earth using Haversine formula
+[24] from latitudes and longitudes of source and destination
+ground stations. Later we find ηLE,OF T N as that distance
+divided by speed of light in the optical fiber (having refractive
+index=1.4675), i.e., 204,287,876 m/s.
+IV. RESULTS
+We consider three inter-continental connections: New York to
+London (low inter-continental distance connection with terrestrial
+distance=5593 km), New York to Istanbul (mid inter-continental
+distance connection with terrestrial distance=8079 km), and
+New York to Hanoi (high inter-continental distance connection
+with terrestrial distance=13164 km). For the metrics λ, ηLE,
+and β, we show bar plots for the four LISL ranges. To clearly
+show both high and low values in the same figure, we use log
+scale in the y-axis in Figs. 1 to 3.
+A. Path Change Rate
+In Fig. 1, we plot λ with LISL range varying along x-axis for
+the three inter-continental connections. For any inter-continental
+connection, we can observe that λ reduces as LISL range
+increases. Also note that for a particular LISL range, the more
+the inter-continental distance, the higher the value of λ.
+B. End-to-End Latency
+Fig. 2 shows end-to-end latency for the three inter-continental
+connections averaged over one hour of simulation period
+without considering ηs and with four ηs values. As LISL range
+increases along x-axis, both ηLE(withoutηs) (black bars) and
+ηLE(withηs) (other bars) decrease. For a certain LISL range, the
+
+102
+New York to London
+New York to Istanbul
+37.5 39.1
+New York to Hanoi
+33.9
+17.5
+14.4
+12.3
+(%)
+9.8
+9.6
+101
+8.7
+8.0
+6.8
+5.8
+100
+1500
+1700
+2500
+5016
+LISL Range (km)(a) New York to London.
+(b) New York to Istanbul.
+(c) New York to Hanoi.
+Figure 2. Average end-to-end latency performance.
+more the value of ηs, the more the overall latency. For example, in
+Fig. 2a with LISL range of 1700 km, ηLE(withηs) is 123.9 ms
+for ηs=1 sec and it reduces to 35.7 ms when ηs is considered to
+be 100 ms. Also, for a certain LISL range with a certain ηs value,
+the more the inter-continental distance, the more the end-to-end
+latency for both the cases: ηLE(withoutηs) and ηLE(withηs).
+It is interesting to note that with the increase of LISL range,
+ηLE(with ηs) reduces faster compared to ηLE(without ηs).
+For example, considering Fig. 2a, ηLE(withoutηs) drops from
+25.9 ms to 24.6 ms when LISL range increases from 1700
+km to 2500 km. If we take the ratio and term the ratio as
+reduction ratio, for this case it will be 25.9
+24.6 =1.053. Similarly,
+for ηLE(with ηs), the reduction ratio will be 123.9
+92.1 = 1.345
+which is greater than that of ηLE(withoutηs).
+C. Impact of ηs
+In Fig. 3, we show the variation of β with LISL range for
+four ηs values in the three inter-continental connections. As we
+see, β reduces as LISL range increases for a certain ηs value.
+Also, at a certain LISL range, β reduces as ηs reduces. For
+example, in Fig. 3b with LISL range of 2500 km, β is 71%
+for ηs=1 sec. However, when ηs reduces to 100 ms, β reduces
+to 19.7%. In addition, for a certain LISL range with a particular
+ηs value, β reduces as inter-continental distance increases. For
+example, assuming 1700 km of LISL range and ηs as 1 sec, β
+reduces from 77% to 73.1% when inter-continental connection
+changes from New York–Istanbul to New York–Hanoi.
+D. Tolerable Value of ηs
+In Fig. 4, we plot ηLE(withηs) and ηLE,OF T N against ηs.
+Note that, ηLE(withηs) is a straight line with a constant slope
+and as LISL range increases, the slope reduces. The significance
+of this figure is where ηLE(withηs) for a certain LISL range
+cuts ηLE,OF T N, the x-coordinate value of the intersection point
+is ηs,max as beyond that point, ηLE(with ηs) will be greater
+than ηLE,OF T N. To show the intersection points clearly, we
+only present ηs values on the x-axis varying from 1 ms to
+100 ms where we mention the coordinates of the intersection
+points. If we substitute ηLE,OF T N =39.55 ms (from Fig. 4b),
+ηLE(without ηs) = 37.9 ms (from Fig. 2b), and λ = 37.5%
+(from Fig. 1) for New York to Istanbul inter-continental
+connection with 1500 km LISL range in (5), we get ηs,max
+as 4.4 ms which matches with Fig. 4b. Also, we should observe
+from Fig. 4 that, as LISL range increases, ηs,max also increases.
+Interesting point to note in Fig. 4c is that it only shows two
+intersection points because ηLE(withηs) for 1500 km and 1700
+km LISL range straight lines (black and blue lines) never intersect
+with ηLE,OF T N for ηs >1 ms values. For 1500 km LISL range,
+ηLE(withoutηs)=66.5 ms (from Fig. 2c) and ηLE,OF T N=64.44
+ms (from Fig. 4c). Putting these values in (5), we get negative
+ηs,max value which does not exist. Similarly, for 1700 km LISL
+range, ηLE(withoutηs)≈ηLE,OF T N which makes ηs,max ≈0.
+V. INSIGHTS AND DESIGN GUIDELINES
+A. Insights
+1) Path Change Rate
+• As LISL range increases, there will be lesser hops, i.e., lesser
+number of satellites for the signal to reach from source to
+destination. For example, in New York to Istanbul inter-
+continental connection, average number of hops drops from
+7 to 6 when LISL range increases from 1500 km to 1700 km.
+The lesser the number of hops, the lesser is the chance of a
+new shortest path. This in turn reduces λ. Also, when the LISL
+range increases, two satellites remain in communication range
+for a longer time span. One of the reasons for the shortest
+path to change is satellites going out of LISL range, and a
+shortest path tends to change lesser with longer LISL range.
+Due to these two reasons, λ reduces as LISL range increases.
+• For a certain LISL range, the longer the inter-continental
+distance, the higher the average number of hops which leads
+to more chances of a new shortest path, and this increases
+the path change rate, λ.
+2) End-to-End Latency
+• Increase in LISL range reduces the number of hops which
+reduces total node delays. This decreases ηLE(withoutηs)
+with the increase in LISL range. Now, as both λ and
+ηLE(withoutηs) decrease with the increase of LISL range,
+from (2), it is clear that ηLE(withηs) also decreases.
+• From (2), we can see that if ηs reduces, ηLE(with ηs)
+reduces for a certain LISL range.
+• For a certain LISL range with a certain ηs value, when
+inter-continental distance increases, λ increases. From (2), we
+can say that ηLE(withηs) will increase with the increase of
+λ. For longer inter-continental connections, propagation delay
+as well as node delay (due to more number of hops) is more
+which increases ηLE(withoutηs) for longer inter-continental
+connections.
+• We consider that at LISL range la and lb (lb > la), path
+change rates are λa and λb, respectively. Also, the average
+end-to-end latencies without ηs are ηLEa(without ηs)
+
+103
+Without ns
+With ns=1 sec
+With ns=100 ms
+365.1
+Withns=10 ms
+With ns=1 ms
+(sw) 3u
+123.9
+102
+92.1
+81.6
+60.5
+35.7
+30.0
+31.4
+29.2
+26.8.
+25.3.
+26.6
+27.0
+25.9
+25.9
+24.6
+23.9.
+24.7
+23.3
+23.4
+101
+1500
+2500
+1700
+5016
+LiSL Range (km)103
+Without ns
+With ns=l sec
+With ns=100 ms
+413.0
+With ns=10 ms
+With ns=1 ms
+160.2
+(sw) 
+122.8
+113.0
+102
+75.4
+49.3
+44.3
+41.6
+38.2
+41.4
+36.5
+37.9
+38.3
+37.1
+36.9
+34.2.
+35.6
+35.7
+33.4
+33.5
+101
+1500
+1700
+2500
+5016
+LISL Range (km)Without ns
+With ns=1 sec
+With ns=100 ms
+103
+With ns=10 ms
+With ns=1 ms
+457.8
+(ms)
+239.6
+205.3
+nLE
+153.1
+105.7
+102
+82.0
+70.4
+75.3
+66.2
+66.3
+66.5
+62.3
+66.9
+64.4
+64.6
+60.9
+57.6 56.8
+61.0
+56.7
+101
+1500
+1700
+2500
+5016
+LiSL Range (km)(a) New York to London.
+(b) New York to Istanbul.
+(c) New York to Hanoi.
+Figure 3. Impact of ηs on end-to-end latency.
+(a) New York to London.
+(b) New York to Istanbul.
+(c) New York to Hanoi.
+Figure 4. Maximum tolerable value of ηs.
+and ηLEb(without ηs), respectively. From Figs. 1 and 2
+values we observe that λa
+λb > ηLEa(withoutηs)
+ηLEb(withoutηs) . For example,
+considering New York to Istanbul inter-continental connection,
+assuming la=1500 km and lb=1700 km, λa
+λb = 37.5
+12.3 = 3.049
+and ηLEa(withoutηs)
+ηLEb(withoutηs) = 37.9
+36.9 =1.027. Now, we can write the
+following:
+λa
+λb
+> ηLEa(withoutηs)
+ηLEb(withoutηs) ,
+(6)
+λa ηs/100
+ηLEa(withoutηs) >
+λb ηs/100
+ηLEb(withoutηs),
+(7)
+1+
+λa ηs/100
+ηLEa(withoutηs) >1+
+λb ηs/100
+ηLEb(withoutηs),
+(8)
+ηLEa(withoutηs)+ λa
+100 ηs
+ηLEa(withoutηs)
+> ηLEb(withoutηs)+ λb
+100 ηs
+ηLEb(withoutηs)
+.
+(9)
+Assuming
+average
+end-to-end
+latency
+with
+ηs
+as
+ηLEa(withηs) and ηLEb(withηs) for LISL range la and lb,
+respectively and using (2) we can rewrite (9) as follows:
+ηLEa(withηs)
+ηLEb(withηs) > ηLEa(withoutηs)
+ηLEb(withoutηs) .
+(10)
+3) Impact of ηs
+• We have seen that ηLE(with ηs) reduces faster compared
+to ηLE(withoutηs) as LISL range increases. Thus, the ratio
+ηLE(withoutηs)
+ηLE(withηs)
+increases as LISL range increases. From (3),
+as we can see that β is proportional to
+�
+1− ηLE(withoutηs)
+ηLE(withηs)
+�
+,
+β reduces with the increase of LISL range.
+• ηLE(withηs) reduces when ηs reduces but ηLE(withoutηs)
+remains the same which causes the ratio ηLE(withoutηs)
+ηLE(withηs)
+to
+increase. As β is proportional to
+�
+1 − ηLE(withoutηs)
+ηLE(withηs)
+�
+, it
+decreases when ηs reduces.
+• Let us consider that for inter-continental distance dx and dy
+(dy >dx), path change rates are λx and λy, respectively. Also,
+average end-to-end latencies without ηs are ηLEx(withoutηs)
+and ηLEy(without ηs), respectively. From Figs. 1 and 2
+values, we also observe that λx
+λy > ηLEx(withoutηs)
+ηLEy (withoutηs) (note that
+in this discussion, we are varying inter-continental distance, not
+LISL range). For example, at 1700 km LISL range, for New
+York to Istanbul and New York to Hanoi inter-continental con-
+nection, λx, λy, ηLEx(without ηs), and ηLEy(without ηs)
+are 12.3%, 17.5%, 36.9 ms, and 64.4 ms, respectively from
+which we get λx
+λy =0.703 and ηLEx(withoutηs)
+ηLEy (withoutηs) =0.573. Using
+the approach in (6) – (10), we can come to the conclusion that
+ηLEx(withoutηs)
+ηLEx(withηs)
+<
+ηLEy (withoutηs)
+ηLEy (withηs)
+where ηLEx(withηs) and
+ηLEy(withηs) are average end-to-end latencies considering ηs
+for inter-continental distance dx and dy, i.e., ηLE(withoutηs)
+ηLE(withηs)
+in-
+creases as inter-continental distance increases which reduces β.
+4) Tolerable Value of ηs
+• (2) represents an equation of a straight line with slope
+proportional to λ considering ηLE(withηs) as y variable and
+ηs as the x variable. As LISL range increases, λ decreases
+which makes the slope of the straight lines to reduce. In
+addition to λ, ηLE(without ηs) also reduces with the
+
+110
+FSOSN with LiSL Range=1500km
+FSOSN with LISL Range=1700km
+100
+FSOSN with LISL Range=2500km
+FSOSN with LISL Range=5016km
+90
+OFTN
+(ms)
+80
+37u
+70
+(24.58,64.44)
+(80.63,64.44)
+60
+50
+20
+40
+60
+80
+100
+ns (ms)92.7
+102
+79.1
+73.3
+71.4
+56.0
+27.5
+21.5
+20.0
+11.3
+101
+(%)
+3.7
+B
+2.7
+2.4
+1.3
+100
+0.4
+0.3
+0.2
+1500
+1700
+2500
+5016
+LISL Range (km)Ns=l sec
+ns=100 ms
+ns=10 ms
+ns=1 ms
+90.8
+102
+77.0
+71.0
+70.4
+49.8
+25.0
+19.7
+19.2
+101
+9.0
+(%)
+3.2
+B
+2.4
+2.3
+1.0
+100
+0.3
+0.2
+0.2
+1500
+1700
+2500
+5016
+LiSL Range (km)85.5
+102
+73.1
+70.3
+63.0
+37.0
+21.4
+19.2
+14.5
+101
+5.6
+2.6
+2.3
+β
+1.7
+100
+0.6
+0.3
+0.2
+0.17
+1500
+2500
+5016
+1700
+LISL Range (km)60
+FSOSN with LISL Range=1500km
+55
+FSOSN with LISLRange=1700km
+FSOSN with LISL Range=2500km
+50
+FSOSN with LISL Range=5016km
+OFTN
+45
+(ms)
+40
+37
+35
+(2.30,27.38)
+30
+(15.10,27.38)
+(40.88,27.38)
+(70.34,27.38)
+25
+20
+10
+20
+30
+40
+50
+60
+70
+80
+ns (ms)80
+FSOSN with LISL Range=1500km
+FSOSN with LISL Range=1700km
+70
+FSOSN withLISL Range=2500km
+FSOSN with LISL Range=5016km
+OFTN
+60
+(sw) u
+50
+(4.40,39.55)
+(21.54,39.55)
+(45.40,39.55)
+40
+(76.87,39.55)
+30
+20
+40
+60
+80
+100
+Ns (ms)increase of LISL range, and from (5) we can say that ηs,max
+increases with increase in LISL range.
+B. Design Guidelines
+The values we get from (5) are exactly same as we
+get from intersection points shown in Fig. 4. Given that
+ηLE,OF T N, ηLE(withoutηs), and λ are known for a particular
+inter-continental connection for a certain LISL range, (5) can
+be used to design LCTs in order to exploit full potential of
+NNG-FSOSNs. For example:
+• For New York to London inter-continental connection with
+1500 km of LISL range, ηLE,OF T N, ηLE(withoutηs), and λ
+are 27.38 ms, 26.6 ms, and 33.9%, respectively. Putting these
+values in (5), we get ηs,max as 2.3 ms (same as in Fig. 4a).
+• With 1700 km LISL range for New York to Istanbul
+inter-continental connection, putting ηLE,OF T N=39.55 ms,
+ηLE(without ηs)=36.9 ms, and λ=12.3% in (5), we get
+ηs,max=21.54 ms, i.e., exactly as shown in Fig. 4b.
+• Considering New York to Hanoi inter-continental connection
+with 5016 km of LISL range, values of ηLE,OF T N,
+ηLE(withoutηs), and λ are 64.44 ms, 56.7 ms, and 9.6%,
+respectively. Using these values in (5), we get ηs,max equal
+to 80.63 ms (same as in Fig. 4c).
+VI. CONCLUSION AND FUTURE WORK
+Dynamic LISLs are essential to leverage the full potential
+of NNG-FSOSNs due to their on-demand flexibility. However,
+whenever a new LISL is established, LISL setup delay is added to
+the end-to-end latency. To model the end-to-end latency including
+LISL setup delay, we study the quantification of LISL setup delay,
+and calculate the end-to-end latencies for low, medium, and high
+inter-continental distance connections for different LISL setup
+delay values. We find that the end-to-end latency depends on path
+change rate which reduces as LISL range increases but increases
+as inter-continental distance increases. We also highlight the
+impact of LISL setup delay on total end-to-end latency which
+clearly indicates that LISL setup delay cannot be ignored. We
+observe that the impact of LISL setup delay reduces as LISL
+range or inter-continental distance increases. We also deduce
+the formula to find maximum tolerable value of LISL setup
+delay which represents design guidelines for LCT manufacturers
+so that FSOSNs can have better latency performance compared
+to OFTN. We see that for some LISL range, there does not
+exist any such value of ηs,max. An interesting takeaway point is
+that higher LISL range has two major benefits. Firstly, highest
+possible LISL range has the best latency performance. Secondly,
+it has the highest value of ηs,max which can be attainable.
+However, with high LISL range, the penalty is more satellite
+transmission power and energy consumption.
+It is evident that due to change of shortest path with time
+slots, LISL setup delay is introduced which negatively impacts
+the latency of an FSOSN using dynamic LISLs. In order to
+minimize end-to-end latency, we need to minimize the path
+change rate so that LISL setup delay is introduced less often.
+In future, we plan to develop algorithms to minimize the path
+change rate for a better latency performance.
+ACKNOWLEDGMENT
+This work was supported by the High Throughput and Secure
+Networks Challenge Program at the National Research Council
+of Canada. The authors would also like to acknowledge Dr.
+Pablo Madoery for his technical help and feedback.
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+

4tAyT4oBgHgl3EQfpPhP/content/tmp_files/2301.00521v1.pdf.txt

@@ -0,0 +1,3696 @@
+A RL-based Policy Optimization Method Guided by Adaptive Stability
+Certification
+Shengjie Wang 1 Fengbo Lan 1 Xiang Zheng 2 Yuxue Cao 3 Oluwatosin Oseni 4 Haotian Xu 1 Yang Gao 5
+Tao Zhang 1
+Abstract
+In contrast to the control-theoretic methods, the
+lack of stability guarantee remains a significant
+problem for model-free reinforcement learning
+(RL) methods. Jointly learning a policy and a
+Lyapunov function has recently become a promis-
+ing approach to ensuring the whole system with
+a stability guarantee. However, the classical Lya-
+punov constraints researchers introduced cannot
+stabilize the system during the sampling-based
+optimization. Therefore, we propose the Adap-
+tive Stability Certification (ASC), making the sys-
+tem reach sampling-based stability. Because the
+ASC condition can search for the optimal policy
+heuristically, we design the Adaptive Lyapunov-
+based Actor-Critic (ALAC) algorithm based on
+the ASC condition. Meanwhile, our algorithm
+avoids the optimization problem that a variety of
+constraints are coupled into the objective in cur-
+rent approaches. When evaluated on ten robotic
+tasks, our method achieves lower accumulated
+cost and fewer stability constraint violations than
+previous studies.
+1. Introduction
+Learning-based (especially Reinforcement-Learning-based)
+controllers have become increasingly popular and have
+achieved excellent performance in non-linear dynamic sys-
+tems (Hwangbo et al., 2019; Andrychowicz et al., 2020).
+However, a lack of some safety notions introduces addi-
+tional risks to the agents and environments. Stability is a
+crucial notion of safety, whose violation can cause unsafe
+behaviours (Jin et al., 2020). Fortunately, there exists an
+effective tool to assess the stability, Lyapunov functions, in
+1Department of Automation, Tsinghua University 2Department
+of Computer Science, City University of Hong Kong 3Beijing
+Institute of Control Engineering 4Covenant University 5Institute
+for Interdisciplinary Information Sciences,Tsinghua University.
+Correspondence to: Tao Zhang <taozhang@tsinghua.edu.cn>.
+State space
+𝑐! 𝑠 ≤ 𝑏
+𝑠"
+𝑐! 𝑠 = 0
+Diverging trajectory
+Optimal trajectory w/ stability
+Sub-optimal trajectory w/ stability
+Unstable trajectory
+Figure 1. Intuitive example showing the mean cost stabil-
+ity according to Definition 3.1. The figure shows the re-
+lationship between three sets, like the whole state space,
+{s | cπ(s) ≤ b} and {s | cπ(s) = 0}. The red line repre-
+sents a diverging trajectory. The yellow line represents a
+trajectory without the mean cost stability. The trajectories
+coloured cyan and purple remain stable, whereas the state in
+the purple trajectory reaches the set {s | cπ(s) = 0} more
+quickly. The task aims to obtain the policy that can generate
+the purple trajectory.
+control-theoretic approaches. Lyapunov functions can be
+designed for a linear system with specific criteria in the form
+of a quadratic positive-definite function. But how to find
+a suitable Lyapunov function remains an open challenge
+in the non-linear dynamic system (La Salle & Lefschetz,
+2012).
+Thanks to the emergence of deep learning, researchers no-
+ticed that the neural Lyapunov function had an outperform-
+ing representation ability, thereby making it possible to
+search for a feasible Lyapunov function. According to Lya-
+punov’s second method for stability, the neural Lyapunov
+function is trained by forcing the updating direction toward
+decreasing the Lyapunov function along an episode’s state
+trajectories (Chang et al., 2019). In the early stage of time,
+these methods are only suitable for the model-based setting
+because the difference of Lyapunov function with respect to
+time brings a requirement of dynamic model (Berkenkamp
+arXiv:2301.00521v1  [cs.RO]  2 Jan 2023
+
+Adaptive Stability Certification
+et al., 2017; Richards et al., 2018). In contrast to model-
+based methods, model-free reinforcement learning meth-
+ods have achieved superior performance on many complex
+robotic systems (Hwangbo et al., 2019; Andrychowicz et al.,
+2020). Therefore, researchers proposed some methods to
+overcome the above issue. Firstly, they applied the discrete
+Lyapunov stability condition directly in the optimization.
+Then, the Lyapunov conditions can be viewed as multiple
+constraints. Prior studies added those constraints into the ob-
+jective function. Specifically, the objective function enables
+the system to reach optimality and stability together. Fur-
+thermore, it is worth noting that current RL-based methods
+with stability guarantee have been applied in some practi-
+cal problems successfully, such as monitoring the security
+of interconnected microgrids (Huang et al., 2021), power
+system control (Zhao et al., 2021), automatic assembly (Li
+et al., 2022) and motion planning of autonomous vehicles
+(Zhang et al., 2021).
+Admittedly, RL-based approaches with the help of the Lya-
+punov function achieve promising performance. However,
+the discrete Lyapunov stability condition should be satisfied
+in the whole state space, which means infinity constraints
+for continuous control tasks. Due to the sampling-based
+optimization in RL, current approaches sample some data
+pairs randomly, which causes a serious problem that the
+discrete stability condition is not convincing. Furthermore,
+reaching optimality and stability together remains a chal-
+lenge for their objective function. Because the Lyapunov
+stability condition introduces various constraints in the ob-
+jective function, it is hard to balance the optimality and
+stability of the system. In practical experiments, we find
+that previous constraints are either loose to find ineffective
+Lyapunov functions or tight to make the policy trapped into
+a sub-optimal point. Therefore, to better improve the cur-
+rent research, we hope the Lyapunov-based constraint can
+facilitate the policy to reach the optima within the set of the
+sampling-based stability. To better understand the principle,
+we give an intuitive example in Figure 1.
+To address the above issues, we propose the Adaptive
+Lyapunov-based Actor-Critic algorithm (ALAC). Unlike
+the discrete Lyapunov stability condition, we propose a
+novel sampling-based stability condition called Adaptive
+Stability Certification (ASC). Meanwhile, the certifica-
+tion can guide the policy to search for the optimal point
+heuristically. Thus, based on the ASC condition, we de-
+sign the Adaptive Lyapunov-based Actor-Critic algorithm
+(ALAC) to reach the optimality and stability of the system.
+Thanks to the supervised learning for a Lyapunov candidate
+and the Lagrangian-based policy optimization, our method
+eliminates the coupling relationship between various con-
+straints and the objective function. Experiments show that
+our method provides promising results under stability con-
+straints on some robotic control problems, like walking of
+legged robots, trajectory planning of a free-floating space
+robot, and so on. Another interesting finding is that, in
+contrast to traditional RL algorithms, our method facilitates
+the controller to enhance the robustness, generalization, and
+efficiency of the whole system 1.
+2. Related works
+Due to the difficulty of a manual design, constructing a
+Lyapunov neural network has become increasingly popu-
+lar in a non-linear dynamic system. For the model-known
+situation, the approaches jointly learn a Lyapunov func-
+tion and a controller (Richards et al., 2018; Chang et al.,
+2019; Mittal et al., 2020; Dai et al., 2020; Lechner et al.,
+2021; Donti et al., 2020; Gaby et al., 2021). But it re-
+stricts the application of complex systems which are hard
+to obtain accurate models. Therefore, some researchers
+present model-learned methods with a stability guarantee,
+in which Gaussian Process or Neural Network approximates
+the model. The model-learned method can be separated into
+two types. The first one is learning dynamics models guided
+by a learnable Lyapunov function, in which policies are
+inherently included or learned by LQR method (Kashima
+et al., 2022; Zhou et al., 2022; Chen et al., 2021; Lawrence
+et al., 2020; Schlaginhaufen et al., 2021). Another approach
+is to construct a learnable policy network updated by a neu-
+ral Lyapunov function, thereby satisfying the stability of
+system (Berkenkamp et al., 2017; Dawson et al., 2022; Zhou
+et al., 2022; Dai et al., 2021; Lale et al., 2022). However, we
+notice that most model-learned methods are only verified
+in relatively easy environments. A possible reason is that
+the coupling of the Lyapunov function and dynamic model
+makes learning unstable or incompatible due to interdepen-
+dency.
+A promising direction is to study model-free methods with
+a stability guarantee. Recently, a large variety of methods
+have been proposed to address the issue. One method is that
+the policy is updated by a mixed objective with respect to the
+neural Lyapunov function and Q function. POLYC (Chang
+& Gao, 2021) introduced the necessary conditions required
+by the Lyapunov function into objectives to optimize the
+policy network. LBPO (Sikchi et al., 2021) applied the log-
+arithmic barrier function based on the form of the Lyapunov
+function. TNLF (Xiong et al., 2022) constructed Lyapunov
+V and Q functions trained by the stability certification. The
+other form is policy optimization with Lyapunov constraints.
+Chow et al. (2018) designed a constrained RL algorithm to
+project a policy in a trust region with Lyapunov stability.
+Han et al. (2020a) provided a novel constrained RL-based
+approach called LAC, using the prime-dual method to mod-
+ify the constraint. Their latter work verified them in both
+1See our project page at https://sites.google.com/view/adaptive-
+lyapunov-actor-critic.
+
+Adaptive Stability Certification
+on-policy and off-policy settings (Han et al., 2021; 2020b).
+In the previous study, there still exist two main drawbacks
+to obtaining the optimal policy and a suitable Lyapunov
+function. The discrete Lyapunov condition they used did
+not meet the demand for a sampling-based stability guar-
+antee in RL. Furthermore, a combined objective function
+via a simple addition causes a sub-optimal policy or invalid
+Lyapunov function, especially for multiple and complex
+constraints. In contrast, our method proposes the Adap-
+tive Lyapunov-based Actor-Critic algorithm to satisfy the
+sampling-based stability and search for the optimal policy
+heuristically.
+3. Problem Formulation
+The pre-process of implementing RL-based algorithms is to
+formulate robotic control problems as a Markov Decision
+Process (MDP). In our paper, MDP mainly consists of four
+elements, such as S, A, P, and C. Here, S is the state space,
+A is the action space, P is the dynamic transition function,
+and C is the cost function. At timestep t, st ∈ S represents
+the state the robot observes. Then, at ∈ A is the action
+executed by the agent(robot). Note that at is sampled from
+the agent’s policy π(at|st). According to P(st+1|st, at),
+the state of system transfers to the next state st+1 with a
+certain probability. At the same time, the agent receives the
+cost cπ(st) = Ea∼πC(st, at). Besides, the distribution of
+starting state denotes s0 ∼ ρ. And then, we can define the
+state distribution T :
+T (s|ρ,π, t + 1) =
+�
+S
+�
+A
+π(at|st)P(st+1|st, at) daT (s|ρ, π, t) ds
+(1)
+Note that T (s|ρ, π, 0) = ρ holds.
+The optimal objec-
+tive is to find the optimal polity π∗, which can minimize
+Eπ[�∞
+t=0 γtcπ(t)|s0 = s] where γ is a discounting factor.
+An MDP system corresponds to a deterministic, continuous-
+state, discrete-time dynamical system st+1 = f(s, a) with
+state space S and action space A. Therefore, the system’s
+stability can be verified by a classical tool, Lyapunov’s Sta-
+bility Function (see Appendix A.3). If a Lyapunov function
+exists, a discrete-time system can achieve stability in the
+sense of Lyapunov as depicted in Appendix A.1 and A.2.
+In order to search for a Lyapunov function, current ap-
+proaches attempt to train a Lyapunov network by min-
+imizing the requirements in Appendix A.3.
+However,
+optimization-based methods meet a common problem in
+that the trained Lyapunov function remains close to 0 along
+the whole trajectories, thus resulting in ineffective guidance
+for the policy. More importantly, the classical definition
+is based on the whole state space, so it is unsuitable for
+RL-based methods due to sampling optimization.
+Therefore, we should extend the stability to a reasonable
+case in RL. We notice that for most robotic tasks, cost
+functions are related to the stability of the closed-loop
+system. For example, the goal of stabilization tasks is
+to make the norm of state equal to zero finally, where
+the cost function is C(s, a) = EP (·|st,at)∥st+1∥. Another
+main task is the tracking task, which aims to achieve the
+reference state r. The cost function can be denoted as
+C(s, a) = EP (·|st,at)∥st+1 − r∥. In this context, we in-
+troduce a Mean Cost Stability to connect the stability and
+the cost.
+Definition 3.1 (Mean Cost Stability). A robotic system
+remains stable in mean cost when satisfying the following
+equation, where b is a constant (Han et al., 2020a).
+lim
+t→∞ Est∼T cπ(st) = 0, cπ(s0) ≤ b
+(2)
+More importantly, the above definition corresponds to the
+mean square stability when the cost function is chosen to
+be the norm of the state; it is also equivalent to the partial
+stability in tracking tasks (Han et al., 2020a).
+In the other direction, the stability definition with regard
+to cost avoids the inconsistent scale between the objective
+and constraint. Specifically, the problem formulation can be
+represented as follows:
+min
+π Eρ,π,P[
+∞
+�
+t=0
+γtcπ(st)]
+s.t. lim
+t→∞ Est∼T cπ(st) = 0
+(3)
+4. Adaptive Lyapunov-based Actor-Critic
+Algorithm
+To target the above-mentioned problem, we propose the
+Adaptive Lyapunov-based Actor-Critic algorithm (ALAC)
+in this section. The main contents are as follows: 1) we
+design an Adaptive Stability Certification (ASC), a union
+of two objectives guaranteeing a system’s mean cost sta-
+bility and guiding the policy to search the optimal point
+heuristically (Section 4.1); 2) we construct an Actor-Critic
+optimizing framework based on ASC and use a supervised
+learning method to update the parameters of the Lyapunov
+critic network(Section 4.2); 3) we apply the prime-dual
+method to update the parameters of the policy network and
+tune the parameters of ASC adaptively (Section 4.3).
+4.1. Adaptive Stability Certification
+In order to achieve the mean cost stability in Definition
+3.1, we first give a reasonable assumption to ensure that the
+starting state is sampled in the region of attraction, which is
+represented as the starting state space.
+
+Adaptive Stability Certification
+Assumption 4.1 (Region of Attraction). There exists a pos-
+itive constant b such that ρ(s) > 0, ∀s ∈ {s|cπ(s) ≤ b}.
+Another assumption is the existence of the stationary state
+distribution, which is generally exploited in the RL litera-
+ture.
+Assumption 4.2 (Ergodic Property). The Markov Chain
+driven by the policy π is ergodic, which means the following
+equation holds.
+ωπ(s) = lim
+t→∞ T (s|ρ, π, t)
+(4)
+Furthermore, we define a new variable Uπ for further deriva-
+tion.
+Uπ = lim
+T →∞
+1
+T
+T
+�
+t=0
+T (s | ρ, π, t)
+(5)
+Based on these mild assumptions, we formalize the
+sampling-based Lyapunov stability that meets the mean
+cost stability in Theorem 4.3 below, proven in Appendix
+B.1.
+Theorem 4.3 (Sampling-based Lyapunov Stability). An
+MDP system is stable with regard to the mean cost, if there
+exists a function L : S → R meets the following conditions:
+αcπ(s) ≤ L(s) ≤ βcπ(s)
+(6)
+L(s) ≥ cπ(s) + λEs′∼PπL(s′)
+(7)
+Es∼Uπ[Es′∼PπL(s′) − L(s)]
+≤ −k[Es∼Uπ[L(s) − λEs′∼PπL(s′)]]
+(8)
+where α, β, λ and k is positive constants. Among them,
+Pπ(s′|s) =
+�
+A π(a|s)P(s′|s, a) da holds.
+In practice, our method constructs sampling-based require-
+ments for the Lyapunov function. Taking advantage of the
+sampling-based stability, we can learn the policy that guar-
+antees the system’s stability in RL framework. Moreover,
+finding a Lyapunov function directly remains a challenge
+when the search space becomes large due to the increas-
+ing dimension of the state. To mitigate the issue, we use a
+Lyapunov candidate which is related to the sum of cost or
+value function. it has been proven to be a valid Lyapunov
+candidate in a previous study for stability analysis (Mayne
+et al., 2000). Thus, we can denote a Lyapunov candidate as:
+Lπ(s) = Eπ[
+∞
+�
+t=0
+γtcπ(st)|s0 = s]
+(9)
+It is noteworthy that the Lyapunov candidate naturally meets
+the constraints in Equation (6) and (7) when λ ≤ γ holds.
+Please see Appendix A.2 for a detailed demonstration.
+∞
+{𝑊(𝑡)}!"#
+$
+{𝑅(𝑡)}!"#
+$
+𝑡
+𝑡 + 1
+𝑘(𝑊(𝑡) − 𝑅(𝑡))
+𝑊(𝑡 + 1)
+𝜆 ↓
+𝜆 ↓
+𝑅(𝑡)
+𝑊(𝑡)	
+Figure 2. Illustrative example of how ASC condition can
+minimize L(s).
+Note that {W(t)}∞
+t=0 = {EL(s)}∞
+t=0
+and {R(t)}∞
+t=0 = = {λEL(s′)}∞
+t=0. The blue point de-
+notes W(t), and the orange point denotes R(t), respectively.
+W(t + 1) should decrease by k(W(t) − R(t)) according to
+the ASC condition. Besides, as {R(t)}∞
+t=0 decreases with
+λ, {W(t)}∞
+t=0 decreases together. When λ remains at the
+minimum value, the ASC condition will help the policy
+minimize L(s).
+Recalling the optimization problem (3), we find that
+the constraint part equals Equation (8) with respect to
+Lπ(s), as well as the objective part can be rewritten as
+minπ Es∼ρLπ(s). The Lyapunov candidate builds a bridge
+between the constraint and the objective. A common ap-
+proach is to use the lagrangian-based method to achieve the
+trade-off. However, making the constraints soft poses a chal-
+lenge to the guarantee of mean cost stability. Accordingly,
+we hope to propose a method which guides the policy to find
+the optimal point on the premise of the stability guarantee.
+Fortunately, we find that Equation (8) can achieve the goal
+when finding the minimum value of λ adaptively. We call
+this the Adaptive Stability Certification (ASC) that pro-
+vides promising insight into searching for the optimal policy
+heuristically without violating stability. In the following,
+we illustrate the underlying reasons based on an interesting
+lemma in a continuous-time system.
+Lemma 4.4 (Finite Tracking Time). In a continuous-time
+system, a trajectory W(t) tracks the reference R(t). W(t)
+can track the reference within a finite time T, such that
+R(t) = W(t), t ≥ T, if the following conditions holds.
+∇tW(t) ≤ −k(W(t) − R(t)), ∀t ∈ [0, T]
+(10)
+Note that the gradient of R(t) should be bounded, meaning
+that ∇tR(t) ≤ µ. The proof see Appendix B.2
+It can be seen that Equation (8) is very similar to Equation
+(10). Although the discrete-time condition does not have a
+finite tracking time unlike the continuous-time setting, the
+same principle comes from the feedback control. To some
+extent, we can think Equation (8) as a special tracking task
+
+Adaptive Stability Certification
+that the sequence {W(t)}∞
+t=0 = {EL(s)}∞
+t=0 tracks the ref-
+erence sequence {R(t)}∞
+t=0 = {λEL(s′)}∞
+t=0. To explain
+it intuitively, Figure 2 depicts an illustrative example. As
+we can see, the value of W(t + 1) needs to decrease by
+k(W(t) − R(t)) at time t + 1. If λ decreases, {R(t)}∞
+t=0
+forces {W(t)}∞
+t=0 to minimize L(s) along the whole se-
+quence. Because of the introduction of the Lyapunov can-
+didate, minimizing L(s) equals the previous objective part
+in Equation (3). Therefore, we can find the optimal policy
+based on the certification by finding the minimum value of λ
+adaptively. To sum up, the optimization problem is modified
+to find the minimum value of λ as well as make Equation (8)
+holds. Note that we do not provide an in-depth theoretical
+analysis of the heuristic method. Thus, it leaves room for
+our future work, indicating that meeting ASC corresponds
+to finding the optimal policy.
+4.2. Lyapunov Critic Learning
+We leverage a traditional Actor-Critic framework to solve
+the optimization problem mentioned above. First, we con-
+struct two neural networks, namely actor πφ(a|s) and Lya-
+punov critic Lθ(s, a). Among them, φ and θ represent
+the parameters of two networks, respectively. The actor
+πφ(a|s) maps a given state s to a distribution over action.
+The action distribution is modelled as a Gaussian, with a
+state-dependent mean µφ(s) and diagonal covariance matrix
+Σφ(s). Unlike traditional Actor-Critic methods, our critic
+network is related to the Lyapunov candidate Lπ(s). As a
+matter of fact, Lπ(s) is the expectation of Lθ(s, a) over the
+distribution of actions. Specifically, EπLθ(s, a) = Lπ(s)
+holds. In the context of this property, the above theoretical
+results about Lπ(s) are also suitable for our critic network.
+For the training of Lθ(s, a), because we choose the value
+function as the Lyapunov candidate illustrated in Equation
+(9), we can update θ according to the TD error:
+θk+1 = θk + αθ(∇θ(Lθ(s, a) − (cπ + γL′(s′, π′(·|s′))))2)
+(11)
+where k is the number of iterations. L′ and π′ are the target
+networks parameterized by θ′ and φ′, respectively. In the
+Actor-Critic method, the parameters θ′ and φ′ are usually
+updated through exponentially moving average of weights
+controlled by a hyper-parameter σ ∈ (0, 1), which is shown
+as:
+θ′
+k+1 ← σθk + (1 − σ)θ′
+k; φ′
+k+1 ← σφk + (1 − σ)φ′
+k
+(12)
+Furthermore, we introduce an interesting mechanism on
+the Lyapunov critic network to speed up the learning pro-
+cess. Admittedly, the Lyapunov candidate Lπ(s) meets
+some requirements sufficiently shown in Appendix A.2,
+which means the output of the critic network can guarantee
+them eventually based on TD-like updating. In order to
+encourage accurate and efficient learning, we construct a
+constrained critic network. Concretely, we denote the output
+of a neural network as f(s, a). And then, Lθ(s, a) can be
+described by:
+Lθ(s, a) = (Gs(f(s, a)))(Gs(f(s, a)))⊤
+(13)
+where Gs is a linear transformation, which guarantees
+Gs=se(f(s = se, a)) = 0 (se is an equilibrium point de-
+fined in Definition A.1.). Note that Gs contains no parame-
+ters to be learned, so the operator does not cause harm to the
+representation ability of the neural network. For the details,
+see Appendix B.3. To this end, the constrained network
+ensures that the output is non-negative and the Lyapunov
+value should be zero when the state is an equilibrium point.
+4.3. Lagrangian-based Policy Learning
+Policy learning aims to search feasible parameters of the
+policy network to make the output of the Lyapunov network
+meet the requirements of the ASC condition.
+For optimizing the policy πφ(a|s), we denote the specific
+constrained condition as follows according to Equation (8).
+∆Lπφ(s, a) = Lθ(s′, πφ(· | s′)) − Lθ(s, a)
++ k[Lθ(s, a) − λLθ(s′, πφ(· | s′))] ≤ 0
+(14)
+We can see that the current policy’s parameters can be up-
+dated due to the embedding of πφ. Meanwhile, because of
+the sampling-based stability theorem (Theorem 4.3), it is
+convenient to efficiently implement the random sampling in
+the replay buffer D that stores the interaction data. More-
+over, the optimal policy can be obtained, when we maxi-
+mize ∆Lπφ(s, a) by finding the minimum λ according to
+the ASC condition. Specifically, The optimization problem
+in Equation (3) can be reformulated as follows.
+max
+λ,πφ
+ED∆Lπφ(s, a)
+s.t. ED∆Lπφ(s, a) ≤ 0
+(15)
+First, we focus on the sub-problem of finding πφ under
+satisfying the constraint with arbitrary λ, which is shown as:
+find πφ
+s.t. ED∆Lπφ(s, a) ≤ 0
+(16)
+Applying the Lagrangian-based method (Stooke et al.,
+2020), the parameters of πφ are updated by:
+φk+1 = φk + αφ(λl∇a∆Lπφ(s, a)∇φπφ(s, a))
+(17)
+where αφ is the learning rate of φ. λl represents the La-
+grange multiplier of the constraint. During the training, λl
+
+Adaptive Stability Certification
+is updated by gradient ascent to maximize ∆Lπφ(s, a).
+λk+1
+l
+= λk
+l + αλl∆Lπφ(s, a)
+(18)
+Note that λl should always be positive. αλl is the learning
+rate. It is worth noting that λl is clipped by 0 and 1, to
+bound the value. In addition, to improve the exploration
+efficiency, we add a constraint about the minimum entropy
+as the same as the maximum entropy RL algorithms.
+4.3.1. THE CHOICE OF λ
+However, how to maximize ∆Lπφ(s, a) in Equation (15)
+remains to be an unsolved problem. Finding the maximum
+value of ∆Lπφ(s, a) equals finding the minimum value of
+λ. First, The range of λ is from γ to 0 in the ASC condition,
+and γ is very close to 1 in practical usage. Moreover, we
+find the Lagrange multiplier λl ranges from 0 to 1, and it
+decreases with the constraint’s satisfaction. Therefore, we
+can update λ by the following method.
+λ ← min(λl, γ)
+(19)
+To be specific, when satisfying the constraint, λ decreases
+toward 0 to maximize ∆Lπφ(s, a). When λ remains at a
+stable level, the minimum value of λ realizes the aim of
+maximizing ∆Lπφ(s, a). Until now, we have constructed
+the whole training process for the optimization problem in
+Equation (15).
+4.3.2. THE CHOICE OF k
+For the choice of k, we also use a heuristics approach to
+adjust the value of k dynamically, k ← 1 − λl. There is
+an essential advantage to the approach. As we can see, λl
+is close to 1 at the early stage of training; thus, k leads to
+slow tracking according to Lemma 4.4. When λl decreases,
+it indicates the current constraint is loose. In the setup,
+kl increases instead, and the constraint forces the tracking
+process to be performed more quickly. Besides, It offers
+another benefit we explain in Section 4.4.
+We have designed the complete ALAC algorithm, and the
+pseudo-code of the proposed algorithm is shown in Ap-
+pendix C.
+4.4. Theoretical Analysis
+A common fact we notice is that there exists a bias be-
+tween the practical computing and theoretical analysis about
+Uπ in Theorem 4.3. To estimate the distribution Uπ, we
+need an infinite number of trajectories with infinite time
+steps, whereas in practice, only M trajectories of T time
+steps are accessible. To better illustrate the issue, we de-
+fine a finite sampling distribution UT
+π , apparently where
+UT
+π = 1
+T
+�T
+t=0 T (s | ρ, π, t). Besides, we introduce a new
+variable ∆Lπ(s) about the Lyapunov candidate to simplify
+the expression of derivation.
+∆Lπ(s) =Est+1∼PπLπ(st+1) − Lπ(st)
++ k[Lπ(st) − λEst+1∼PπLπ(st+1)]
+(20)
+First, we provide a quantitative bound of expectation from
+Uπ and UT
+π .
+Theorem 4.5. Suppose that the length of sampling trajecto-
+ries is T, then the bound can be expressed as:
+|Es∼Uπ∆Lπ(s) − Es∼UT
+π ∆Lπ(s)| ≤ 2(k + 1)cπ
+1 − γ
+T q−1
+(21)
+where cπ is the maximum of cost and q is a constant in
+(0, 1). For proof see Appendix B.4.
+Next, we take the number of trajectories into considera-
+tion and derive the probabilistic bound of the difference of
+∆Lπ(s) estimated by UT
+π distribution and M trajectories.
+Theorem 4.6. Suppose that the length of sampling trajec-
+tories is T and the number of trajectories is M, then there
+exists the following upper bound:
+P(| 1
+MT
+M
+�
+m=1
+T
+�
+t=1
+∆Lπ(sm
+t ) − Es∼UT
+π ∆Lπ(s)| ≥ α)
+≤ 2 exp(−
+Mα2(1 − γ)2
+((1 − kλ)2 + (k − 1)2)cπ2 )
+(22)
+where sm
+t represents the state in the m-th trajectory at the
+timestep t. For proof see Appendix B.5.
+Theorem 4.5 suggests that if k is close to 0, the gap becomes
+small in practice. Alternatively, Theorem 4.6 indicates that
+k has better to be 1. That means the best choice of k ranges
+from 0 to 1. Thus, the updating method of k in Section 4.3.2
+balances the impacts of each other efficiently. Furthermore,
+two theorems illustrate the theoretical gap between infinite
+and finite samples in practical usage, thus making Theorem
+4.3 more complete.
+5. Experiments
+In this section, we demonstrate empirical evidence that
+ALAC captures an improved trade-off between optimal-
+ity and stability compared to the baseline approaches.
+We test our method and baselines in ten robotic control
+environments, including Cartpole-cost,Point-circle-cost,
+Halfcheetah-cost, Swimmer-cost, Ant-cost, Humanoid-
+cost, Minitaur-cost, Spacereach-cost, Spacerandom-
+cost and Spacedualarm-cost. Details of the environments
+are given in Appendix D.1. Furthermore, we benchmark
+the ALAC method against five algorithms with a neu-
+ral Lyapunov function. The algorithms can be separated
+
+Adaptive Stability Certification
+Table 1. Performance evaluations of the cultivated costs and stability constraint violations on ten environments compared with six baselines.
+All quantities are provided in a scale of 0.1. Standard errors are provided in brackets. (if the mean constraints are less than 0.2, the sign is
+↓ else ↑.‘-’ indicates the algorithm does not contain the stability constraints.)
+Task
+Metrics
+ALAC
+SAC-cost
+SPPO
+LAC
+LAC∗
+POLYC
+LBPO
+TNLF
+Cartpole-cost
+Cost Return
+26.2(7.0)
+22.7(12.6)
+102.3(59.3)
+31.0(10.1)
+31.5(5.1)
+104.8(70.7)
+205.3(27.0)
+33.5(24.5)
+Violation
+↓
+-
+↑
+↓
+↓
+↓
+-
+↓
+Point-circle-cost
+Cost Return
+111.1(4.5)
+111.8(2.4)
+247.9(58.2)
+958.6(15.5)
+112.0(5.0)
+207.0(62.4)
+722.1(126.1)
+145.8(38.0)
+Violation
+↓
+-
+↑
+↓
+↑
+↓
+-
+↓
+Halfcheetah-cost
+Cost Return
+1.7(0.7)
+16.6(25.2)
+144.0(14.6)
+119.5(37.3)
+1.8(0.5)
+168.8(10.7)
+37.8(24.8)
+6.5(1.4)
+Violation
+↓
+-
+↑
+↓
+↓
+↓
+-
+↓
+Swimmer-cost
+Cost Return
+44.6(4.8)
+53.7(12.4)
+52.5(4.2)
+47.5(1.3)
+44.8(3.0)
+104.7(11.0)
+52.3(11.3)
+46.5(2.4)
+Violation
+↓
+-
+↑
+↓
+↑
+↓
+-
+↓
+Ant-cost
+Cost Return
+101.0(42.1)
+155.2(29.9)
+255.0(31.2)
+166.9(13.6)
+125.6(12.5)
+259.8(37.1)
+114.6(26.1)
+186.8(11.0)
+Violation
+↓
+-
+↑
+↓
+↑
+↓
+-
+↓
+Humanoid-cost
+Cost Return
+354.6(97.1)
+441.9(18.3)
+531.8(22.9)
+431.3(14.9)
+368.3(76.6)
+490.4(32.5)
+452.4(13.9)
+317.7(31.1)
+Violation
+↓
+-
+↑
+↓
+↑
+↓
+-
+↓
+Minitaur-cost
+Cost Return
+493.0(67.9)
+692.2(93.0)
+950.0(72.3)
+612.2(47.8)
+666.6(306.7)
+608.3(65.6)
+838.3(237.0)
+382.9(62.6)
+Violation
+↓
+-
+↑
+↓
+↑
+↓
+-
+↓
+Spacereach-cost
+Cost Return
+1.6(0.2)
+8.9(8.8)
+19.4(2.5)
+35.2(1.6)
+1.8(0.4)
+125.7(20.8)
+31.0(19.1)
+112.1(53.0)
+Violation
+↓
+-
+↓
+↓
+↓
+↓
+-
+↓
+Spacerandom-cost
+Cost Return
+2.3(0.3)
+38.4(28.6)
+53.2(32.7)
+33.9(3.5)
+2.8(0.9)
+112.8(19.4)
+35.82.9)
+85.9(42.3)
+Violation
+↓
+-
+↓
+↓
+↓
+↓
+-
+↓
+Spacedualarm-cost
+Cost Return
+26.1(3.5)
+36.1(8.3)
+201.9(48.8)
+66.3(10.6)
+63.6(62.1)
+140.6(17.4)
+37.87.5)
+280.1(99.3)
+Violation
+↓
+-
+↓
+↓
+↓
+↓
+-
+↓
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+300
+350
+400
+450
+500
+550
+Cost Return
+Humanoid-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+5
+10
+15
+20
+25
+30
+Violation
+Humanoid-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+400
+600
+800
+1000
+Cost Return
+Minitaur-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+2
+4
+6
+8
+10
+12
+Violation
+Minitaur-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+Figure 3. Ablation studies of the ASC condition. ALAC(original) shows comparable or the best performance compared with
+other certifications on each task.
+into two categories, optimizing policy with a mixed ob-
+jective and Lyapunov-like constraints. The first one con-
+tains POLYC (Chang & Gao, 2021), LBPO(Sikchi et al.,
+2021) and TNLF(Xiong et al., 2022). The second one in-
+cludes SPPO(Chow et al., 2019), LAC (Han et al., 2020a)
+2.
+We also take the SAC-cost (Haarnoja et al., 2018)
+method into account because the method is very close to
+our method without the ASC condition. For the detailed
+hyper-parameter settings see Appendix D.2.1 and D.2.2.
+5.1. Comparing with Baselines
+In this part, we evaluate the optimality and stability of our
+methods and baselines. The results demonstrate ALAC out-
+2We find LAC with a large α3 (see Appendix D.2.1) performs
+better, so we call it LAC∗ for the distinction between them.
+performs baselines. To fairly evaluate the performance of
+the methods mentioned above, we run the experiments over
+5 rollouts and 5 seeds for all algorithms. We use the accu-
+mulated cost in a testing episode as the metric of optimality
+and the stability constraint violations as the stability metric.
+Although the metric of stability violations depends on each
+algorithm’s design, the best value should be close to 0. Ta-
+ble 1 shows the performance on all tasks, and the training
+curves for different algorithms are in Appendix D.3. Results
+confirm that ALAC outperforms state-of-the-art baselines
+on all tasks except for Cartpole-cost, Humanoid-cost and
+Minitaur-cost where our method doesn’t outperform SAC-
+cost and TNLF, respectively. Furthermore, although LAC∗
+using tighter constraints achieves comparable performance
+with our method in contrast to LAC, the stability violations
+in LAC∗ remain at a high level on some tasks. Admittedly,
+TNLF achieves lower cost than ALAC on Minitaur-cost,
+
+Adaptive Stability Certification
+Cartpole-cost
+HalfCheetah-cost
+Minitaur-cost
+𝑔𝑜𝑎𝑙!
+𝑔𝑜𝑎𝑙"
+𝑔𝑜𝑎𝑙#
+Spacerandom-cost
+Steps
+𝜑
+𝜑 ̇
+ℒ!!
+Values
+Figure 4. Visualization of states for ALAC method by t-SNE and phase trajectory techniques. The top row of the figure
+depicts the t-SNE dimension reduction technique. (Cartpole-Cost is visualized within 2 dims while others within 3 dims.)
+The bottom row shows the phase trajectories and Lyapunov-value surfaces of environments. ψ and ˙ψ denotes the angular
+position and velocity respectively.
+Table 2. Average evaluation score and standard deviation on our environments for ALAC with and without the feedback under different
+biases of goals. (w/ errors means using errors between the desired and achieved goals as extra states for the agent)
+Task
+Point-circle-cost
+Halfcheetah-cost
+Spacereach-cost
+Biases of goals
+-20%
+0%
+20%
+-20%
+0%
+20%
+-20%
+0%
+20%
+ALAC w/ errors
+85.2(4.5)
+110.1(3.9)
+148.5(12.2)
+3.9(0.8)
+2.5(0.6)
+8.3(4.7)
+6.4(1.7)
+2.4(1.4)
+8.7(1.7)
+ALAC w/o errors
+178.8(7.8)
+118.9(11.4)
+247.8(11.9)
+10.1(2.1)
+3.3(1.2)
+13.4(2.1)
+11.9(0.4)
+1.6(0.2)
+11.5(0.3)
+SAC-cost w/ errors
+84.2(4.2)
+109.3(2.2)
+140.1(3.0)
+60.1(27.5)
+81.6(50.2)
+129.5(85.6)
+21.9(12.3)
+22.1(16.9)
+20.6(18.1)
+SAC-cost w/o errors
+180.9(6.3)
+115.3(4.0)
+240.3(3.7)
+15.9(15.7)
+16.7(25.5)
+33.5(34.0)
+16.1(6.2)
+8.8(8.8)
+15.0(7.2)
+but TNLF converges to suboptimal policies on many tasks.
+According to Figure 11, we notice that TNLF satisfies the
+stability constraints quickly during the training. Neverthe-
+less, the reason is that the trained Lyapunov function is close
+to 0 quickly. To this end, it does not provide dense guid-
+ance for the policy, thus leading it to a suboptimal solution.
+To recap, ALAC strikes an efficient balance between the
+optimality and stability of the system.
+5.2. Ablation Studies
+To better demonstrate the effectiveness of the ASC condi-
+tion, we do the ablation studies about different certifications
+in ALAC. We compare the performance of the original
+ALAC with a version that uses ∆L1
+πφ and ∆L2
+πφ, and with
+a version where k is a constant throughout training. The de-
+tails of ∆L1
+πφ and ∆L2
+πφ are given in Appendix D.4. In con-
+trast to ∆Lπφ, ∆L1
+πφ and ∆L2
+πφ represents the upper bound
+and lower bound of the constraint, respectively. Figure 3 and
+Figure 10 (see Appendix D.4) depict the accumulated cost
+and constraint violations on all tasks, where the algorithms
+are modified from ALAC directly. ALAC(∆L2
+πφ) achieve
+lower performance than ALAC, while ALAC(∆L1
+πφ) per-
+forms the tasks comparably with ALAC. Nevertheless, more
+strict constraints (ALAC(∆L1
+πφ)) negatively affect its per-
+formance on constraint violations, as shown in Figure 3.
+This is because there doesn’t exist a reasonable policy
+that meets such strict constraints. Moreover, the results of
+ALAC(k = 0.1) comparing with ALAC demonstrate that
+the heuristics updating of k is effective during the training.
+The slight gap between ALAC(Tanh) with the Lyapunov
+function activated by the Tanh function and ALAC shows
+our method is not sensitive to the form of the Lyapunov
+function.
+
+Adaptive Stability Certification
+5.3. Evaluation Results
+In this section, we describe the impacts of the stability condi-
+tion more concretely by using various visualization methods.
+Furthermore, we verify that ALAC achieves excellent ro-
+bustness, generalization, and efficiency with the aid of the
+ASC condition.
+5.3.1. ANALYSIS OF VISUALIZATION
+First, we use the t-SNE method to indicate the visualization
+of the state in 3 dimensions in order to illustrate better the
+stability of the system learned by ALAC (Cartpole-cost
+in 2 dimensions). As we can see, the top row of Figure 4
+shows the states in the final stage of an episode converge
+to a point or circle. Basically, we recognize that those
+patterns happen in a stable system. Furthermore, experts
+can judge a system’s stability from a phase space of the
+system. Therefore, we also introduce various phase space
+trajectories of the system to analyze stability. The second
+row of Figure 4 shows the phase trajectories with variance
+according to the state pairs of joint angular position and
+velocity. Concretely, ψ and ˙ψ represent an angular position
+and velocity, respectively. It can be found that the angular
+velocity starts from 0 to 0, and the angular position starts
+from the beginning to an equilibrium point. (For more
+details see Figure 6). Based on the above phenomenons, it
+suggests the trained systems using the ALAC method satisfy
+focal stability or stable limit cycles. More importantly, we
+observe that the Lyapunov value decreases while learning
+and consistently falls to its lowest point at the end of an
+episode, as shown in Figure 4 (bottom row). Furthermore,
+it can facilitate the policy to achieve promising results on
+robustness and generalization, which is demonstrated in
+further analysis. More implementation details for t-SNE
+and phase trajectories are given in Appendix D.5.
+5.3.2. VERIFICATION OF PROPERTIES
+Robustness
+Generally speaking, stability has a potential
+relationship with robustness to some extent (Chang & Gao,
+2021). Thus, we add external disturbances with different
+magnitudes in each environment and observe the perfor-
+mance difference. To be concrete, we introduce periodic
+external disturbances with different magnitudes in each task.
+Furthermore, we omit the algorithms which do not converge
+to a reasonable solution in each task. Figure 8 (see Appendix
+D.6.1) shows in all scenarios, ALAC enjoys superior per-
+formance over other methods.
+Generalization
+Previous studies focus on the robustness
+of a system to demonstrate the effectiveness of Lyapunov
+constraints. In this paper, we build some interesting exper-
+iments to verify whether the policy can generalize well to
+follow previously unseen reference signals or not. We in-
+Table 3. Comparisons on the performance of ALAC and SAC-cost
+methods under different actor structures.
+Task
+Halfcheetah-cost
+Actor structure
+[64,64]
+[32,32]
+[16,16]
+ALAC
+1.7(0.7)
+2.9(1.7)
+5.0(1.9)
+SAC-cost
+16.6(25.2)
+94.1(38.4)
+86.6(57.8)
+Task
+Minitaur-cost
+Actor structure
+[64,64]
+[32,32]
+[16,16]
+ALAC
+492.9(67.9)
+403.5(85.7)
+571.7(116.1)
+SAC-cost
+692.1(92.9)
+664.9(115.3)
+934.1(149.3)
+Task
+Spaceramdom-cost
+Actor structure
+[64,64]
+[32,32]
+[16,16]
+ALAC
+3.6 (1.0)
+8.3(6.0)
+12.1(4.9)
+SAC-cost
+28.7(9.5)
+30.6(13.9)
+38.4(28.6)
+troduce the error between the desired and achieved goals as
+additional information in the state. Because the Lyapunov
+function is significantly related to the error, ALAC w/ er-
+rors gains remarkable performance improvement on gen-
+eralization as Table 2 illustrated. We choose the SAC-cost
+algorithm as a comparison since SAC-cost is very similar
+to our method without the ASC condition. For more ex-
+perimental results, see Appendix D.6.2. In particular, the
+gap between each other enlarges with the increasing biases.
+Furthermore, we observe that the errors have a negative
+impact on the performance of SAC-cost on complex tasks
+like Halfcheetah-cost and Spacereach-cost. The reason
+can be that SAC-cost does not efficiently capture the error
+information without the guidance of a Lyapunov function.
+Efficiency
+Our method also offers a positive effect on the
+limited network size. Results in Table 3 suggest ALAC
+method consistently achieves comparable performance un-
+der different actor structures. Compared with the SAC-cost
+method, the ALAC method brings another benefit, perform-
+ing the task more efficiently with limited parameters of
+controllers. It is because the adaptive stability certification
+provides efficient guidance for policy optimization.
+6. Discussion and Future Work
+We propose the Adaptive Stability Certification (ASC),
+which meets the sampling-based stability of mean cost.
+Meanwhile, it guides the current policy to approaches to
+the optimal point heuristically in the context of classical
+feedback control. Based on the Actor-Critic framework and
+Lagrangian-based optimization, We present a practical algo-
+rithm, namely the Adaptive Lyapunov-based Actor-Critic
+algorithm (ALAC). Furthermore, empirical results show that
+our method outperforms baselines on diverse robotic tasks
+with two metrics, stability constraint violations and mean
+costs. Furthermore, the controller trained by our method
+
+Adaptive Stability Certification
+achieves higher generalization ability compared with the
+method without the stability guarantee. By making a heuris-
+tic formulation, we provide an interesting method to com-
+bine the policy’s optimality with the system’s stability in
+model-free RL. In this work, we focus on the system’s sta-
+bility; an exciting direction for future work would be to
+combine ASC with some safe RL algorithms.
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+Adaptive Stability Certification
+A. Preliminary Remarks
+A.1. Lyapunov function
+Definition A.1 (Equilibrium Point). A state se is an equilibrium point if ∃ action ae ∈ A such that f(se, ae) = se. (Murray
+et al., 2017)
+Definition A.2 (Stabilizable in the sense of Lyapunov). A system is stabilizable if ∀ϵ > 0, ∃δ such that for all s0 ∈ S such
+that ||s0 − se|| ≤ δ, there exists {at}∞
+t=0 such that the resulting {st}∞
+t=0 satisfies ||st − se|| ≤ ϵ ∀t ≥ 0.(Murray et al., 2017)
+Definition A.3 (Lyapunov Function). A continuous and radially unbounded function L : S → R is a Lyapunov function if
+the following conditions hold:
+1. ∀s ∈ S, ∃a ∈ A, s.t. L(s) ≥ L(f(s, a)),
+2. ∀s ̸= 0, L > 0; L(0) = 0.
+If a Lyapunov function exists, a discrete-time system can achieve stability in the sense of Lyapunov without considering the
+physical energy.
+A.2. Lyapunov Candidate Bound
+In this part, we show that the Lyapunov candidate Lπ(s) meets the property in Theorem 4.3, which can be formulated as:
+cπ(s) + λEs′∼PπL(s′) ≤ L(s) ≤ βcπ(s)
+(23)
+where we omit the lower bound αcπ(s) which is naturally satisfied by Lπ(s).
+Firstly, according to the definition of Lπ(s), we have
+Lπ(s) = Eπ[
+∞
+�
+t=0
+γtcπ(st)|s0 = s]
+= Eπ[cπ(s0) +
+∞
+�
+t=1
+γtcπ(st)|s0 = s]
+= cπ(s) + Eπ[
+∞
+�
+t=1
+γtcπ(st)]
+= cπ(s) + γEπ,s′∼Pπ[
+∞
+�
+t=0
+γtcπ(st)|s0 = s′]
+= cπ(s) + γEs′∼PπL(s′)
+(24)
+Considering the left-hand side of Equation (6), we can find that if λ ≤ γ holds, the lower bound of the Lyapunov function
+can be satisfied. This is because the Lyapunov candidate Lπ(s) is positive at each state. Furthermore, the right-hand side of
+Equation 6 illustrates the higher bound of the Lyapunov function exists. The condition is also guaranteed for our Lyapunov
+candidate shown in the following process.
+Lπ(s) = Eπ[
+∞
+�
+t=0
+γtcπ(st)|s0 = s]
+≤
+∞
+�
+t=0
+γtEπ[cπ(st)|s0 = s]
+≤
+cπ
+1 − γ
+(25)
+Note that cπ denotes the maximum cost. The second row of the inequality holds due to Jensen inequality. Only if the
+maximum cost exists, ∃β ∈ R+,
+cπ
+1−γ ≤ βcπ(s) holds.
+
+Adaptive Stability Certification
+B. Details of Theoretical Analysis
+B.1. Proof of Theorem 4.3
+Theorem B.1 (Sampling-based Lyapunov Stability). An MDP system is stable with regard to the mean cost, if there exists a
+function L : S → R meets the following conditions:
+αcπ(s) ≤ L(s) ≤ βcπ(s)
+(26)
+L(s) ≥ cπ(s) + λEs′∼PπL(s′)
+(27)
+Es∼Uπ[Es′∼PπL(s′) − L(s)] ≤ −k[Es∼Uπ[L(s) − λEs′∼PπL(s′)]]
+(28)
+where α, β, λ and k is positive constants. Among them, Pπ(s′|s) =
+�
+A π(a|s)P(s′|s, a) da holds.
+Proof. Firstly, we simplify the left side of the Equation (28) with reference to (Han et al., 2020a). Introducing the definition
+of Uπ(s) leads to
+Es∼Uπ[Es′∼PπLπ(s′) − Lπ(s)]
+=
+�
+S
+lim
+T →∞
+1
+T
+T
+�
+t=0
+T (s | ρ, π, t)(
+�
+S
+Pπ(s′|s)Lπ(s′)ds′ − Lπ(s))ds
+(29)
+Due to the boundedness of Lπ, we apply the Lebesgue’s Dominated convergence theorem. To be specific, when |Fn(s)| ≤
+B(s), ∀s ∈ S, ∀n holds, we have
+lim
+n→∞
+�
+S
+Fn(s)ds =
+�
+S
+lim
+n→∞ Fn(s)ds
+(30)
+Hence, we get
+Es∼Uπ[Es′∼PπLπ(s′) − Lπ(s)]
+=
+�
+S
+lim
+T →∞
+1
+T
+T
+�
+t=0
+T (s | ρ, π, t)(
+�
+S
+Pπ(s′|s)Lπ(s′)ds′ − Lπ(s))ds
+= lim
+T →∞
+�
+S
+1
+T
+T
+�
+t=0
+T (s | ρ, π, t)(
+�
+S
+Pπ(s′|s)Lπ(s′)ds′ − Lπ(s))ds
+= lim
+T →∞
+1
+T (
+T +1
+�
+t=1
+ET (s|ρ,π,t)Lπ(s) −
+T
+�
+t=0
+ET (s|ρ,π,t)Lπ(s))
+= lim
+T →∞
+1
+T (ET (s|ρ,π,T +1)Lπ(s) − ET (s|ρ,π,t=0)Lπ(s))
+(31)
+Note that T (s|ρ, π, t = 0) is equal to ρ. Since the expectation of Lπ(s) is a finite value, the left side of Equation (28) is
+zero.
+Now, we turn to the right side of Equation (28). According to the Equation (31), we have
+−k[Es∼Uπ[L(s) − λEs′∼PπL(s′)]] ≥ 0
+Es∼Uπ[L(s) − λEs′∼PπL(s′)] ≤ 0
+(32)
+
+Adaptive Stability Certification
+Since L(s) ≥ cπ(s) + λEs′∼PπL(s′) holds, we get
+Es∼Uπcπ(s) ≤ 0
+(33)
+Based on the Abelian theorem, we know there exists
+Uπ(s) = lim
+T →∞
+1
+T
+T
+�
+t=0
+T (s | ρ, π, t)
+= lim
+t→∞ T (s|ρ, π, t)
+= ωπ(s)
+(34)
+Thus, we get
+Es∼ωπ[cπ(s)] ≤ 0
+(35)
+The last row of inequality holds because of Equation (34). Based on the definition of ωπ(s), we have
+lim
+t→∞ ET (s|ρ,π,t)cπ(s) ≤ 0
+(36)
+Suppose that there exists a starting state s0
+∈
+{s0
+|
+cπ(s0)
+≤
+b} and a positive constant d such that
+limt→∞ ET (s|ρ,π,t)cπ(s) = d or limt→∞ ET (s|ρ,π,t)cπ(s) = ∞. Consider that ρ(s0) > 0 for all starting states in
+{s0 | cπ(s0) ≤ b} (Assumption 4.1), then limt→∞ Es∼T (·|ρ,π,t)cπ(s) > 0 , which is contradictory with Equation (36).
+Thus ∀s0 ∈ {s0 | cπ(s0) ≤ b}, limt→∞ ET (s|ρ,π,t)cπ(s) = 0. Thus the system meets the mean cost stability by Definition
+3.1.
+Furthermore, we find that when L(s) = cπ(s) + λEs′∼PπL(s′) holds, our theorem is corresponding to the Theorem 1 in
+(Han et al., 2020a). That means we extend the previous method to a more general case. To be specific, the introduction of
+λ enlarges the solution space of the policy. Thus, it facilitates the policy to find the optimal point as well as maintain the
+system’s stability.
+B.2. Proof of Lemma 4.4
+Lemma B.2 (Finite Tracking Time). In a continuous-time system, a trajectory W(t) tracks the reference R(t). W(t) can
+track the reference within a finite time T, such that R(t) = W(t), t ≥ T, if the following conditions holds.
+∇tW(t) ≤ −k(W(t) − R(t)), ∀t ∈ [0, T]
+(37)
+Note that the gradient of R(t) is bounded, meaning that ∇tR(t) ≤ µ holds.
+Proof. First, we build the mean square error V (t) between them.
+V = 1
+2(W(t) − R(t))2
+(38)
+Then, we can derive the difference of V (t) as follows
+∇tV = (W − R)(∇tW − ∇tR)
+≤ (W − R)(−k(W − R) − ∇tR)
+≤ −k|W − R|2 − (W − R)∇tR
+(39)
+
+Adaptive Stability Certification
+Introducing the Assumption that the bounded gradient of R(t) , we have
+∇tV ≤ −2k |W − R|2
+2
+−
+√
+2µ|W − R|
+2
+≤ −2kV −
+√
+2µ
+√
+V
+(40)
+Observe that the above formulation belongs to a form of the Bernoulli differential equation. In this case, we can reduce the
+Bernoulli equation to a linear differential equation by substituting z =
+√
+V . Then, the general solution for z is
+z =
+√
+V ≤ −
+√
+2
+2
+µ
+k + Ce−kt
+(41)
+Applying the initial condition V (t = 0) = vt0, we have
+C = √vt0 +
+√
+2
+2
+µ
+k
+(42)
+Finally, the convergence time T can be represented as:
+T = 1
+k ln
+� √
+2
+2
+µ
+k + √vt0
+√
+2
+2
+µ
+k
+�
++ t0
+(43)
+B.3. Constrained Lyapunov Critic Network
+In this work, the output of the neural network of the Lyapunov critic is described by:
+f(s, a) = hO(hO−1(· · · h2(h1(< s, a >))))
+(44)
+where each ho(z) has the same form:
+ho(z) = ψo(Woz + bo)
+(45)
+Here, O represents the number of layers, and ψo is the non-linear activation function used in the o-th layer. Furthermore,
+{Wo, bo} is the weight and bias of the o-th layer.
+First of all, to meet the demand of Lθ(se, a) = 0, we introduce a linear transformation Gs, one of whose possible forms can
+be
+Gs(f) =
+1
+�I
+i δsi + ϵ
+�δs1
+δs2
+· · ·
+δsI
+�
+�
+��
+f1
+f1
+· · ·
+fv
+f1
+f1
+· · ·
+fv
+· · ·
+· · ·
+· · ·
+· · ·
+f1
+f1
+· · ·
+fv
+�
+��
+(46)
+where I denotes the number of elements of the state, and v is the number of units of the output layer. ϵ is a constant close to 0
+to avoid singularity. Note that δs = s − se, which indicates the difference between the current state and an equilibrium point.
+As we can see, when each element of δs is zero, the multiplication of matrices is zero. Thus, Gs=se(f(s = se, a)) = 0
+holds. Furthermore, it brings another benefit having no impact on the training of networks.
+B.4. Proof of Theorem 4.5
+Theorem B.3. Suppose that the length of sampling trajectories is T, then the bound can be expressed as:
+|Es∼Uπ∆Lπ(s) − Es∼UT
+π ∆Lπ(s)| ≤ 2(k + 1)cπ
+1 − γ
+T q−1
+(47)
+where q is a constant in (0, 1).
+
+Adaptive Stability Certification
+Proof. First, we can get the following equation by introducing the definitions of Uπ and UT
+π .
+Es∼Uπ∆Lπ(s) − Es∼UT
+π ∆Lπ(s)
+=
+�
+S
+(Uπ(s) − 1
+T
+T
+�
+t=1
+T (s | ρ, π, t))∆Lπ(s)ds
+= 1
+T
+T
+�
+t=1
+�
+S
+(Uπ(s) − T (s | ρ, π, t))∆Lπ(s)ds
+(48)
+Then, eliminating the integral operator, we obtain
+|Es∼Uπ∆Lπ(s) − Es∼UT
+π ∆Lπ(s)|
+≤ 1
+T
+T
+�
+t=1
+∥Uπ(s) − T (s | ρ, π, t)∥1∥∆Lπ(s)∥∞
+(49)
+Thus, the next step is to get the bounds of ∥Uπ(s) − T (s | ρ, π, t)∥1 and ∥∆Lπ(s)∥∞.
+For the first part, we introduce the assumption that first is mentioned in (Zou et al., 2019), shown as follows:
+T
+�
+t=1
+∥Uπ(s) − T (s | ρ, π, t)∥1 ≤ 2T q, ∀T ∈ Z+, ∃q ∈ (0, 1)
+(50)
+Frankly speaking, the assumption is easily satisfied because the L1 distance between two distributions is bounded by 2. At
+the same time, T (s | ρ, π, t) converges to Uπ(s) with time approaching.
+For the second part, we can get the bound of ∆Lπ(s) according to Equation 25.
+∆Lπ(s) = Es′∼PπLπ(s′) − Lπ(s) + k(Lπ(s) − λEs′∼Pπ(s′))
+≤
+cπ
+1 − γ − 0 + k( cπ
+1 − γ − 0)
+(51)
+Then, we have
+∥∆Lπ(s)∥∞ ≤ (k + 1) cπ
+1 − γ
+(52)
+Adding results in Equation 53, we finally get
+|Es∼Uπ∆Lπ(s) − Es∼UT
+π ∆Lπ(s)| ≤ 2(k + 1)cπ
+1 − γ
+T q−1
+(53)
+B.5. Proof of Theorem 4.6
+Theorem B.4. Suppose that the length of sampling trajectories is T and the number of trajectories is M, then there exists
+the following upper bound:
+P(| 1
+MT
+M
+�
+m=1
+T
+�
+t=1
+∆Lπ(sm
+t ) − Es∼UT
+π ∆Lπ(s)| ≥ α)
+≤ 2 exp(−
+Mα2(1 − γ)2
+((1 − kλ)2 + (k − 1)2)cπ2 )
+(54)
+where sm
+t represents the state in the m-th trajectory at the timestep t.
+
+Adaptive Stability Certification
+Proof. First, eliminating ∆Lπ(s) by Equation 20, we rewrites the left side of Equation 54 as
+δ = P(| 1
+MT
+M
+�
+m=1
+T
+�
+t=1
+∆Lπ(sm
+t ) − Es∼UT
+π ∆Lπ(s)| ≥ α)
+= P(| 1
+MT
+M
+�
+m=1
+T
+�
+t=1
+(Lπ(st+1) − Lπ(st) + kl(Lπ(st) − λLπ(st+1))) − Es∼UT
+π ∆Lπ(s)| ≥ α)
+= P(| 1
+MT
+M
+�
+m=1
+T
+�
+t=1
+((1 − kλ)Lπ(st+1) + (k − 1)Lπ(st)) − Es∼UT
+π ∆Lπ(s)| ≥ α)
+(55)
+Here Es∼UT
+π ∆Lπ(s) is expected value of
+1
+MT
+�M
+m=1
+�T
+t=1 ∆Lπ(sm
+t ). In addition, the bounds of (1 − kλ)Lπ(st+1) and
+(k − 1)Lπ(st) can be obtained easily by Equation 25. Thus, we obtain the Theorem 4.6 by applying Hoeffding’s inequality.
+δ ≤ 2 exp(−
+2M 2α2
+M((1 − kλ)2 + (k − 1)2)
+cπ2
+(1−γ)2
+)
+≤ 2 exp(−
+Mα2(1 − γ)2
+((1 − kλ)2 + (k − 1)2)cπ2 )
+(56)
+C. Details of Algorithms
+As mentioned in the main text, we introduce a minimum entropy as a constraint in policy optimization and apply the
+primal-dual method to update the policy and the Lagrange multiplier λe. To be specific, the constraint can be expressed as
+log πφ(a|s) ≤ −Ze
+(57)
+where Ze is the minimum value of policy entropy, usually, Ze corresponds to the dimension of action space in the
+environment.
+Algorithm 1: Adaptive Lyapunov-based Actor-Critic Algorithm (ALAC)
+Orthogonal initialize the parameters of actor and critic networks with φ, θ
+Initialize replay buffer D and λl, λe
+Initialize the parameters of target network with φ′ ← φ and θ′ ← θ
+for episode m = 1, M do
+Sample an initial state s0
+for step t = 0, T − 1 do
+Sample an action from πφ(at|st)
+Execute the action at and observe a new state st+1
+Store < st, at, ct, st+1 > into D
+end for
+for iteration n = 1, N do
+Sample a minibatch B from the replay buffer D
+Update θ according to Eq.(11) using minibatch B
+Update φ, λl, λe according to Eq.(17),(18),(57) using minibatch B
+Update the parameters of target networks, θ′, φ′, according to Eq. (12)
+end for
+end for
+D. Details of Experiments
+We test our method and baselines in ten robotic control environments, including Cartpole-cost,Point-circle-cost,
+Halfcheetah-cost, Swimmer-cost, Ant-cost, Humanoid-cost, Minitaur-cost, Spacereach-cost, Spacerandom-cost and
+
+Adaptive Stability Certification
+Cartpole-cost
+Point-circle-cost
+Swimmer-cost
+Halfcheetah-cost
+Ant-cost
+Humanoid-cost
+Minitaur-cost
+Spacereach-cost
+Spacerandom-cost
+Spacedualarm-cost
+Figure 5. Overview of our environments.
+Spacedualarm-cost. Most tasks in ten environments are goal-oriented, tracking a target position or speed, which corre-
+sponds to most control tasks. Furthermore, the latter four environments involve models of practical robots like a quadruped
+robot and a robotic arm, making them relatively more difficult. It is worth noting that the task of Spacedualarm-cost is
+trajectory planning of a free-floating dual-arm space robot. The coupling property of the base and the robotic arms brings
+hardship for both traditional control and RL-based methods (Wang et al., 2022).
+D.1. Environmental Design
+Cartpole-cost
+This task aims to maintain the pole vertically at a target position. The environment is inherited from (Han
+et al., 2020b). The state and action space are the same as the default settings in OpenAI Gym(Brockman et al., 2016), so we
+omit the description. The cost function is c =
+�
+x
+xthreshold
+�2
++ 20 ∗
+�
+θ
+θthreshold
+�2
+, where xthreshold = 10 and θthreshold = 20◦. The
+other settings can be found in Table 4.
+Point-circle-cost
+This task aims to allow a sphere to track a circular trajectory. The environment is inherited from (Achiam
+et al., 2017). The sphere is initialized at the original point. The cost function is represented as c = d, where d denotes the
+distance between the current position and the reference. The other settings can be found in Table 4.
+Table 4. Hyper-parameters of non-linear dynamic environments
+Hyper-parameters
+Cartpole-cost
+Point-circle-cost
+State shape
+4
+7
+Action shape
+2
+2
+Length of an episode
+250 steps
+65 steps
+Maximum steps
+3e5 steps
+3e5 steps
+Actor network
+(64, 64)
+(64, 64)
+Critic network
+(64, 64, 16)
+(64, 64, 16)
+Halfcheetah-cost
+The goal of this task is to make a HalfCheetah (a 2-legged simulated robot) to track the desired velocity.
+The environment is inherited from (Han et al., 2020b). The state and action space are the same as the default settings in
+
+A
+c300mm
+300mm
+300mmAdaptive Stability Certification
+Table 5. Hyper-parameters of mujoco environments
+Hyper-parameters
+Swimmer-cost
+Halfcheetah-cost
+Ant-cost
+Humanoid-cost
+State shape
+8
+17
+27
+376
+Action shape
+2
+6
+8
+8
+Length of an episode
+250 steps
+200 steps
+200 steps
+500 steps
+Maximum steps
+3e5 steps
+1e6 steps
+1e6 steps
+1e6 steps
+Actor network
+(64, 64)
+(64, 64)
+(64, 64)
+(256, 256)
+Critic network
+(64, 64, 16)
+(256, 256, 16)
+(64, 64, 16)
+(256, 256, 128)
+OpenAI Gym(Brockman et al., 2016), so we omit the description. The cost function is c = (v − 1)2, where 1 represents the
+desired velocity. The other settings can be found in Table 5.
+Swimmer-cost
+This task aims to make a multi-joint snake robot to track the desired velocity. The environment is inherited
+from (Han et al., 2020b). The state and action space are the same as the default settings in OpenAI Gym(Brockman et al.,
+2016), so we omit the description. The cost function is c = (v − 1)2, where 1 represents the desired velocity. The other
+settings can be found in Table 5.
+Ant-cost
+This task aims to make an Ant (a quadrupedal simulated robot) track the desired velocity. The environment is
+inherited from (Brockman et al., 2016). The state and action space are the same as the default settings in OpenAI Gym
+(Brockman et al., 2016), so we omit the description. The cost function is c = (v − 1)2, where 1 represents the desired
+velocity. The other settings can be found in Table 5.
+Humanoid-cost
+This task aims to make a humanoid robot to track the desired velocity. The environment is inherited from
+(Brockman et al., 2016). The state and action space are the same as the default settings in OpenAI Gym (Brockman et al.,
+2016), so we omit the description. The cost function is c = (v − 1)2, where 1 represents the desired velocity. The other
+settings can be found in Table 5.
+Minitaur-cost
+This task aims to control the Ghost Robotics Minitaur quadruped to run forward at the desired velocity.
+The environment is inherited from (Coumans & Bai, 2016). The state and action space are the same as the default settings
+in PyBullet environment(Coumans & Bai, 2016), so we omit the description. The cost function is c = (v − 1)2, where 1
+represents the desired velocity. The other settings can be found in Table 6.
+Spacereach-cost
+This task aims to make a free-floating single-arm space robot’s end-effector reach a fixed goal position.
+Since the base satellite is uncontrolled, collisions will cause system instability once collisions occur. Therefore, it is critical
+to plan a collision-free path while maintaining the stability of the base. The agent can obtain the state, including the angular
+positions and velocities of joints, the position of the end-effector, and the position of the reference point. Then, the agent
+outputs the desired velocities of joints. In low-level planning, a PD controller converts the desired velocities into torques,
+and then controls the manipulator. The cost function is defined as c = d, where d is the distance between the goal and
+end-effector. The other settings can be found in Table 6.
+Spacerandom-cost
+This task aims to make a free-floating single-arm space robot’s end-effector reach a random goal
+position. The agent can obtain the state, including the angular positions and velocities of joints, the position of the
+end-effector, and the position of the reference point. Then, the agent outputs the desired velocities of joints. In low-level
+planning, a PD controller converts the desired velocities into torques to control the manipulator. The cost function is defined
+as c = d, where d is the distance between goal and end-effector. The other settings can be found in Table 6.
+Spacedualarm-cost
+This task aims to make a free-floating dual-arm space robot’s end-effectors reach random goal
+positions. The complexity of the task increases dramatically due to two arms’ coupling effects on the base. The agent can
+obtain the state, including the angular positions and velocities of joints, the positions of end-effectors, and the position
+of target points of two manipulators. Then, the agent outputs the desired velocities of joints. In low-level planning, a PD
+controller converts the desired velocities into torques to control the manipulators. The cost function is defined as follows:
+c = d0 + d1, where di is the distance between goal and end-effector of Arm-i. The other settings can be found in Table 6.
+
+Adaptive Stability Certification
+Table 6. Hyper-parameters of robotic environments
+Hyper-parameters
+Minitaur-cost
+Spacereach-cost
+Spacerandom-cost
+Spacedualarm-cost
+State shape
+27
+18
+18
+54
+Action shape
+8
+6
+6
+12
+Length of an episode
+500 steps
+200 steps
+200 steps
+200 steps
+Maximum steps
+1e6 steps
+3e5 steps
+5e5 steps
+5e5 steps
+Actor network
+(256, 256)
+(256, 256)
+(256, 256)
+(512, 512)
+Critic network
+(256, 256, 16)
+(256, 256, 128)
+(256, 256, 128)
+(512, 512, 256)
+D.2. Implementation Details
+D.2.1. BASELINES
+SAC-cost
+Soft Actor-Critic(SAC) is an off-policy maximum entropy actor-critic algorithm (Haarnoja et al., 2018). The
+main contribution is to add a maximum entropy objective into standard algorithms. The soft Q and V functions are trained
+to minimize the soft Bellman residual, and the policy can be learned by directly minimizing the expected KL-divergence.
+The only difference between SAC and SAC-cost is replacing maximizing a reward function with minimizing a cost function.
+The hyper-parameters of SAC-cost is illustrated in Table 7.
+Table 7. Hyper-parameters of SAC-cost
+Hyper-parameters
+SAC-cost
+Learning rate of actor
+1.e-4
+Learning rate of critic
+3.e-4
+Optimizer
+Adam
+ReplayBuffer size
+106
+Discount (γ)
+0.995
+Polyak (1 − τ)
+0.995
+Entropy coefficient
+1
+Batch size
+256
+SPPO
+Safe proximal policy optimization (SPPO) is a Lyapunov-based safe policy optimization algorithm. The neural
+Lyapunov network is constructed to prevent unsafe behaviors. Actually, the safe projection method is inspired by the TRPO
+algorithm (Schulman et al., 2015). In this paper, we modify it to apply the Lyapunov constraints on the MDP tasks, similar
+to the process in (Han et al., 2020a). The hyper-parameters of SPPO is illustrated in Table 8.
+Table 8. Hyper-parameters of SPPO
+Hyper-parameters
+SPPO
+Learning rate of actor
+1.e-4
+Learning rate of Lyapunov
+3.e-4
+Optimizer
+Adam
+Discount (γ)
+0.995
+GAE parameter (λ)
+0.95
+Clipping range
+0.2
+KL constraint (δ)
+0.2
+Fisher estimation fraction
+0.1
+Conjugate gradient steps
+10
+Conjugate gradient damping
+0.1
+Backtracking steps
+10
+Timesteps per iteration
+2000
+
+Adaptive Stability Certification
+LAC
+Lyapunov-based Actor-Critic(LAC) algorithm is an actor-critic RL-based algorithm jointly learning a neural
+controller and Lyapunov function (Han et al., 2020a). Particularly, they propose a data-driven stability condition on the
+expected value over the state space. Moreover, they have found that the method achieves high generalization and robustness.
+The hyper-parameters of LAC is illustrated in Table 9. Among them, α3 is 0.1 in LAC, while it is changed as 1 in LAC∗.
+Table 9. Hyperparameters of LAC
+Hyperparameters
+LAC
+Learning rate of actor
+1.e-4
+Learning rate of Lyapunov
+3.e-4
+Learning rate of Larange multiplier
+3.e-4
+Optimizer
+Adam
+ReplayBuffer size
+106
+Discount (γ)
+0.995
+Polyak (1 − τ)
+0.995
+Parameter of Lyapunov constraint (α3)
+0.1
+Batch size
+256
+POLYC
+Policy Optimization with Self-Learned Almost Lyapunov Critics (POLYC) algorithm is built on the standard
+PPO algorithm (Schulman et al., 2017). Introducing a Lyapunov function without access to the cost allows the agent to
+self-learn the Lyapunov critic function by minimizing the Lyapunov risk. The hyper-parameters of POLYC is illustrated in
+Table 10.
+Table 10. Hyper-parameters of POLYC
+Hyper-parameters
+POLYC
+Learning rate of actor
+1.e-4
+Learning rate of critic
+3.e-4
+Learning rate of Lyapunov
+3.e-4
+Optimizer
+Adam
+Discount (γ)
+0.995
+GAE parameter (λ)
+0.95
+Weight of Lyapunov constraint (β)
+0.1
+Clipping range
+0.2
+Timesteps per iteration
+2000
+LBPO
+Lyapunov Barrier Policy Optimization (LBPO) algorithm (Sikchi et al., 2021) is built on SPPO algorithm (Chow
+et al., 2019). However, the core improvement uses a Lyapunov-based barrier function to restrict the policy update to a safe
+set for each training iteration. Compared with the SPPO algorithm, the method avoids backtracking to ensure safety. For
+the implementation in our paper, the process is similar to that of the SPPO algorithm. The hyperparameters of LBPO is
+illustrated in Table 11.
+TNLF
+Twin Neural Lyapunov Function (TNLF) algorithm is proposed to deal with safe robot navigation in (Xiong et al.,
+2022). Different from other approaches, the TNLF method defines a Lyapunov V function and Lyapunov Q function, which
+are trained by minimizing the Lyapunov risk. In effect, the Lyapunov risk is similar to that of (Chang & Gao, 2021). Since
+the Lyapunov function strictly decreases over time, the robot starting with any state in a Region of Attraction (RoA) will
+always stay in the RoA in the future. It should be pointed out that as our environments only support the cost function,
+the objective, except for Lyapunov risk, is to minimize the cumulative return of cost. The hyper-parameters of TNLF is
+illustrated in Table 12.
+D.2.2. OUR METHOD
+ALAC
+Our method offers a significant advantage in contrast to baselines, which is to use fewer hyperparameters. The
+main hyperparameters are illustrated in Table 13. We notice that these parameters control networks’ learning without
+
+Adaptive Stability Certification
+Table 11. Hyperparameters of LBPO
+Hyperparameters
+LBPO
+Learning rate of actor
+1.e-4
+Learning rate of critic
+1.e-4
+Learning rate of Lyapunov
+3.e-4
+Optimizer
+Adam
+Discount (γ)
+0.99
+GAE parameter (λ)
+0.97
+Clipping range
+0.2
+KL constraint
+0.012
+Fisher estimation fraction
+0.1
+Conjugate gradient steps
+10
+Conjugate gradient damping
+0.1
+Backtracking steps
+10
+Weight of Lyapunov constraint (β)
+0.01
+Timesteps per iteration
+2000
+Table 12. Hyper-parameters of TNLF
+Hyper-parameters
+TNLF
+Learning rate of actor
+1.e-4
+Learning rate of critic
+3.e-4
+Learning rate of Lyapunov V functiob
+3.e-4
+Learning rate of Lyapunov functiob
+3.e-4
+Optimizer
+Adam
+ReplayBuffer size
+106
+Discount (γ)
+0.995
+Polyak (1 − τ)
+0.995
+Weight of Lyapunov constraint (α)
+0.1
+Variance of noise distribution
+1
+Batch size
+256
+including the parameters of constraints. The reason is they are automatically updated according to Lagrange multipliers, λl,
+and λe. The initial value of Lagrange multipliers is set to 1, common usage in previous constrained methods.
+D.3. More Results on Comparison
+Figure 11 shows the learning curves of the accumulated cost and constraint violations of ALAC and other baselines in ten
+environments.
+D.4. More Results on Ablation Study
+We provide the specific formulation of ∆L1
+πφ and ∆L2
+πφ. Compared with ∆Lπφ in Equation 14, we intuitively find
+that ∆L1
+πφ and ∆L2
+πφ are lower and higher bound of ∆Lπφ respectively. In other words, ∆L1
+πφ represents the strongest
+constraint, while ∆L2
+πφ represents the loosest constraint. The comparison between them can demonstrate that the ASC
+condition (∆Lπφ) has a positive effect on the performance of optimality and stability.
+∆L1
+πφ(s, a) = Lθ(s′, πφ(·|s′)) − Lθ(s, a) + k[Lθ(s, a) − 0]
+∆L2
+πφ(s, a) = Lθ(s′, πφ(·|s′)) − Lθ(s, a) + k[Lθ(s, a) − Lθ(s′, πφ(·|s′))]
+(58)
+The ablation experiments on other tasks are shown in Figure 10.
+
+Adaptive Stability Certification
+Table 13. Hyper-parameters of ALAC
+Hyper-parameters
+ALAC
+Learning rate of actor
+1.e-4
+Learning rate of Lyapunov
+3.e-4
+Learning rate of Lagrange multipliers (λl and λe)
+3.e-4
+Optimizer
+Adam
+ReplayBuffer size
+106
+Discount (γ)
+0.995
+Polyak (1 − τ)
+0.995
+Batch size
+256
+1
+n_components=2 or 3 ,
+2
+e a r l y _ e x a g g e r a t i o n =12 ,
+3
+l e a r n i n g _ r a t e =200.0 ,
+4
+n _ i t e r =1000 ,
+5
+n _ i t e r _ w i t h o u t _ p r o g r e s s =300 ,
+6
+min_grad_norm=1e −7 ,
+7
+p e r p l e x i t y =30 ,
+8
+me tric ="euclidean" ,
+9
+n_jobs =None ,
+10
+random_state =42 ,
+11
+verbose =True ,
+12
+i n i t =’pca’
+Table 14. Other hyper-parameters of t-SNE method.
+D.5. Details of Visualization
+Our RL-based policy optimization method guided by adaptive stability is difficult to express the latent laws of states in the
+convergent process of different environments as the high-dimension states-space. To find and show the state’s change laws
+in the convergent process:
+• We use the t-SNE dimension reduction technique to visualize the state-space.
+• We plot the phase trajectory with variance according to the state pairs of joint angular position and velocity.
+• We plot the Lyapunov-value surface and its shadow with the phase trajectory and values in the convergence process.
+T-SNE Visualization
+The top row of Figure 4 shows the results of the t-SNE state plotting with SciKit-Learn
+tools(i.e.sklearn.manifold.TSNE function) with varying parameters(e.g. early_exaggeration, min_ grad_norm). Cartpole-
+Cost is visualized with n_components=2 while other environments with n_components=3. The hyper-parameters for t-SNE
+are shown in Table 14.
+Phase Trajectories of Systems
+We select the angular position and velocity of a joint in the state space in each environment
+and plot the phase trajectory with variance in Figure 6. The convergent process is shown as the angular velocity starts from
+0 to 0, and the joint angle starts from the beginning to the convergence position.
+Lyapunov Functions of Systems
+We visualize the change of Lyapunov-value in 3 dimensions based on the phase
+trajectory. The second row of Figure 4 shows the Lyapunov-value surface. The curves of values along the phase trajectory
+are mapped to the whole plane with down-sampled and smoothed by a Gaussian filter; we add the values and the phase
+trajectory shadows correspondingly simultaneously.
+
+Adaptive Stability Certification
+0.10
+0.05
+0.00
+0.05
+0.10
+0.15
+Angular Position
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+Angular Velocity
+Cartpole-cost phase trajectory
+0.30
+0.35
+0.40
+0.45
+0.50
+Angular Position
+0
+1
+2
+3
+4
+Angular Velocity
+Halfcheetah-cost phase trajectory
+1.6
+1.8
+2.0
+2.2
+2.4
+Angular Position
+15
+10
+5
+0
+5
+10
+Angular Velocity
+Minitaur-cost phase trajectory
+1.75
+1.50
+1.25
+1.00
+0.75
+0.50
+0.25
+0.00
+Angular Position
+0.3
+0.2
+0.1
+0.0
+0.1
+Angular Velocity
+Spacerandom-cost phase trajectory
+Figure 6. Phase trajectories of the systems trained by ALAC. (we report the results of 20 trials and select a joint to graph the
+phase trajectory in each task.)
+D.6. More Results on Evaluation
+D.6.1. ROBUSTNESS
+We verify that ALAC achieves excellent robustness on most tasks. It is worth noting that we introduce periodic external
+disturbances with different magnitudes in each task. Furthermore, we omit the algorithms which do not converge to a
+reasonable solution in each task.
+D.6.2. GENERALIZATION
+We verify that ALAC achieves excellent generalization with the feedback of errors. In particular, the gap between each other
+enlarges with the increasing biases. Furthermore, we observe that the errors bring a negative impact on the performance of
+SAC-cost. The reason can be that SAC-cost does not capture the error information without the guidance of a Lyapunov
+function. Note that the number of environment steps in Halfcheetah-cost is 5e5 in this section.
+D.6.3. EFFICIENCY
+We verify that ALAC achieves comparable performance under different network structures of the actor on three tasks. By
+contrast, the network structure significantly impacts the performance of SAC-cost.
+
+Adaptive Stability Certification
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+25
+50
+75
+100
+125
+150
+175
+Cost Return
+Cartpole-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+200
+400
+600
+800
+1000
+Cost Return
+Pointcircle-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+20
+40
+60
+80
+100
+120
+Cost Return
+Swimmer-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+25
+50
+75
+100
+125
+150
+175
+Cost Return
+HalfCheetah-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+50
+100
+150
+200
+250
+300
+350
+Cost Return
+Ant-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+100
+200
+300
+400
+500
+600
+Cost Return
+Humanoid-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+200
+400
+600
+800
+1000
+Cost Return
+Minitaur-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Cost Return
+Spacereach-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+20
+40
+60
+80
+100
+120
+140
+Cost Return
+Spacerandom-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Magnitude
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Cost Return
+Spacedualarm-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SAC-cost
+SPPO
+TNLF
+Figure 7. Performance of ALAC method and other baselines under persistent disturbances with different magnitudes. (The
+X-axis indicates the magnitude of the applied disturbance. We evaluate the trained policies for 20 trials in each setting.)
+-20%
+-10%
+0%
+10%
+20%
+Biases of goals
+75
+100
+125
+150
+175
+200
+225
+250
+Cost Return
+Pointcircle-cost
+ALAC w/ error
+ALAC w/o error
+SAC-cost w/ error
+SAC-cost w/o error
+-20%
+-10%
+0%
+10%
+20%
+Biases of goals
+0
+50
+100
+150
+200
+Cost Return
+HalfCheetah-cost
+ALAC w/ error
+ALAC w/o error
+SAC-cost w/ error
+SAC-cost w/o error
+-20%
+-10%
+0%
+10%
+20%
+Biases of goals
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Cost Return
+Spacereach-cost
+ALAC w/ error
+ALAC w/o error
+SAC-cost w/ error
+SAC-cost w/o error
+Figure 8. Evaluation of ALAC and SAC-cost methods in the presence of different biases of goals. (The X-axis indicates the
+magnitude of the applied shifting. We evaluate the trained policies for 20 trials in each setting.)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+50
+100
+150
+200
+Cost Return
+HalfCheetah-cost
+ALAC(64x64)
+ALAC(32x32)
+ALAC(16x16)
+SAC-cost(64x64)
+SAC-cost(32x32)
+SAC-cost(16x16)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+400
+600
+800
+1000
+1200
+Cost Return
+Minitaur-cost
+ALAC(64x64)
+ALAC(32x32)
+ALAC(16x16)
+SAC-cost(64x64)
+SAC-cost(32x32)
+SAC-cost(16x16)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+50
+100
+150
+200
+Cost Return
+HalfCheetah-cost
+ALAC(64x64)
+ALAC(32x32)
+ALAC(16x16)
+SAC-cost(64x64)
+SAC-cost(32x32)
+SAC-cost(16x16)
+Figure 9. Learning curves of ALAC and SAC-cost methods with different network structures of the actor on Halfcheetah-cost,
+Minitaur-cost, and Spaceramdom-cost tasks.
+
+Adaptive Stability Certification
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+50
+100
+150
+200
+250
+Cost Return
+Cartpole-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+200
+400
+600
+800
+1000
+Cost Return
+Pointcircle-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Cost Return
+HalfCheetah-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+45
+50
+55
+60
+65
+70
+75
+80
+Cost Return
+Swimmer-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Violation
+Cartpole-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0
+1
+2
+3
+4
+5
+6
+Violation
+Pointcircle-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Violation
+HalfCheetah-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+Violation
+Swimmer-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+60
+80
+100
+120
+140
+160
+180
+200
+220
+Cost Return
+Ant-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Cost Return
+Spacereach-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+10
+20
+30
+40
+50
+60
+70
+Cost Return
+Spacerandom-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+25
+50
+75
+100
+125
+150
+175
+200
+225
+Cost Return
+Spacedualarm-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+1
+2
+3
+4
+5
+Violation
+Ant-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Violation
+Spacereach-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Violation
+Spacerandom-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+1
+2
+3
+4
+5
+6
+7
+Violation
+Spacedualarm-cost
+ALAC(original)
+ALAC(
+1 )
+ALAC(
+2 )
+ALAC(Tanh)
+ALAC(kl = 0.1)
+Figure 10. Ablation studies of the ASC condition. ALAC(original) shows comparable or the best performance compared
+with other certifications on each task.
+
+Adaptive Stability Certification
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+50
+100
+150
+200
+250
+300
+Cost Return
+Cartpole-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+200
+400
+600
+800
+1000
+Cost Return
+Pointcircle-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+50
+100
+150
+200
+250
+300
+Cost Return
+HalfCheetah-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+40
+60
+80
+100
+120
+Cost Return
+Swimmer-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0
+2
+4
+6
+8
+10
+Violation
+Cartpole-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0
+2
+4
+6
+8
+Violation
+Pointcircle-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Violation
+HalfCheetah-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Violation
+Swimmer-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+50
+100
+150
+200
+250
+300
+Cost Return
+Ant-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+300
+350
+400
+450
+500
+550
+600
+Cost Return
+Humanoid-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+400
+600
+800
+1000
+1200
+Cost Return
+Minitaur-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0
+25
+50
+75
+100
+125
+150
+175
+Cost Return
+Spacereach-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+1
+2
+3
+4
+5
+Violation
+Ant-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+2
+4
+6
+8
+10
+12
+Violation
+Humanoid-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Timestep
+1e6
+0
+1
+2
+3
+4
+5
+6
+7
+Violation
+Minitaur-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Violation
+Spacereach-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Cost Return
+Spacerandom-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Violation
+Spacerandom-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+50
+100
+150
+200
+250
+300
+350
+400
+Cost Return
+Spacedualarm-cost
+ALAC(ours)
+LAC
+LAC*
+LBPO
+POLYC
+SAC-cost
+SPPO
+TNLF
+0
+1
+2
+3
+4
+5
+Timestep
+1e5
+0
+1
+2
+3
+4
+5
+6
+7
+Violation
+Spacedualarm-cost
+ALAC(ours)
+LAC
+LAC*
+POLYC
+SPPO
+TNLF
+Figure 11. Performance comparison on ten tasks. The ALAC method finds a good trade-off between minimizing the
+accumulated cost and constraint violations in contrast to their rivals.
+

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+Noname manuscript No.
+(will be inserted by the editor)
+Przemys�law Ko´scik
+On the Exponential Decay of Strongly Interacting
+Cold Atoms from a Double-Well Potential
+the date of receipt and acceptance should be inserted later
+Abstract In this article, we study an exponential decay for the gas of bosons with strong repulsive
+delta interactions from a double-well potential. We consider an exactly solvable model comprising an
+infinite wall and two Dirac delta barriers. We explore its features both within the exact method and
+with the resonance expansion approach. The study reveals the effect of the splitting barrier on the
+decay rate in dependence on the number of particles. Among other things, we find that the effect of the
+splitting barrier on the decay rate is most pronounced in systems with odd particle numbers. During
+exponential decay, the spatial correlations in an internal region are well captured by the ”radiating
+state”.
+1 Introduction
+Over the past few years there has been a growing interest in understanding the decay properties
+of unstable quantum states [1,2,3,4,5,6,7,8]. In particular, recent progress in fabricating systems of
+interacting particles has inspired the theoretical community to study the decay properties of unstable
+many-particle states [9,10,11,12,13,14]. A simple model to study the decay process of many-particle
+states is the system of bosons with infinitely strong delta-contact interactions, i.e., the so-called Tonks-
+Girardeau (TG) gas [15]. Considerable effort has been made already to understand the decay properties
+of such systems. Among other works, the relevant for exponential and long-time decays were presented
+in [12] and [13], respectively. A recent paper [14] went even further and explained the decay mechanism
+of TG gases at intermediate stages of the time evolution (between exponential and long-time regimes).
+In the present paper, we provide a deeper insight into the exponential decay of TG gases from the
+double-well trap. As a model, we consider the potential in the form [14]
+V (x) =
+�
+∞,
+if x ≤ −L
+αδ(x) + ηδ(x − L)
+otherwise.
+(1)
+Note that the model is a modification of the celebrated Winter model [3], to which a Delta barrier at
+x = 0 was added, see Fig.1. The remainder of this article is structured as follows. Section 2 discusses
+the theoretical tools for studying the time evolution of the decaying TG gas and focuses on the results.
+Section 3 presents some concluding remarks.
+2 Results
+The scenario we consider is typical of controllable studies of the tunnelling phenomena in modern
+experiments. In the case studied, the system is initially prepared (t < 0) in the ground state of the
+University of Applied Sciences, Department of Computer Sciences, ul. Mickiewicza 8, PL-33100 Tarn´ow, Poland
+arXiv:2301.03082v1  [cond-mat.quant-gas]  8 Jan 2023
+
+2
+Fig. 1 Ilustrative diagram of double-well structure in Eq. (1)
+TG gas in a hard-wall split trap (η = ∞). At t = 0, the strength of the right barrier is changed to a
+finite value of η. As a result, the initial state is no longer stationary and begins to evaluate in time.
+According to Bose-Fermi mapping [15], the time-dependent TG wave function is given by
+Ψ(x1, x2, ..., xN, t) =
+= Πk<lsgn(xk − xl)
+1
+√
+N!
+detN
+i,j[φi(xj, t)],
+(2)
+where the one-particle state φk(x, t) is governed by the Schr¨odinger equation,
+Iℏ∂φk(x, t)
+∂t
+= [− ℏ2
+2m
+∂2
+∂x2 + V (x)]φk(x, t),
+(3)
+with the initial condition as the bound-state eigenfunction of the hard-wall split trap, ϕk(x), that
+is, φk(x, t)|t=0 = ϕk(x). From here we set L = ℏ = m = 1 so that the spatial coordinates, time
+coordinates, and energies are measured in units of L and mL2/ℏ, in and ℏ2/(mL2), respectively. The
+system under consideration has a nice feature where both eigenfunctions of the hard-wall split trap,
+ϕk(x) and continuum wave functions (η < ∞) ψp(x) (normalised to a delta Dirac distribution) can be
+obtained in closed analytical forms. For further detail, we refer readers to the papers [14,16], in which
+the relevant formulas are reported. Thanks to those, the solutions to Eq. (3) can be condensed in a
+Fourier series as follows:
+φk(x, t) =
+� ∞
+0
+ck(p)ψp(x)e− Itp2
+2 dp,
+(4)
+where ck(p),
+ck(p) =
+� 1
+−1
+ϕk(x)ψp(x)dx,
+(5)
+is given in closed analytical form [14]. Nonetheless, numerical computations are required to evaluate
+the integrals in Eq. (4). We conduct our analysis in terms of the non-escape probability,
+P (N)(t) =
+�
+∆N |Ψ(x1, x2, ..., xN, t)|2dx1...dxN,
+(6)
+
+4
+3.5
+V(x)
+3
+2.5
+8
+n (x-L)
+2
+1.5
+α (x)
+1
+0.5
+0
+x=-L
+x=0
+x=L
+x3
+∆N = [−1, 1]N (internal region), which informs us of the probability that N bosons remain in the
+internal region at time t. For the TG wavefunction in Eq. (2), the N-particle non-escape probabil-
+ity can be reduced to the matrix form [12,13] P (N)(t) = detN
+k,l[Pkl(t)] with the entries Pkl(t) =
+� 1
+−1 ψ∗
+k(x, t)ψl(x, t)dx. The diagonal elements Pkk(t) are nothing but the non-escape probabilities
+of the one-particle states. We denote Pk(t) = Pkk(t). Within the resonance expansion method [4],
+the one-particle state that experiences an exponential decay follows in the region ∆ of the approx-
+imation: ψk(x, t) ≈ Mk(x)e−Γkt/2−Iεkt with Γk = −Im{p2
+k}, εk = (Im{pk}2 − Re{pk}2)/2, and
+Mk(x) = 2πIrespk{ck(p)ψp(x)}, where pk are the roots of the denominator in the integrand in Eq.(4)
+on the fourth quadrant of the complex p-plane (the proper poles) and respk{f} stands for the residue
+of f at a pole pk. Then, the corresponding non-escape probability is Pk(t) ≈ mke−Γkt, where mk =
+� 1
+−1 |Mk(x)|2dx. If mk ≈ 1(Pk(0) ≈ 1), then the exponential decay starts at about t = 0 and the state
+|ψk(t)⟩ can well be approximated by the so-called ”radiating state”: |ψk(t)⟩ ≈ e−Γkt/2−Iεkt |ψk(0)⟩.
+When taking this approach, the time-dependent TG wavefunction in the region ∆N takes the form:
+Ψ(x1, x2, ..., xN, t) ≈ e−Γ (N)t/2−Iε(N)tΠk<lsgn(xk − xl)
+1
+√
+N!
+detN
+i,j[φi(xj, 0)],
+(7)
+with Γ (N) = �N
+k=1 Γk, ε(N) = �N
+k=1 εk. Consequently its validity is expected to hold when mk ≈ 1
+for k = 1, ..., N ( M (N) = ΠN
+k=1mk ≈ 1). The corresponding N-particle non-escape probability is:
+P (N)(t) ≈ e−Γ (N)t.
+(8)
+Now, we focus on examining when the above approximations are satisfied and what follows from their
+applicability. Our resuls are summarised in Fig. 2. Fig. 2 (a) shows the behaviour of M (N) as a function
+of N. We can observe a local minimum with a value slightly smaller than one. As a result, there is a
+value where N = Nc(greater than where the minimum occurs) such that M (Nc) ≈ 1. Starting from
+N = Nc the deviation of M (N)from 1 rapidly increases as N increases. This suggests that the point
+N = Nc can be viewed as a transition point to the regime in which the non-escape probability begins to
+diverge significantly from its approximation in Eq. (8). To clarify this, Fig. 2 (b) offers a comparison of
+the results obtained from Eq. (8) with the results of the exact numerical calculations, where to support
+the presentation only case η = 10, α = 0 is shown. As the results indicate, the period in which the
+decay is consistent with the ”radiating state” shrinks with increasing N. When N exceeds the critical
+value Nc = 10 (see Fig. 2 (a)), the decay of the N-particle state switches to the non-exponential regime
+at a very small t value. When the splitting barrier is present an effect of the parity of a number of
+particles appears. This is demonstrated in Fig. 2 (c) which displays the behaviour of a relative change
+defined as γ(N) = (Γ (N) − Γ (N)
+α=0) : Γ (N)
+α=0, where Γ (N)
+α=0 represents the decay rate for a system without
+the splitting barrier. We conclude that the change in the decay process caused by the addition of the
+splitting barrier is most pronounced for systems with odd numbers of particles and in the small N
+regime, i. e. where γ(N) exhibits its most rapid variation. When there is instead an even number of
+particles, the decay rate becomes almost insensitive to changes in α. It is worth mentioning that the
+coherence of the initial state depends on α in the opposite way [16]. That is to say, it strongly depends
+on α only when N is even. To determine whether Eq. (7) can capture the correlation in the internal
+region, we tested its ability to reproduce a function n(x, t) defined as P (N)(t) =
+� 1
+−1 n(x, t)dx,
+n(x, t) =
+�
+∆N−1 |Ψ(x, x2, ..., xN, t)|2dx2...dxN.
+(9)
+If n(x, t) is divided by the corresponding value of P (N)(t), then the resulting quantity can be interpreted
+as the probability density of finding a particle, provided that all the particles remain in the region ∆.
+The correctness of the ”radiating state” in reproducing the spatial correlation is confirmed in Fig. 2 (d),
+where the results for n(x, t) obtained using it and the exact wave function are compared. The results
+imply that during exponential decay, spatial correlations in the internal region are indeed consistent
+with those in the initial state. More details regarding the correlation in the initial state (i.e. in the TG
+ground state in the hard-wall split trap) can be found in [16].
+
+4
+Fig. 2 Results for different quantities as discussed in the text, when investigating transparent control pa-
+rameter values. (a) Results for M (N) as a function of N. (b) N- particle non-escape probability, where the
+continuous lines represent the results of Eq.(8) and the markers the results of exact calculations. (c) Behaviour
+of the relative change γ(N). (d) Results for n(x, t), obtained for N = 3 particles at t = 10 from the numerically
+exact wave function (markers) and its approximation in Eq. (7)(continuous lines)
+
+4
+(a)
+n=10
+n=15
+3.5
+--α=0
+-α=0
+*:α=5
+0:α=5
+3
+0
+(N)
+2.5
+M
+2
+1.5
+*
+5
+10
+15
+20
+25
+30
+N0
+(b)
+N=5
+-50
+N=8
+-100
+X
++
+N=15
+:11
+X
+-150
+X
+-200
+n=10
+α=0
+0
+2
+4
+6
+8
+10
+t(c)
+0.2
+n=10
+x:α=1
+0.15
+0:α=3
+*:α=5
+(N)
+0.1
+0.05
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+N0.04 N=3,t=10
+(d)
+0.035
+0.03
+0.025
+('x)u
+X
+n=10
+0.02
+x:α=0
+0:α=3
+0.015
+0.01
+x
+0
+0.005
+x
+x5
+3 Conclusions
+In this article, we have studied the exponential decay of N bosons with strong delta interactions from
+the double-well structure. Using the resonance expansion approach, we have analysed the effect of the
+splitting barrier on the decay rate of the N-particle system in dependence on N. We have found that
+the splitting barrier strongly affects the decay rates of systems with odd numbers of particles. In such
+systems, the coherence of the initial state behaves differently and is insensitive to changes in the height
+of the splitting barrier. Our results have shown that in the exponential regime the ”radiating state”
+effectively reproduces the spatial correlations in the internal region.
+References
+1. G. Gamow,Z. Phys. 51, 204 (1928)
+2. E. U. Condon, R. W. Gurney, Nature London 112, 439 (1928)
+3. R. G. Winter, Phys. Rev. 123, 1503 (1961)
+4. G. Garcia-Calder´on, J. L. Mateos, M. Moshinsky, Phys. Rev. Lett. 74, 337 (1995)
+5. A.Wyrzykowski, Acta Physica Polonica B, 51,11 (2020)
+6. F. Giacosa, P. Ko´scik, and T. Sowi´nski, Phys. Rev. A 102, 022204 (2020)
+7. D. R. Jim´enez, N. G. Kelkar,Phys. Rev. A 104.2, 022214 (2021)
+8. G. Garcia-Calder´on, R. Romo, Annals of Physics, 424, 168348 (2021)
+9. G. Garcia-Calder´on, L. G. Mendoza-Luna, Phys. Rev. A 84, 032106 (2011)
+10. J. Dobrzyniecki, T. Sowi´nski, Phys. Rev. A 98, 013634 (2018)
+11. I. S. Ishmukhamedov, Physica E: Low-dimensional Systems and Nanostructures 142, 11522 (2022)
+12. M. Pons, D. Sokolovski, A. del Campo, Phys. Rev. A 85, 022107 (2012)
+13. A. del Campo, Phys. Rev. A 84, 012113 (2011)
+14. P. Ko´scik, Phys. Rev. A 102, 033308 (2020)
+15. M. D. Girardeau, J. Math. Phys. 1, 516 (1960)
+16. X. Yin, Y. Hao, S. Chen, and Y. Zhang, Phys. Rev. A 78, 013604 (2008)
+

79E1T4oBgHgl3EQfTwOd/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf,len=215
+page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  (will be inserted by the editor)  Przemys�law Ko´scik  On the Exponential Decay of Strongly Interacting  Cold Atoms from a Double-Well Potential  the date of receipt and acceptance should be inserted later  Abstract In this article, we study an exponential decay for the gas of bosons with strong repulsive  delta interactions from a double-well potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We consider an exactly solvable model comprising an  infinite wall and two Dirac delta barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We explore its features both within the exact method and  with the resonance expansion approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The study reveals the effect of the splitting barrier on the  decay rate in dependence on the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Among other things, we find that the effect of the  splitting barrier on the decay rate is most pronounced in systems with odd particle numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' During  exponential decay, the spatial correlations in an internal region are well captured by the ”radiating  state”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  1 Introduction  Over the past few years there has been a growing interest in understanding the decay properties  of unstable quantum states [1,2,3,4,5,6,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' In particular, recent progress in fabricating systems of  interacting particles has inspired the theoretical community to study the decay properties of unstable  many-particle states [9,10,11,12,13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' A simple model to study the decay process of many-particle  states is the system of bosons with infinitely strong delta-contact interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', the so-called Tonks-  Girardeau (TG) gas [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Considerable effort has been made already to understand the decay properties  of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Among other works, the relevant for exponential and long-time decays were presented  in [12] and [13], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' A recent paper [14] went even further and explained the decay mechanism  of TG gases at intermediate stages of the time evolution (between exponential and long-time regimes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  In the present paper, we provide a deeper insight into the exponential decay of TG gases from the  double-well trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' As a model, we consider the potential in the form [14]  V (x) =  �  ∞,  if x ≤ −L  αδ(x) + ηδ(x − L)  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  (1)  Note that the model is a modification of the celebrated Winter model [3], to which a Delta barrier at  x = 0 was added, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The remainder of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Section 2 discusses  the theoretical tools for studying the time evolution of the decaying TG gas and focuses on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  Section 3 presents some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  2 Results  The scenario we consider is typical of controllable studies of the tunnelling phenomena in modern  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' In the case studied, the system is initially prepared (t < 0) in the ground state of the  University of Applied Sciences, Department of Computer Sciences, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Mickiewicza 8, PL-33100 Tarn´ow, Poland  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='03082v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='quant-gas]  8 Jan 2023  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 1 Ilustrative diagram of double-well structure in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (1)  TG gas in a hard-wall split trap (η = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' At t = 0, the strength of the right barrier is changed to a  finite value of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' As a result, the initial state is no longer stationary and begins to evaluate in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  According to Bose-Fermi mapping [15], the time-dependent TG wave function is given by  Ψ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', xN, t) =  = Πk<lsgn(xk − xl)  1  √  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  detN  i,j[φi(xj, t)],  (2)  where the one-particle state φk(x, t) is governed by the Schr¨odinger equation,  Iℏ∂φk(x, t)  ∂t  = [− ℏ2  2m  ∂2  ∂x2 + V (x)]φk(x, t),  (3)  with the initial condition as the bound-state eigenfunction of the hard-wall split trap, ϕk(x), that  is, φk(x, t)|t=0 = ϕk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' From here we set L = ℏ = m = 1 so that the spatial coordinates, time  coordinates, and energies are measured in units of L and mL2/ℏ, in and ℏ2/(mL2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The  system under consideration has a nice feature where both eigenfunctions of the hard-wall split trap,  ϕk(x) and continuum wave functions (η < ∞) ψp(x) (normalised to a delta Dirac distribution) can be  obtained in closed analytical forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' For further detail, we refer readers to the papers [14,16], in which  the relevant formulas are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Thanks to those, the solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (3) can be condensed in a  Fourier series as follows:  φk(x, t) =  � ∞  0  ck(p)ψp(x)e− Itp2  2 dp,  (4)  where ck(p),  ck(p) =  � 1  −1  ϕk(x)ψp(x)dx,  (5)  is given in closed analytical form [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Nonetheless, numerical computations are required to evaluate  the integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We conduct our analysis in terms of the non-escape probability,  P (N)(t) =  �  ∆N |Ψ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', xN, t)|2dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='dxN,  (6)  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  V(x)  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  8  n (x-L)  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  α (x)  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  0  x=-L  x=0  x=L  x3  ∆N = [−1, 1]N (internal region), which informs us of the probability that N bosons remain in the  internal region at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' For the TG wavefunction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (2), the N-particle non-escape probabil-  ity can be reduced to the matrix form [12,13] P (N)(t) = detN  k,l[Pkl(t)] with the entries Pkl(t) =  � 1  −1 ψ∗  k(x, t)ψl(x, t)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The diagonal elements Pkk(t) are nothing but the non-escape probabilities  of the one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We denote Pk(t) = Pkk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Within the resonance expansion method [4],  the one-particle state that experiences an exponential decay follows in the region ∆ of the approx-  imation: ψk(x, t) ≈ Mk(x)e−Γkt/2−Iεkt with Γk = −Im{p2  k}, εk = (Im{pk}2 − Re{pk}2)/2, and  Mk(x) = 2πIrespk{ck(p)ψp(x)}, where pk are the roots of the denominator in the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (4)  on the fourth quadrant of the complex p-plane (the proper poles) and respk{f} stands for the residue  of f at a pole pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Then, the corresponding non-escape probability is Pk(t) ≈ mke−Γkt, where mk =  � 1  −1 |Mk(x)|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' If mk ≈ 1(Pk(0) ≈ 1), then the exponential decay starts at about t = 0 and the state  |ψk(t)⟩ can well be approximated by the so-called ”radiating state”: |ψk(t)⟩ ≈ e−Γkt/2−Iεkt |ψk(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  When taking this approach, the time-dependent TG wavefunction in the region ∆N takes the form:  Ψ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', xN, t) ≈ e−Γ (N)t/2−Iε(N)tΠk<lsgn(xk − xl)  1  √  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  detN  i,j[φi(xj, 0)],  (7)  with Γ (N) = �N  k=1 Γk, ε(N) = �N  k=1 εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Consequently its validity is expected to hold when mk ≈ 1  for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', N ( M (N) = ΠN  k=1mk ≈ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The corresponding N-particle non-escape probability is:  P (N)(t) ≈ e−Γ (N)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  (8)  Now, we focus on examining when the above approximations are satisfied and what follows from their  applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Our resuls are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 (a) shows the behaviour of M (N) as a function  of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We can observe a local minimum with a value slightly smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' As a result, there is a  value where N = Nc(greater than where the minimum occurs) such that M (Nc) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Starting from  N = Nc the deviation of M (N)from 1 rapidly increases as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' This suggests that the point  N = Nc can be viewed as a transition point to the regime in which the non-escape probability begins to  diverge significantly from its approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' To clarify this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 (b) offers a comparison of  the results obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (8) with the results of the exact numerical calculations, where to support  the presentation only case η = 10, α = 0 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' As the results indicate, the period in which the  decay is consistent with the ”radiating state” shrinks with increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' When N exceeds the critical  value Nc = 10 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 (a)), the decay of the N-particle state switches to the non-exponential regime  at a very small t value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' When the splitting barrier is present an effect of the parity of a number of  particles appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' This is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 (c) which displays the behaviour of a relative change  defined as γ(N) = (Γ (N) − Γ (N)  α=0) : Γ (N)  α=0, where Γ (N)  α=0 represents the decay rate for a system without  the splitting barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We conclude that the change in the decay process caused by the addition of the  splitting barrier is most pronounced for systems with odd numbers of particles and in the small N  regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' where γ(N) exhibits its most rapid variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' When there is instead an even number of  particles, the decay rate becomes almost insensitive to changes in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' It is worth mentioning that the  coherence of the initial state depends on α in the opposite way [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' That is to say, it strongly depends  on α only when N is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' To determine whether Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (7) can capture the correlation in the internal  region, we tested its ability to reproduce a function n(x, t) defined as P (N)(t) =  � 1  −1 n(x, t)dx,  n(x, t) =  �  ∆N−1 |Ψ(x, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=', xN, t)|2dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='dxN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  (9)  If n(x, t) is divided by the corresponding value of P (N)(t), then the resulting quantity can be interpreted  as the probability density of finding a particle, provided that all the particles remain in the region ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  The correctness of the ”radiating state” in reproducing the spatial correlation is confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 (d),  where the results for n(x, t) obtained using it and the exact wave function are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' The results  imply that during exponential decay, spatial correlations in the internal region are indeed consistent  with those in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' More details regarding the correlation in the initial state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' in the TG  ground state in the hard-wall split trap) can be found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' 2 Results for different quantities as discussed in the text, when investigating transparent control pa-  rameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (a) Results for M (N) as a function of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (b) N- particle non-escape probability, where the  continuous lines represent the results of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (8) and the markers the results of exact calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (c) Behaviour  of the relative change γ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (d) Results for n(x, t), obtained for N = 3 particles at t = 10 from the numerically  exact wave function (markers) and its approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' (7)(continuous lines)  4  (a)  n=10  n=15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  --α=0  α=0  :α=5  0:α=5  3  0  (N)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5  M  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='5 5  10  15  20  25  30  N0  (b)  N=5  50  N=8  100  X  +  N=15  :11  X  150  X  200  n=10  α=0  0  2  4  6  8  10  t(c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='2  n=10  x:α=1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='15  0:α=3  :α=5  (N)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='05  0  2  4  6  8  10  12  14  16  18  20  N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='04 N=3,t=10  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content="025  ('x)u  X  n=10  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='02  x:α=0  0:α=3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='01  x  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content='005  x  x5  3 Conclusions  In this article, we have studied the exponential decay of N bosons with strong delta interactions from  the double-well structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' Using the resonance expansion approach, we have analysed the effect of the  splitting barrier on the decay rate of the N-particle system in dependence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' We have found that  the splitting barrier strongly affects the decay rates of systems with odd numbers of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
+page_content=' In such  systems, the coherence of the initial state behaves differently and is insensitive to changes in the height  of the splitting barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E1T4oBgHgl3EQfTwOd/content/2301.03082v1.pdf'}
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9dE1T4oBgHgl3EQfoAQF/content/tmp_files/2301.03314v1.pdf.txt

@@ -0,0 +1,818 @@
+.
+SIGN INVOLUTIONS ON PARA-ABELIAN VARIETIES
+JAKOB BERGQVIST, THUONG DANG, AND STEFAN SCHR¨OER
+9 January 2023
+Abstract. We study the so-called sign involutions on twisted forms of abelian
+varieties, and show that such a sign involution exists if and only if the class in the
+Weil–Chˆatelet group is annihilated by two. If these equivalent conditions hold,
+we prove that the Picard scheme of the quotient is ´etale and contains no points
+of finite order. In dimension one, such quotients are Brauer–Severi curves, and
+we analyze the ensuing embeddings of the genus-one curve into twisted forms of
+Hirzebruch surfaces and weighted projective spaces.
+Contents
+Introduction
+1
+1.
+The scheme of sign involutions
+3
+2.
+The Picard scheme of the quotient
+6
+3.
+Morphisms to Brauer–Severi curves
+9
+References
+13
+Introduction
+Recall that an abelian variety A over a ground field k is a group scheme that
+is proper, smooth, and with h0(OA) = 1. It then follows that the group law is
+commutative, such that A comes with a canonical automorphism x �→ −x. This
+sign involution plays a prominent role in the theory of abelian varieties, because it
+gives rise to the notion of symmetric invertible sheaves. Furthermore, one can form
+the quotient A/G for the corresponding group G = {±1} of order two. In dimension
+g = 1 this gives the projective line, whereas for g = 2 we get Kummer surfaces. In
+characteristic p ̸= 2 this is a K3 surface with rational double points. The case p = 2
+requires extra attention, because than A/G may also be a rational surface with an
+elliptic singularity ([35] and [16]).
+In this paper we investigate the existence of sign involutions σ on twisted forms
+X of abelian varieties A, over general ground fields k of arbitrary characteristic
+p ≥ 0. These σ are involutions on X that become a sign involutions with respect to
+a suitable group law that arises on some base-change. The following point of view,
+developed by Laurent and the third author [18], is most suitable: A para-abelian
+variety is a proper scheme X such that X ⊗ k′ admits the structure of an abelian
+variety, for some field extension k ⊂ k′. It then turns out that the the subgroup
+scheme A ⊂ AutX/k that acts trivially on the numerically trivial part Picτ
+X/k is an
+abelian variety, and that the canonical A-action on X is free and transitive. In
+2010 Mathematics Subject Classification. 14L30, 14K15, 14K30, 14J26.
+1
+arXiv:2301.03314v1  [math.AG]  9 Jan 2023
+
+SIGN INVOLUTIONS
+2
+turn, one may view the scheme X as a torsor with respect to the abelian variety A,
+and obtains a class [X] in the Weil–Chˆatelet group H1(k, A). Our first main result
+relates these cohomology classes with the kernel A[2] for the multiplication-by-two
+map and the existence of sign involutions on X:
+Theorem. (See Thm. 1.2) Let X be a para-abelian variety. Then the following are
+equivalent:
+(i) There is a sign involution σ : X → X.
+(ii) We have 2 · [X] = 0 in the Weil–Chˆatelet group H1(k, A).
+(iii) There is an torsor P with respect to H = A[2] such that X ≃ P ∧H A.
+Here P ∧H A denotes the quotient of P × A by the diagonal H-action, usually
+called contracted product or associated fiber bundle. The main idea idea for the above
+result is to introduce the scheme of sign involutions Invsgn
+X/k ⊂ AutX/k, analyze the
+effect of the conjugacy action on this subscheme, and derive consequences using the
+general machinery of twisted forms and non-abelian cohomology.
+Now suppose X is a para-abelian variety admitting a sign involution σ : X → X.
+We then can form the quotient B = X/G with respect to the cyclic group G = {e, σ}
+of order two. In particular in characteristic two, not much seems to be known on
+this proper normal scheme. Our second main result is concerned with Picτ
+B/k, the
+numerically trivial part of the Picard scheme:
+Theorem. (See Thm. 2.1) In the above situation, the group scheme Picτ
+B/k is trivial.
+This relies on Grothendieck’s two spectral sequences abutting to equivariant coho-
+mology groups [11]. The result is not difficult in the tame case p ̸= 2, but requires a
+careful analysis in the wild case p = 2. In dimension g = 1 the para-abelian varieties
+X are usually called genus-one curves; we like to call them para-elliptic curves. The
+above shows that the quotient by any sign involution is a Brauer–Severi curve, that
+is, a twisted form of P1.
+Our third main result deals with the converse situation: Suppose there is a degree-
+two morphism f : X → B from a para-elliptic curve X to some Brauer–Severi
+curve B. Then the projectivization S = P(E ) of the rank-two sheaf E = f∗(OX)
+is a twisted form of a Hirzebruch surface with invariant e = 2, and comes with
+a contraction to a normal surface S′, having a unique singularity, which is often
+factorial. The geometry of the situation is as follows:
+Theorem. (See Section 3) Assumptions as above. Then f : X → B is the quotient
+by some sign involution σ on the para-elliptic curve X, and the latter embeds into
+both S and S′ as an anti-canonical curve. Moreover, S′ is the anti-canonical model
+of S, and also a twisted form of the weighted projective space P(1, 1, 2).
+We also show that if there are two different sign involutions σ1 ̸= σ2, the ensuing
+diagonal map gives an embedding X ⊂ B1 × B2 into a product of Brauer–Severi
+curves.
+Such products where studied by Koll´ar [17] and Hogadi [15].
+Again X
+becomes an anti-canonical curve, and it turns out that B1 × B2 embeds into P3 if
+and only if the factors are isomorphic.
+The paper is structured as follows: In Section 1 we recall the theory of para-abelian
+varieties X, introduces the scheme of sign involutions Invsgn
+X/k ⊂ AutX/k, analyze
+
+SIGN INVOLUTIONS
+3
+the conjugacy action, and establish the link between sign involutions, cohomology
+classes, and structure reductions. Section 2 is devoted to the Picard scheme of the
+quotient B = X/G of a para-abelian variety X of arbitrary dimension g ≥ 0 by a
+sign involution. In Section 3 we consider the case g = 1, and unravel the geometry
+attached to degree-two maps X → B from a para-elliptic curve X to a Brauer–Severi
+curve B.
+Acknowledgement. The research was conducted in the framework of the research
+training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and
+Topology. The first two authors where financially supported by the Deutsche For-
+schungsgemeinschaft with a PhD grant in GRK 2240/1, the first author also with a
+PhD grant in GRK 2240/2.
+1. The scheme of sign involutions
+Let k be a ground field of characteristic p ≥ 0, and X be a proper scheme. Then
+the group scheme AutX/k is locally of finite type, and the connected component
+Aut0
+X/k of the neutral element e = idX is of finite type ([19], Theorem 3.7). By
+the Yoneda Lemma, the map σ �→ σ2 defines a morphism of the scheme AutX/k to
+itself, which usually disrespects the group law. The scheme of involutions InvX/k is
+defined via a cartesian diagram
+InvX/k
+−−−→ AutX/k
+���
+���σ�→σ2
+Spec(k) −−−→
+e
+AutX/k .
+It contains the neutral element and is stable under the inverse map σ �→ σ−1, but
+otherwise carries no further structure in general.
+Now suppose that X can be endowed with the structure of an abelian variety.
+Recall that for each rational point x0 ∈ X, there is a unique group law that turns X
+into an abelian variety, with origin 0 = x0. Fix such a datum, and write A for the
+abelian variety obtained by endowing X with the ensuing group law. Note that A
+can also be regarded as the pair (X, x0). The automorphism group scheme becomes
+a semidirect product
+AutX/k = A ⋊ AutA/k,
+where the normal subgroup on the left acts on X by translations x �→ a + x. The
+cokernel AutA/k on the right is an ´etale group scheme with countably many points,
+acting on A in the canonical way. Its rational points are the automorphisms σ :
+X → X fixing the origin x0. It contains a canonical element, namely the standard
+sign involution x �→ −x. This defines a morphism (−1) : Spec(k) → AutA/k. Its
+fiber with respect to the canonical projection A ⋊ AutA/k → AutA/k is denoted by
+A ⊗ κ(−1).
+Lemma 1.1. The closed subscheme A ⊗ κ(−1) ⊂ AutX/k is invariant under the
+conjugacy action of AutX/k, lies inside InvX/k, and does not depend on the choice
+of the origin x0 ∈ X.
+
+SIGN INVOLUTIONS
+4
+Proof. Let x, a, b ∈ A(R) and ϕ ∈ AutA/k(R) be R-valued points, for some k-algebra
+R. Then x �→ a − x is some R-valued point of A ⊗ κ(−1). Conjugation by (b, id) is
+(1)
+x �−→ −b + x �−→ a − (−b + x) �−→ (a + 2b) − x,
+whereas conjugation by (0, ϕ) takes the form
+x �−→ ϕ−1(x) �−→ a − ϕ−1(x) �−→ ϕ(a) − x.
+Both are R-valued points of A⊗κ(−1). Furthermore, the composition x �→ a−x �→
+a − (a − x) is the identity. With the Yoneda Lemma, we see that A ⊗ κ(−1) is
+invariant under conjugacy, and must be contained in InvX/k.
+Now let a0 ∈ X be another origin. The ensuing new group law and negation are
+given by
+x ⊕ y = x + y − x′
+0
+and
+⊖ x = −x + 2a0,
+and thus a ⊖ x = (a + a0) − x. This shows that the closed subscheme A ⊗ κ(−1) ⊂
+AutX/k does not depend on the choice of origin.
+□
+Recall that a proper scheme X is called a para-abelian variety if there is a field
+extension k ⊂ k′ such that the base-change X′ = X ⊗ k′ admits the structure
+of an abelian variety. This notation was introduced and studied by Laurent and
+the third author [18]. According to loc. cit., Proposition 5.2, the closed subscheme
+A ⊂ AutX/k that acts trivial on Picτ
+X/k is an abelian variety, and the canonical
+A-action on X is free and transitive. The resulting class
+[X] ∈ H1(k, A)
+in the Weil–Chˆatelet group is called the cohomology class of the para-abelian variety.
+Note that since A is smooth, the ´etale and fppf topology yield the same cohomology
+groups ([13], Theorem 11.7). Consequently, the class [X] has some finite order; this
+number is usually called period per(X) ≥ 1.
+Conversely, if H is any commutative group scheme, with a torsor P and a homo-
+morphism H → A, we get a para-abelian variety X = P ∧H X0. The latter denotes
+the quotient of P × X0 by the diagonal action h · (p, x) = (h · p, h + x), and X0 is
+the underlying scheme of the abelian variety A. By construction, this X is a twisted
+form of X0.
+Recall that the index ind(X) ≥ 1 is the greatest common divisor of the degrees
+[κ(a) : k] for the closed points a ∈ X. This is indeed the index for the image of the
+degree map CH0(X) → Z on the Chow group of zero-cycles. Note that in dimension
+one this can also be seen as the degree map on the Picard group. According to [21],
+Proposition 5 the divisibility property per(X) | ind(X) holds, and both numbers
+have the same prime factors.
+As explained in [36], Section 3, the group scheme AutX/k is a twisted form of
+AutX0/k with respect to the conjugacy action.
+In turn, the conjugacy-invariant
+closed subscheme A ⊗ κ(−1) ⊂ AutX0/k becomes a closed subscheme
+Invsgn
+X/k ⊂ AutX/k,
+which we call the scheme of sign involutions.
+Any automorphism σ : X → X
+belonging to Invsgn
+X/k is called a sign involution.
+
+SIGN INVOLUTIONS
+5
+Theorem 1.2. For each para-abelian variety X of dimension g ≥ 0, the following
+three conditions are equivalent:
+(i) There is a sign involution σ : X → X.
+(ii) We have 2 · [X] = 0 in the Weil–Chˆatelet group H1(k, A).
+(iii) There is an torsor P with respect to H = A[2] such that X ≃ P ∧H A.
+It these conditions hold we have the divisibility property ind(X) | 4g.
+Proof. We start with some general observations: The first projection
+AutX0/k = A ⋊ AutA/k −→ A
+identifies the scheme of sign involutions Z0 = Invsgn
+X0/k = A ⊗ κ(−1) with a copy of
+X0 = A. According to (1), the kernel for the conjugacy homomorphism A → AutZ0/k
+is A[2], so this factors over multiplication-by-two map A
+2→ A. It is now convenient
+to write X = T ∧A X0 for some A-torsor T. Note that since the X0 is the trivial
+A-torsor, one actually has T = X. What is important now is that the scheme of sign
+involutions Z = Invsgn
+X/k coincides with Z = T ∧A Z0, and the latter is the quotient
+of T × Z0 by the A-action a · (t, z0) = (a + t, 2a + z0).
+This quotient can be computed as successive quotients, first for the action of
+H = A[2] and then for the induced action of A/A[2]. The group H acts trivially on
+the second factor, hence H\(T × X0) = (H\T) × X0. In light of the short exact
+sequence
+(2)
+0 −→ H −→ A
+2
+−→ A −→ 0,
+we may regard ¯T = H\T as the A-torsor induced from T with respect to A
+2→ A.
+In other words Z = ¯T ∧ ¯
+A Z0, where we write ¯A = A/H = A to indicate the nature
+of the action. By construction, the ¯A-action on Z0 is free and transitive, so the
+projection ¯T ⊗ κ(−1) → Z is an isomorphism. We conclude that there is a rational
+point σ ∈ Z if and only if the torsor ¯T is trivial.
+From the short exact sequence (2) we get a long exact sequence
+H0(k, A)
+2
+−→ H0(k, A) −→ H1(k, H) −→ H1(k, A)
+2
+−→ H1(k, A).
+It follows that the element [X] = [T] in H1(k, A) is annihilated by two if and only
+if there is an H-torsor P such that such that X ≃ P ∧H X0, giving the equivalence
+of (ii) and (iii). Similarly, we see that [X] = [T] is annihilated by two if and only if
+¯T is trivial. Together with the previous paragraph this gives the equivalence of (i)
+and (ii).
+It remains to verify the divisibility property of the index. This is just a special
+case of general fact: Suppose X has period n ≥ 1. From the long exact sequence
+for the multiplication-by-n map we see that the quotient of X by A[n] contains a
+rational point, so its fiber Z ⊂ X is a torsor with respect to A[n]. According to
+[24], page 147 the kernel A[n] is finite of length l = n2g. Clearly, the torsor Z has
+the same length, hence X contains a zero-cycle of degree n2g. Now if (ii) holds, we
+have n | 2, and thus ind(X) | 4g.
+□
+Recall that for each m ≥ 1 there is an identification H1(k, µm) = k×/k×m. Sup-
+pose now that k contains a primitive m-th root of unity, such that µn ≃ (Z/mZ)k.
+Let us recall the following result of Lang and Tate ([21], Theorem 8): Assume that
+
+SIGN INVOLUTIONS
+6
+the ground field k, the abelian variety A, and the integer m ≥ 0 satisfies the fol-
+lowing conditions: The Z/mZ-module k×/k×m contains a free module of infinite
+rank, the quotient A(k)/mA(k) is finite, and A(k) contains an element of order m.
+Then the Weil–Chˆatelet group H1(k, A) contains infinitely many elements X whose
+period and index equals m. Note that for global fields k, the first two conditions
+are automatic, and the third can be obtained after a finite extension, provided the
+abelian variety has dimension g ≥ 1 and the characteristic exponent p ≥ 1 of k is
+prime to m.
+2. The Picard scheme of the quotient
+Let X be a para-abelian variety having a sign involution σ : X → X. Write
+G ⊂ Aut(X) the corresponding subgroup of order two. The quotient B = X/G is
+a projective scheme that is geometrically integral and geometrically normal, with
+h0(OB) = 1. Following [9], Section 2, we write Sing(B/k) for the locus of non-
+smoothness. In contrast to the locus of non-regularity Sing(B), it comes with a
+scheme structure, defined via Fitting ideals for K¨ahler differentials.
+Let Picτ
+B/k be the open-and-closed subgroup scheme inside the Picard scheme
+comprising numerically trivial invertible sheaves. Its Lie algebra is H1(B, OB), and
+the group scheme of connected components is the torsion part of the N´eron–Severi
+group scheme. It therefore encodes important information on B.
+Theorem 2.1. The group scheme Picτ
+B/k is trivial. Moreover, Sing(B/k) is finite,
+and is contained in the image of the fixed scheme Xσ.
+Proof. It suffices to treat the case that k is algebraically closed, and we choose the
+origin so that X = A is an abelian variety with σ(x) = −x. Write q : A → B for
+the quotient map, let U ⊂ A be the complement of the fixed scheme, and V = q(U)
+be its image. The induced map q : U → V is a G-torsor, in particular smooth.
+According to [12], Theorem 17.11.1 the smoothness of U ensures the smoothness of
+V . Thus Sing(B/k) is contained in the image of Aσ = A[2], and is therefore finite.
+The structure sheaf OA has a G-linearization, and thus comes with equivariant
+cohomology groups Hi(A, G, OA), and likewise we get Hi(A, G, O×
+A). According to
+[11], Section 5.2 there are two spectral sequences
+(3)
+Ers
+2 = Hr(G, Hs(A, O×
+A))
+and
+Ers
+2 = Hr(B, Hs(G, O×
+A),
+both with equivariant cohomology Hr+s(A, G, O×
+A) as abutment.
+This gives two
+exact sequences forming a diagram
+(4)
+0
+Pic(B)
+Pic(A)G
+H2(G, k×)
+H1(A, G, O×
+A)
+0
+H1(G, k×)
+H0(B, F)
+H2(B, O×
+B),
+where the abelian sheaf F = H1(G, O×
+A) is supported by the singular locus of B.
+Recall that the cohomology groups for the cyclic group G = {e, σ} are given by
+H2j+1(G, M) = Ker(σ + id)
+Im(σ − id)
+and
+H2j+2(G, M) = Ker(σ − id)
+Im(σ + id) ,
+
+SIGN INVOLUTIONS
+7
+for any G-module M. It follows that H2(G, k×) vanishes, because G acts trivially
+on k×, and k× = k×2, whereas H1(G, k×) = µ2(k) = {±1}.
+According to (4)
+the kernel for Picτ(B) → Picτ(A) is the intersection of Pic(B) ∩ H1(G, k×) inside
+the equivariant cohomology group, whereas the image is contained in Pic(A)[2] =
+Pic(A)G. This already shows that the group scheme Picτ
+B/k must be finite. It also
+solves the case of dimension g = 1: Now B is a normal curve with finite Picard
+scheme. The latter is smooth, according to [23], Section 27 because H2(B, OB) = 0.
+Consequently B = P1, and thus Picτ
+B/k = 0.
+From now on, we assume that we are in dimension g ≥ 2. At each a ∈ A[2], the
+induced G-action on the local ring OA,a is ramified only at the origin, and it follows
+from [22], Proposition 3.2 that the local ring at the image b ∈ B is singular, and that
+the finite degree-two extension OB,b ⊂ OA,a is not flat. Consequently, the quotient
+map q : A → B induces a bijection between A[2] and Sing(B). Furthermore, the
+short exact sequence 0 → OB → q∗(OA) → F → 0 defines a coherent sheaf F that
+is invertible on the open set V = Reg(B), but not at the points b ∈ Sing(B).
+We claim that the canonical map Pic(B) → Pic(A)[2] is injective. Equivalently,
+the intersection Pic(B) ∩ H1(G, k×) inside H1(A, G, O×
+A) is trivial.
+The group
+H1(G, k×) = µ2(k) vanishes in characteristic two, so only the case p ̸= 2 requires
+attention. Then the trace map gives a splitting q∗(OA) = OB ⊕ F, thus F satis-
+fies Serre’s Condition (S2). The canonical identification FV ⊗ F ∨
+V = OV yields an
+element in Γ(V, q∗(OA) ⊗ F ∨) = Γ(U, q∗(F ∨)) without zeros, and it follows that
+the invertible sheaf F|V becomes trivial on U. Using the diagram (4) for the quo-
+tient V = U/G instead of B = A/G, we conclude that F|V generates the kernel
+of Pic(V ) → Pic(U). Seeking a contradiction, we now assume that there is a non-
+trivial invertible sheaf L on B that becomes trivial on Y , we therefore must have
+L |V = F|V . Using that both L and F satisfies Serre’s Condition (S2) together
+with [14], Theorem 1.12 we infer that L = F, contradicting that F is not invert-
+ible. This establishes our claim. We therefor may regard the canonical map as an
+inclusion Picτ(B) ⊂ Pic(A)[2].
+We next check that for p ̸= 2 the finite group scheme Picτ
+B/k is reduced. Equiv-
+alently, its Lie algebra H1(B, OB) vanishes. To see this, consider the spectral se-
+quences (3) with the additive sheaf OA instead the multiplicative sheaf O×
+A. For
+i ≥ 1, the vector spaces Hi(G, k) are annihilated by the group order |G| = 2. For
+p ̸= 2 they consequently vanish, and we obtain inclusions
+H1(B, OB) ⊂ H1(A, G, OA) ⊂ H1(A, OA)G.
+Moreover, the term on the right also vanishes because G acts via the sign involution
+on the cohomology group, according ([27], proof of Proposition 2.3). This establishes
+the claim.
+To proceed we use the fact that for any finite commutative group scheme N the
+isomorphism classes of N-torsors B′ → B corresponds to homomorphisms of group
+schemes N ∗ → PicB/k, where N ∗ = Hom(N, Gm) denotes the Cartier dual (see [26],
+Proposition 6.2.1, and also the discussion in [33], Section 4).
+The constant group scheme N = (Z/2Z)k has Cartier dual N ∗ = µ2. Suppose we
+have an inclusion µ2 ⊂ Picτ
+B/k such that the composite map µ2 → Picτ
+A/k remains
+a monomorphism. The corresponding N-torsor B′ → B thus induces a non-trivial
+
+SIGN INVOLUTIONS
+8
+N-torsor A′ → A. According to the Serre–Lang Theorem ([24], page 167), there is
+a unique structure of an abelian variety for A′ so that A′ → A is a homomorphisms.
+This gives an embedding N ⊂ A′ defined by a 2-division point a′ ∈ A′.
+The
+composite A′ → B is the quotient for the action of N ⋊ {±1}. Since this semidirect
+product is actually a direct product, the projection A′ → B′ must be the quotient
+by G = {±1}. Now choose a closed point x′ ∈ A′ with 2x′ = a′. It follows that
+the orbit G · x′ = {±x′}, viewed as a rational point on B′, is fixed by the the N-
+action, contradiction. This settles the case p ̸= 2: Then µ2 = (Z/2Z)k, and we see
+that Picτ(B) ⊂ Pic(A)[2] is trivial. We already saw in the previous paragraph that
+Picτ
+B/k is reduced, and infer that it must be trivial.
+It remains to treat the case p = 2, where the arguments in some sense run parallel
+to the preceding paragraph. At each a ∈ A[2], the local ring at the image b ∈ B
+is singular, with depth(OB,b) = 2, according to [22], Proposition 3.2. Note that is
+in stark contrast to the situation p ̸= 2, when such rings of invariants are Cohen–
+Macaulay. Again we consider the short exact sequence 0 → OB → q∗(OA) → F → 0
+of coherent sheaves on B. According to [8], Section 1 we have F|V = OV . The
+short exact sequence of local cohomology
+H0
+b (B, q∗(OA)) −→ H0
+b (B, F) −→ H1
+b (B, OB)
+reveals that F is torsion-free.
+So the adjunction map F → i∗(F|V ) = OB is
+injective, hence F is a sheaf of ideals.
+Using that F is not invertible we infer
+H0(B, F) = 0. The exact sequence
+H0(B, F) −→ H1(B, OB) −→ H1(A, OA)
+ensures that the map on the right is injective. On the other hand, its kernel is the
+Lie algebra for the kernel of Picτ
+B/k → PicA/k[2]. It follows that this map is actually
+a closed embedding Picτ
+B/k ⊂ PicA/k[2].
+Now we use that the Lie algebra of any group scheme in characteristic p > 0
+carries as additional structure the p-map x �→ x[p] and becomes a restricted Lie
+algebra (see [36], Section 1 for more details). Suppose H1(B, OB) ̸= 0. Then there
+is a p-closed vector x ̸= 0, in other words x[p] is a multiple of x. The case x[p] ̸= 0
+yields an inclusion of µp ⊂ B where the composite map µp → A is injective. We saw
+above that this is impossible. In turn we must have x[p] = 0. This gives an inclusion
+of N ∗ = αp into B where the composite map αp → A remains injective. The Cartier
+dual is N = αp. Thus we get a non-trivial αp-torsor B′ → B for αp whose base-
+change A′ → A remains non-trivial. A similar situation with N ∗ = (Z/2Z)k and
+N = µp arise if there is a point of order two on PicB/k. In both cases the discussion
+in [27], beginning of Section 2 shows that A′ has the structure of an abelian variety
+so that the projection A′ → A is a homomorphism, and we get an inclusion N ⊂ A′.
+The composition A′ → B is the quotient by the group scheme N ⋊ {±1}. Again
+this is actually a direct product. In the cartesian diagram
+A′ −−−→ B′
+���
+���
+A −−−→ B
+
+SIGN INVOLUTIONS
+9
+the vertical maps are quotients by the action of the infinitesimal group scheme N,
+and the horizontal maps are quotients by G = {±1}. Fix some a′ ∈ A′[2], with
+image b′ ∈ Sing(B′), and consider the ring of invariants OB′,b′ ⊂ OA′,a′. According
+to [22], Lemma 3.3 no element f ∈ mA′,a′ ∖ m2
+a′ is G-invariant.
+It follows that
+the infinitesimal neighborhood Spec(OA′,a′/m2
+a′) maps to Z′ = Spec(OB′,b′/mb′), and
+therefore the same holds for the orbit N · {a′}. In light of the above commutative
+diagram, the N-action on B′ is not free, contradiction.
+□
+Para-abelian varieties X of dimension g = 1 are usually called genus-one curves.
+Throughout, we shall prefer the term para-elliptic curves. These are twisted forms
+of elliptic curves. The moduli stack of such curves was studied by the second author
+[6]. Recall that the Brauer–Severi varieties Y are twisted forms of projective space
+Pn, for some n ≥ 0. For more details we refer to [3]. In case n = 1 we also say that
+Y is a Brauer–Severi curve.
+Corollary 2.2. Assumption as in the proposition, and suppose additionally g = 1.
+Then the corresponding quotient B = X/G is a Brauer–Severi curve.
+Proof. The scheme B is geometrically normal and of dimension one, hence smooth.
+According to the theorem, the Picard scheme is discrete. It follows that the tangent
+space H1(B, OB) vanishes. If there is a rational point a ∈ X, the resulting invertible
+sheaf L = OB(a) is very ample, with h0(L ) = 2, and we obtain an isomorphism
+B → P1.
+□
+In dimension g = 2 and characteristic p ̸= 2, the quotient B = A/{±1} is called
+a Kummer surface, and is a K3 surface with rational double points. For p = 2,
+the quotient B is either a K3 surface with rational double points, or a rational
+surface with an elliptic singularity. This was discovered by Shioda [35], see also [16],
+[31], [32] and [20]. The formation of such quotients is studied by the first author
+[5]. Little seems to be know on the quotient in higher dimensions, in particular in
+characteristic two, compare Schilson’s investigation [29], [30].
+3. Morphisms to Brauer–Severi curves
+Let X be a para-elliptic curve over a ground field k. If there is a sign involution
+σ : X → X, the quotient B by the corresponding group of order two is a Brauer–
+Severi curve, according to Corollary 2.2. In this section we conversely assume that
+our para-elliptic curve X admits a morphism f : X → B of degree two to some
+Brauer–Severi curve B, and derive several geometric consequences.
+First note that the corresponding function field extension k(B) ⊂ k(X) has degree
+two. It must be separable, because X and B are smooth of different genus. So this
+is a Galois extension, and the Galois group G is cyclic of order two. Let σ ∈ G be
+the generator.
+Proposition 3.1. The automorphism σ : X → X is a sign involution.
+Proof. It suffices to treat the case that k is algebraically closed. The action is not
+free, because χ(OX) = 0 ̸= 2 = |G|·χ(OB). Choose a fixed point x0 ∈ X, and regard
+E = (X, x0) as an elliptic curve. If Aut(E) is cyclic, there is a unique element of
+order two, and we infer that σ equals the sign involution. Suppose now that Aut(E)
+is non-cyclic. According to [7], Proposition 5.9 this group is either the semi-direct
+
+SIGN INVOLUTIONS
+10
+product Z/3Z ⋊ µ4(k) in characteristic p = 3, or Q ⋊ µ3(k) in characteristic p = 2,
+where Q = {±1, ±i, ±j ± k} denotes the quaternion group. In these groups, the
+respective elements (0, −1) and (−1, 1) are the only ones of order two, and we again
+conclude that σ coincides with the sign involution.
+□
+Proposition 3.2. The cokernel for the inclusion OB ⊂ f∗(OX) is isomorphic to
+ωB, and the resulting extension 0 → OB → f∗(OX) → ωB → 0 of coherent sheaves
+splits.
+Proof. The sheaf f∗(OX) has rank two and is torsion-free, hence is locally free. The
+inclusion of OB is locally a direct summand, so the cokernel L is invertible. We
+have 0 = χ(OX) = χ(OB) + χ(L ) = 2 + deg(L ) and conclude deg(L ) = −2. Since
+deg : Pic(B) → Z is injective, this gives L ≃ ωB. The extension yields a class in
+Ext1(ωB, OB) = H1(X, ω⊗−1
+B
+), which vanishes by Serre Duality. So the extension
+splits.
+□
+Choose a splitting and set E = f∗(OX) = OB ⊕ ωB. The smooth surface
+S = P(E ) = Proj(Sym• E )
+is a twisted form of the Hirzebruch surface S0 = P(E0), where E0 = OP1 ⊕ OP1(−2).
+Let us call S the twisted Hirzebruch surface attached to the Brauer–Severi curve B.
+Since f : X → B is affine, the invertible sheaf OX is relatively very ample, and we
+get a closed embedding X ⊂ S. By abuse of notation we also write f : S → B for
+the extension of our original morphism on X.
+Recall that each invertible quotient E → N defines a section s : B → S, whose
+image D has self-intersection D2 = deg(N ) − deg(N ′), where N ′ ⊂ E is the
+kernel. For more details we refer to [10], Section 6. In particular, pr1 : E → OB
+yields a curve D ⊂ S with D2 = 2, whereas pr2 : E → ωB gives some E ⊂ S
+with E2 = −2, and the two sections are disjoint. The Adjunction Formula gives
+(ωS ·D) = −4 and (ωS ·E) = 0. Hence ωS = f ∗(ω⊗2
+B )⊗OS(−2E), because both sides
+have the same intersection numbers with D and E. In particular c2
+1 = (ωS · ωS) =
+−8 · deg(ωB) + 4 · E2 = 8. Setting
+ω⊗1/2
+S
+= f ∗(ωB) ⊗ OS(−E),
+we get an invertible sheaf whose square is isomorphic to the dualizing sheaf. In other
+words, the surface S comes with a canonical theta characteristic, or spin structure,
+compare [4] and [25].
+Proposition 3.3. The dual sheaf L = ω⊗−1/2
+S
+is globally generated with h0(L ) = 4.
+The image of the resulting r : S → P3 is an integral normal surface S′ ⊂ P3 of degree
+two, and the induced morphism r : S → S′ is the contraction of E. Moreover, the
+image a = r(E) is a rational point, the local ring OS′,a is singular, and the restriction
+r|X is a closed embedding.
+Proof. Our sheaf has intersection numbers (L · L ) = 2 and (L · E) = 0. Serre
+Duality gives h2(L ) = h0(ω⊗3/2
+S
+) = 0, and Riemann–Roch yields
+h0(L ) ≥ χ(L ) = c2
+1/4 + c2
+1/2
+2
++ χ(OS) = (2 + 4)/2 + 1 = 4.
+
+SIGN INVOLUTIONS
+11
+The base locus Bs(L ) is contained in E, because ω⊗−1
+B
+is globally generated. The
+short exact sequence 0 → f ∗(ω⊗−1
+B
+) → L → L |E → 0 yields an exact sequence
+0 −→ H0(S, f ∗(ω⊗−1
+B
+)) −→ H0(S, L ) −→ H0(E, OE),
+consequently h0(L ) ≤ h0(ωB) + h0(OE) = 4. This ensures h0(L ) = 4, and that L
+is globally generated.
+In turn, our spin structure yields a morphism r : S → P3 with r∗(OP3(1)) =
+ω⊗1/2
+S
+. It therefore contracts E. Moreover, the image S′ ⊂ P3 is integral and two-
+dimensional, of some degree n ≥ 1. This image is not a plane, because the morphism
+is defined by the complete linear system H0(S, L ). From 2 = (L ·L ) = deg(S/S′)·n
+we infer that S → S′ is birational and n = 2.
+The Adjunction Formula gives
+ωS′ = OS′(2), consequently r∗(ωS′) = ωS. It follows that the birational morphism
+r : S → S′ is in Stein factorization. Since Pic(S) has rank two, the exceptional
+divisor is irreducible, whence must coincide with E.
+The image a = r(E) is a rational point, because h0(OE) = 1. The local ring OS′,a
+must be singular, because otherwise S = Bla(S′), such that E = r−1(a) must be a
+projective line with E2 = −1, contradiction.
+It remains to verify that the curves X, E ⊂ S are disjoint. Since deg(X/B) = 2
+we have ωS = OS(−X)⊗f ∗(N ) for some invertible sheaf N on B. The Adjunction
+Formula gives
+0 = (ωS · X) + X2 = −X2 + 2 deg(N ) + X2.
+Consequently N
+is trivial, and ωS = OS(−X).
+This gives X2 = c2
+1 = 8, and
+furthermore (X · E) = −(ωS · E) = 0. Thus the integral curves X and E must be
+disjoint, hence r|X is a closed embedding.
+□
+Note that the local ring OS′,a is factorial provided that B ̸≃ P1. The above also
+shows that the image S′ = r(S) can also be viewed as the anti-canonical model
+P(S, −KS) of the scheme S, which is defined as the homogeneous spectrum of the
+anti-canonical ring R(S, −KS) = �
+t≥0 H0(S, ω⊗t
+S ).
+Recall that the weighted projective space P(d0, . . . , dn) is the homogeneous spec-
+trum of k[U0, . . . , Un], where the generators have degrees di = deg(Ui). The case
+d0 = . . . = dn = 1 gives back the standard projective space Pn. Let us say that a
+closed subscheme of a Gorenstein surface is an anti-canonical curve if its sheaf of
+ideals is isomorphic to the dualizing sheaf.
+Proposition 3.4. The anti-canonical model S′ = P(S, −KS) is a twisted form of
+the weighted projective space P(1, 1, 2). Moreover, X ⊂ S and the resulting inclusion
+X ⊂ S′ are anti-canonical curves.
+Proof. It suffices to treat the case that k is algebraically closed. We claim that S′ is
+defined inside P3 = Proj k[T0, . . . , T3] by the equation T 2
+0 − T1T2 = 0, for a suitable
+choice of homogeneous coordinates. The main challenge is the case p = 2: According
+to [1], Satz 2 our quadric X ⊂ P3 must be defined by an equation of the form
+r
+�
+i=1
+(αiX2
+i + XiYi + γiY 2
+i ) +
+s
+�
+j=1
+δjZ2
+j = 0,
+with 1 ≤ 2r + s ≤ 4, and non-zero coefficients δj. Since k is algebraically closed, we
+can make a change of variables and achieve δj = 1, and furthermore αi = γi = 0.
+
+SIGN INVOLUTIONS
+12
+On now immediately sees that only for r = s = 1 the quadric S′ ⊂ P3 is normal
+and singular, and setting T0 = Z1 and T1 = X1 and T2 = Y1 gives the claim. For
+p ̸= 2 our quadric can be defined by an equation of the form �3
+j=0 δjZ2
+j = 0, and
+one argues similarly.
+Consider the graded ring A = k[U0, U1, U2] with weights (1, 1, 2). The Veronese
+subring A(2) is generated by the homogeneous elements U0U1, U 2
+0, U 2
+1, U2, which sat-
+isfy the relation (U0U1)2 = U 2
+0 · U 2
+1. This gives a surjection
+k[T0, T1, T2, T3]/(T 2
+0 − T1T2) −→ A(2),
+defined by the assignments T0 �→ U0U1 and T1 �→ U 2
+0 and T2 �→ U 2
+1 and T3 �→ U2.
+Both rings are integral of dimension three. Using Krull’s Principal Ideal Theorem,
+we infer that the above surjection is bijective. The homogeneous spectrum of A(2)
+coincides with P(1, 1, 2) = Proj(A), and by the above also with S′.
+We already saw in the previous proof that ωS = OS(−X), hence X ⊂ S is an
+anti-canonical curve. From the Theorem of Formal functions one infers f∗(ωS) is
+invertible, and this ensures that the direct image coincides with ωS′. Using X ∩E =
+∅ we infer ωS′ = OS′(−X).
+□
+Now suppose that we have two morphism B1
+f1
+← X
+f2
+→ B2 to Brauer–Severi curves,
+with deg(X/Bi) = 2. According to Proposition 3.1, they comes from sign involutions
+σ1 and σ2, respectively.
+Proposition 3.5. If σ1 ̸= σ2, the diagonal morphism i : X → B1 × B2 is a closed
+embedding, and its image is an anti-canonical curve.
+Proof. Let A ⊂ AutX/k be the subgroup scheme that fixes Picτ
+X/k. As discussed in
+Section 1, this is an elliptic curve, and the action on the para-elliptic curve X is free
+and transitive. Moreover, the dual abelian variety is identified with Pic0
+X/k. But
+note that the principal polarization stemming from the origin also gives A = Pic0
+X/k.
+We saw in the proof of Proposition 1.1 that the two rational points σ1, σ2 ∈ Invsgn
+X/k
+differ by the action of some non-zero a ∈ A(k). In other words, σ2(x) = a + σ1(x).
+It follows that there is no rational point x ∈ X with σ1(x) = σ2(x). In particular,
+the fixed schemes Xσ1 and Xσ2 are disjoint.
+To proceed, we assume that k is algebraically closed. Let x ∈ X be a closed
+point and write y = i(x) = (b1, b2). The inverse image i−1(y) is the intersection
+of the fibers f −1
+1 (b1) ∩ f −1
+2 (b2). This is just the spectrum of κ(x), by the previous
+paragraph. According to [12], Corollary 18.12.6 the finite morphism i : X → B1×B2
+is a closed embedding.
+By construction, we have deg(X/B1) = deg(X/B2) = 2. Set V = B1 × B2. Its
+Picard scheme PicV/k can seen as the Galois module Pic(V ⊗ksep) = Z×Z, compare
+the discussion in [34], Section 1. Obviously, the elements (2, 0) and (0, 2) are fixed
+by Gal(ksep/k), hence the whole Galois action is trivial, and thus PicV/k = (Z × Z)k
+is a constant group scheme. The dualizing sheaf ωV = pr∗
+1(ωB1) ⊗ pr∗
+2(ωB2) has class
+(2, 2), and we infer ωV = OS(−X).
+□
+Note that ωV is anti-ample, so the smooth surface V = B1 ×B2 coincides with its
+anti-canonical model P(V, −KV ). Products of Brauer–Severi curves were studied by
+Koll´ar [17] and Hogadi [15]. Let us close this paper with the following observation:
+
+SIGN INVOLUTIONS
+13
+Proposition 3.6. The surface V = B1 × B2 admits an embedding into P3 if and
+only if B1 ≃ B2.
+Proof. The Picard scheme is given by PicV/k = (Z × Z)k. The classes (−2, 0) and
+(0, −2) come from the preimages of the invertible sheaves on B1 and B2, and thus
+belong to the subgroup Pic(V ) ⊂ PicV/k(k).
+Suppose we have V ⊂ P3, and write d ≥ 1 for its degree. From ωV = OV (d − 4)
+we get 8 = (ωV · ωV ) = d(d − 4)2, and thus d = 2. In particular, V admits the spin
+structure ω⊗1/2
+V
+= OV (−1). The dual sheaf L = OV (1) has h0(L ) = 4, which easily
+follows from the short exact sequence 0 → OP3(−1) → OP3(1) → L → 0. Choose
+some non-zero global section s ̸= 0 from L , and let D ⊂ V the resulting effective
+Cartier divisor. Suppose D is reducible. Since deg(D) = 2 we see that there are
+two components. Since L has class (1, 1) in PicV/k(k), it follows that D = D1 +D2,
+where the summands are preimages of rational points on B1 and B2, respectively.
+Thus both Brauer–Severi curves are copies of P1. Suppose now that D is irreducible.
+Then deg(D/Bi) = 1, so the morphism D → Bi are birational. By Zariski’s Main
+Theorem, it must be an isomorphism, and therefore B1 ≃ B2.
+Conversely, suppose there is an isomorphism h : B1 → B2. Its graph defines
+an effective Cartier divisor D ⊂ B1 × B2 with class (1, 1) ∈ PicV/k(k). Set L =
+OV (D). Passing to the algebraic closure of k, we get L = pr∗
+1(OP1(1))⊗pr∗
+2(OP1(1)),
+and compute h0(L ) = 4. Moreover, L is very ample, and thus defines a closed
+embedding X ⊂ P3.
+□
+Given a sign involution σ : X → X and a non-zero rational point a ∈ A(k), we
+get another sign involution x �→ a + σ(x). We see that the situation B1
+f1
+← X
+f2
+→ B2
+with σ1 ̸= σ2 appears if and only if the set Invsgn
+X/k(k) is non-empty and the group
+A(k) is non-trivial.
+References
+[1] C. Arf: Untersuchungen ¨uber quadratische Formen in K¨orpern der Charakteristik 2. I. J.
+Reine Angew. Math. 183 (1941), 148–167.
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+Analysis, pp. 21–71. Univ. Tokyo Press, Tokyo, 1969.
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+groups in ring theory and algebraic geometry, pp. 194—210, Springer, Berlin-New York,
+1982.
+[4] M. Atiyah: Riemann surfaces and spin structures. Ann. Sci. ´Ecole Norm. Sup. 4 (1971),
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+
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+threefolds, and exceptional Enriques surfaces. ´Epijournal Geom. Alg´ebrique 4 (2020), Art.
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+morphismes de sch´emas. Publ. Math., Inst. Hautes ´Etud. Sci. 32 (1967).
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+cohomologie des sch´emas, pp. 88–189. North-Holland, Amsterdam, 1968.
+[14] R. Hartshorne: Generalised divisors on Gorenstein schemes. K-Theory 8 (1994), 287–339.
+[15] A. Hogadi: Products of Brauer–Severi surfaces. Proc. Amer. Math. Soc. 137 (2009), 45–50.
+[16] T. Katsura: On Kummer surfaces in characteristic 2. In: M. Nagata (ed.), Proceedings of
+the international symposium on algebraic geometry, pp. 525–542. Kinokuniya Book Store,
+Tokyo, 1978.
+[17] J. Koll´ar: Conics in the Grothendieck ring. Adv. Math. 198 (2005), 27–35.
+[18] B. Laurent, S. Schr¨oer: Para-abelian varieties and Albanese maps. Preprint, arXiv:2101.
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+(1958), 659–684.
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+Lectures on curves on an algebraic surface. Princeton University Press,
+Princeton, 1966.
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+(1971), 181–192.
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+(1970), 27–76.
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+(2022), 93–121.
+[28] T. Saito: The discriminant and the determinant of a hypersurface of even dimension. Math.
+Res. Lett. 19 (2012), 855–871.
+[29] B. Schilson: Singularit¨aten von Kummer-Variet¨aten in beliebiger Charakteristik. Disserta-
+tion, D¨usseldorf (2018), https://nbn-resolving.org/urn/resolver.pl?urn=urn:nbn:
+de:hbz:061-20181108-114448-1.
+[30] B. Schilson: Wild singularities of Kummer varieties. J. Singul. 20 (2020), 274–288.
+[31] S. Schr¨oer: Kummer surfaces for the selfproduct of the cuspidal rational curve. J. Algebraic
+Geom. 16 (2007), 305–346.
+[32] S. Schr¨oer: The Hilbert scheme of points for supersingular abelian surfaces. Arkiv Mat. 47
+(2009), 143–181.
+[33] S. Schr¨oer: Enriques surfaces with normal K3-like coverings. J. Math. Soc. Japan. 73 (2021),
+433–496.
+[34] S. Schr¨oer: There is no Enriques surface over the integers. Ann. of Math. 197 (2023), 1–63.
+[35] T. Shioda: Kummer surfaces in characteristic 2. Proc. Japan Acad. 50 (1974), 718–722.
+[36] N. Tziolas, S. Schr¨oer: The structure of Frobenius kernels for automorphism group schemes.
+arXiv:2105.07860, to appear in Algebra Number Theory.
+
+SIGN INVOLUTIONS
+15
+Mathematisches Institut, Heinrich-Heine-Universit¨at, 40204 D¨usseldorf, Ger-
+many
+Email address: Jakob.Bergqvist@hhu.de
+Mathematisches Institut, Heinrich-Heine-Universit¨at, 40204 D¨usseldorf, Ger-
+many
+Email address: dangt@uni-duesseldorf.de
+Mathematisches Institut, Heinrich-Heine-Universit¨at, 40204 D¨usseldorf, Ger-
+many
+Email address: schroeer@math.uni-duesseldorf.de
+

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+1
+1
+FRA: A novel Face Representation Augmentation algorithm for face recognition
+Soroush Hashemifar1, Abdolreza Marefat2, Javad Hassannataj Joloudari3,*, Hamid Hassanpour4
+1School of Computer Engineering, Iran University of Science and Technology, Tehran, Iran
+2Department of Artificial Intelligence, Technical and Engineering Faculty, South Tehran Branch, Islamic 
+Azad University, Tehran, Irans
+3Department of Computer Engineering, Faculty of Engineering, University of Birjand, Birjand 
+9717434765, Iran 
+4Faculty of Computer Engineering & Information Technology, Shahrood University of Technology, P.O. 
+Box 316, Shahrood, Iran
+Corresponding author*: javad.hassannataj@birjand.ac.ir
+Abstract
+A low amount of training data for many state-of-the-art deep learning-based Face Recognition (FR) 
+systems causes a marked deterioration in their performance. Although a considerable amount of research 
+has addressed this issue by inventing new data augmentation techniques, using either input space 
+transformations or Generative Adversarial Networks (GAN) for feature space augmentations, these 
+techniques have yet to satisfy expectations. In this paper, we propose a novel method, named the Face 
+Representation Augmentation (FRA) algorithm, for augmenting face datasets. To the best of our 
+knowledge, FRA is the first method that shifts its focus towards manipulating the face embeddings 
+generated by any face representation learning algorithm in order to generate new embeddings 
+representing the same identity and facial emotion but with an altered posture. Extensive experiments 
+conducted in this study convince the efficacy of our methodology and its power to provide noiseless, 
+completely new facial representations to improve the training procedure of any FR algorithm. Therefore, 
+FRA is able to help the recent state-of-the-art FR methods by providing more data for training FR 
+systems. The proposed method, using experiments conducted on the Karolinska Directed Emotional Faces 
+(KDEF) dataset, improves the identity classification accuracies by 9.52 %, 10.04 %, and 16.60 %, in 
+comparison with the base models of MagFace, ArcFace, and CosFace, respectively.
+Keywords:
+Face Recognition, Face Embeddings, Face Representation Learning, Autoencoder, Vision Transformers, 
+Latent Space Data Augmentation, Facial Pose Reconstruction
+1.Introduction
+Face images are one of the most popular biometric modalities which have been continuously utilized in 
+Face Recognition (FR) systems [1]. It is used in a wide range of contexts with the aim of identity 
+authentication and its applications vary from daily life and finance to military and public security [2]. In 
+fact, in comparison with other biometrics, such as the fingerprint, iris, or retina which are ubiquitously 
+used for authorizing individuals, FR can provide us with the most convenient way to capture visual 
+information without the need for any extra activity from the subject. In recent years, FR has been one of 
+
+2
+2
+the most proactively studied areas in Computer Vision [3]. Particularly, with the advent of deep learning 
+and architectures like Convolutional Neural Networks (CNNs) [4], a large number of efficient facial 
+recognition methods with outstanding performance have been invented to address this challenge [5-12]. 
+These successful algorithms depend heavily on the performance of neural networks which use a cascade 
+of layers comprised of neurons that are able to learn different levels of abstractions and representations 
+from the input data [2]. These representations are more powerful substitutions for hand-crafted features 
+from facial attributes such as Scale-Invariant Feature Transform (SIFT) and Speeded Up Robust Features 
+(SURF) [13, 14]. Their principal advantage is that they obviate the need for manually and exhaustively 
+searching for the best features representing one’s face. Moreover, the process of learning representations 
+via deep learning-based algorithms makes the generated features surprisingly discriminative in that the 
+inter-class diversity and intra-class compactness within the training data are all taken into account by the 
+network itself [15]. 
+However, there are still problematic scenarios in which FR systems fail to realize the expectations. For 
+instance, in real-life situations, the imagery of a person’s face has a high chance of being in a variety of 
+facial expressions, occlusions, poor illumination, low resolution, etc. [16-18], and all these factors cause 
+substantial degradation of the overall performance of the current FR algorithms. Thus, different 
+approaches have been adopted to rectify the negative impact of such barriers in FR systems [19-21]. 
+Some have opted for experimenting and devising new loss functions whose capability to better feedback 
+to their neural network in the backpropagation step, enables the extracted deep features to be more 
+discriminative and clearly separable [2, 6, 9, 22-26]. In addition to these works, different architectures 
+have been implemented to extract feature maps which are more useful in terms of facial representations. 
+Moreover, developing larger and more variant datasets has been one of the main stimuli which have been 
+pushing the boundaries in recent FR systems [27]. Nevertheless, although some of these benchmark 
+datasets can be found in large volumes, we often lack such a training set of images when it comes to real 
+use cases. A typical case would be a situation in which the goal is to train a deep learning-based method 
+on a private, in-house set of identities that have been chosen by a multimedia organization for video 
+indexing purposes. The data-gathering phase can be very time and labor-consuming and sometimes even 
+impossible, and it acts as an impediment in the way of achieving a tailored amount of training datasets. 
+These have motivated researchers to pave the way by introducing different data augmentation techniques. 
+Data augmentation refers to a set of techniques that are used to increase the number of training datasets 
+without the loss of previously annotated data. The benefit of such methods is that it equips the trained 
+model with more generalizability and acts as a regularizer in the case of overfitting which is one of the 
+most frequent complications when dealing with a small amount of training data [28, 29]. Overall, there 
+are two mainstream categories of methods for augmenting data. The first set of methods has the aim of 
+manipulating the data in the input space in that they simply take the input image and apply different 
+geometric transformations such as translations, cropping, vertical and horizontal flipping, rotation, etc 
+[30]. Even though these methods are proven to be extremely useful in some other challenges like image 
+classification, object detection, and image captioning in computer vision, in the case of FR they cannot be 
+as helpful as they expected. The main reason is that in order for any FR system to capture a reliable visual 
+representation of a face crop image, the content should be aligned in terms of facial landmarks. This 
+means that any geometric alteration on these which conspicuously happens when one uses these classical 
+methods, can perturb the overall performance of FR pipeline. These challenges have motivated the 
+researchers to shift their studies’ direction toward more modern and domain-specific solutions [31-33], 
+leading to the second set of methods, which are known to be Generative Adversarial Networks (GANs) 
+
+3
+3
+[34]. These methods are the well-known type of generative models which are used with the objective of 
+transforming the input data in feature space with the aim of generating new augmented image data. This 
+group of models is capable of adjusting the facial attributes existent in a face image such as hair style, 
+expression, posture, skin color, etc. to a target style. However, in most cases, these generative models 
+cannot create realistic outputs and these models deal with the high complexities of reconstructing the 
+feature space to input space, without having any considerable improvement on the downstream task, 
+which in our case, is classification on the identity of the samples.
+In order to address these difficulties, in this paper we propose the Face Representation Augmentation 
+(FRA) algorithm. This algorithm augments the posture of a given face image in the latent space. This 
+means that, given a set of embeddings representing a specific person, the proposed approach alters the 
+embedding to sustain the identity-related features with a transformed pose feature. The FRA algorithm 
+can help the existing facial recognition systems especially when the number of training samples is 
+imbalanced or less than expected. Our main contributions in this paper are itemized in the following:
+1. A novel algorithm for facial posture augmentation inside the latent space to reduce the complexity of 
+the image augmentation problem.
+2. Generating noiseless, non-duplicated embeddings which are proved to be linearly separable.
+3. Extensive experiments were conducted on the Karolinska Directed Emotional Faces (KDEF) [35] 
+dataset and improved the identity classification accuracies in comparison with the base models of 
+MagFace, ArcFace, and CosFace, respectively. 
+The rest of the paper is organized as follows. In Section 2, we briefly review the related works on face-
+specific data augmentation and representation learning. Then, in Section 4, we present the details of our 
+proposed methodology. In Section 4, we demonstrate the results of our experiments in comparison with 
+other related state-of-the-art approaches. Finally, the conclusion will be drawn in Section 5. 
+2.Related works
+In this section, we present an overview of face-specific data augmentation techniques. These are 
+categorized into two groups classical and generative-based methods in 2.1. Additionally, we review the 
+related literature of FR algorithms in 2.2.
+2.1.Face-Specific Data Augmentation
+To begin with, five data augmentation techniques for face photos were reported by Lv et al. [29]. These 
+techniques were landmark perturbation, hairdo synthesis, glasses synthesis, postures synthesis, and 
+lighting synthesis. Vincent et al. [36] tried to synthesize more data by applying different types of noise 
+such as Gaussian and Salt-and-pepper with the objective of training Stacked Denoising Autoencoders on 
+more complicated samples. Wang et al. [37] addressed the issue of data augmentation in picture 
+classification using conventional transformation techniques and GANs. They also suggested a technique 
+for learning network-based augmentations that better enhance the classifier in the setting of generic 
+photos rather than face images. 
+Moreover, although the hair is not an intrinsic part of the human face, it interferes with facial recognition 
+since it obscures the face and changes its appearance. Using DiscoGAN, which was developed to find 
+cross-domain relationships using unpaired data, Kim et al. altered hair color. In addition to the color, Kim 
+et al. in [38], suggested changing the bang by transferring an unsupervised visual characteristic using a 
+reconfigurable GAN. An online compositing technique was used in the face synthesis system proposed by 
+Kemelmacher-Shlizerman et al. [39]. The system might produce a series of fresh photographs with the 
+
+4
+4
+input person's identification and the questioned look using one or more photos of their face and a text 
+query like curly hair. Jiang et al., in [40], proposed Pose and expression resilient Spatial aware GAN 
+(PSGAN). It starts by using Makeup Distill Network to separate the reference image's makeup into two 
+spatially aware makeup matrices. After that, a module called Attentive Makeup Morphing is developed to 
+let users describe how a pixel's appearance in the source picture is altered based on the reference image. 
+In order to ease applications in the real-world setting, PSGAN is the first to concurrently accomplish 
+partial, shade tunable, and pose/expression robust makeup transfer. In order to separate the makeup from 
+the reference picture as two makeup matrices, an MDNet is also included. The flexible partial and shade 
+adjustable transfer is made possible by the spatially aware makeup matrices. To learn all cosmetics 
+attributes [41], including color, form, texture, and position, it comprises an enhanced color transfer branch 
+and a new pattern transfer branch. They present makeup in this work as a combination of color 
+transformation and pattern addition, and they create a thorough makeup transfer technique that works for 
+both delicate and dramatic looks. They suggest using warped faces in the Ultraviolet (UV) space while 
+training two network branches to eliminate the disagreement between input faces in terms of form, head 
+posture, and expression. They also create a new architecture with two branches for color and pattern 
+transfer. They present brand-new cosmetics transfer datasets with extreme fashions that were not taken 
+into account in the earlier datasets.
+2.2.Representation Learning for Face Recognition 
+Representation learning refers to a set of algorithms that are designed to solve a variety of challenges like 
+image retrieval [42-44], the person [45, 46] and vehicle [47, 48] re-identification, landmark detection, and 
+fine-grained object recognition [49, 50]. The task of face recognition in computer vision is heavily 
+dependent on learning representations that have fine intra-class and large inter-class distances [51]. 
+Previous works [6, 22, 25, 52, 53] have mainly adopted different, more robust loss functions with the aim 
+of learning representations that satisfy the aforementioned requirements.
+In [52], a deep convolutional neural network, named FaceNet, was proposed which learns facial 
+representations with the help of triplet loss. The main objective of this work is to achieve an embedding 
+f(x) from an image x into a d-dimensional Euclidean space Rd. The obtained embedding is generated in a 
+way that the squared distance among the embeddings from one class is small and that of the embeddings 
+from different classes is large. This algorithm achieves 99.63% and 95.12% accuracy in LFW [54] and 
+YouTube Faces Database [55] respectively. Liu et al. [53] have proposed a new look at the loss functions 
+which are based on the Euclidean margin between the produced embeddings. For CNNs to learn 
+discriminative facial characteristics with clear and innovative geometric interpretation, they suggest the 
+A-Softmax loss. The assumption that faces also lie on a manifold is fundamentally compatible with the 
+learnt features' discriminative spread on a hypersphere manifold. In order to approximate the learning 
+problem that minimal inter-class distance is greater than maximum intra-class distance, they develop 
+lower the margin set between such classes.
+In [22], the authors have proposed ArcFace, a major modification of the Softmax loss to further improve 
+the robustness of the learned deep features. By utilizing the arc-cosine function to calculate the angle 
+between the current feature and the target weight and adding an additive angular margin to the target 
+angle, the target logit can be obtained. Then, these logits are rescaled by a fixed feature norm followed by 
+exactly the same steps in the Softmax loss function. Their approach has the following advantages over the 
+others. (1) Directly optimizing the geodesic distance margin (2) State-of-the-art performance in several 
+
+5
+5
+benchmark datasets: achieving 99.53% accuracy (3) Easiness in terms of implementation (4) Efficiency in 
+terms of computational complexity.
+In [25], the authors reformulated the Softmax loss as a cosine loss with the aim of introducing a novel 
+loss function, named Large Margin Cosine Loss (LMCL). Their improvement is to further maximize the 
+decision margin in the angular space by introducing and training a deep model called CosFace. In this 
+deep model, LMCL guides the convolutional layers to learn features with huge cosine margins. Their 
+results demonstrate that they have achieved 97.96% accuracy in face verification on the MegaFace 
+benchmark, which has been a major improvement in comparison to previous works.  
+Meng et al. [6] proposed a new set of losses that enable the network to learn embeddings whose 
+magnitude represents the quality of the given face. By extending ArcFace [22] and introducing the 
+MagFace loss function, they demonstrate that the more likely the subject is to be recognized, the bigger 
+the magnitude of the generated embedding becomes. MagFace learns to generate these universal 
+embeddings by pulling the easier samples within a class of identities to the class center and pushing them 
+away from the origin. This makes the embeddings robust to ambiguity and the absence of high 
+discriminative features which prevalently exist in unconstrained face images in real scenarios. They have 
+achieved 99.83% verification accuracy in the LFW benchmark dataset. In Table 1, a comparison of these 
+works is depicted.
+Table 1. Verification accuracy of MagFace, CosFace, ArcFace, and SphereFace. These models are 
+evaluated on CALFW, CPLFW, AgeDB, LFW, and CFP-FP datasets.
+Method
+CALFW [56]
+CPLFW [57]
+AgeDB [58]
+LFW [54]
+CFP-FP [59]
+MagFace [6]
+96.15
+92.87
+98.17
+99.83
+98.46
+CosFace [25]
+96.18
+92.18
+98.17
+99.78
+98.26
+ArcFace [22]
+95.96
+92.72
+98.05
+9981
+98.40
+SphereFace [53]
+95.58
+91.27
+97.05
+99.67
+96.84
+Moreover, although these approaches have significant performance, directly applying GAN approaches 
+appears to have a few disadvantages. Models collapse, difficulty in training and convergence problems, 
+and poor image generation effect, along with the unreliable results of the generator for unconstrained 
+input images, cause the generated image examples to be incapable of being utilized for industrial data 
+augmentation tasks ]60 ,61[.
+3.
+Proposed approach
+3.1.Overview
+This section presents the proposed FRA algorithm. As can be inferred from Figure 1, our method includes 
+four steps. These are as follows: face detection and alignment, input preparation: facial landmark and 
+representation extraction, pose feature extraction, and representation augmentation. Steps 1 and 2 
+comprise our data preprocessing pipeline which is explained in Section 3.2. Steps 3 and 4 represent our 
+main contribution to this paper and are explained in Section 3.4.
+
+6
+6
+Figure 1. The overall procedure of FRA. FRA is composed of four steps to generate a new representation 
+vector with identity i, emotion e, and posture p, by applying a target posture p on a base image with 
+identity i and emotion e.
+3.2. Dataset preprocessing and preparation
+Our data preprocessing step includes three main phases. These three phases are depicted in Figure 1. As is 
+seen in the first phase, we feed the raw face images to the Multi-task Cascaded Convolutional Networks 
+(MTCNN) algorithm [53] which is a robust face and landmark detector. MTCNN provides us with 5 
+landmark points, including the center of both eyes, the tip of the nose, and the left and right corners of the 
+lips, and a bounding box that perfectly encloses the face area within the image without any padding. In 
+this phase, we also align the face images by feeding the acquired facial landmarks along with the face 
+image itself to the method of warp affine which exists in OpenCV [62], a famous library with ready-to-
+use computer vision-related algorithms. 
+In the second phase, we feed the aligned face images to MLXTEND1 so as to determine more facial 
+landmarks. As is shown in Figure 1, MLXTEND outputs 68 facial key points which we use to construct 
+binarized images with pixel value 0 (completely black) for the background and 1 (completely white) for 
+facial landmarks. On the other hand, we need to have fixed-size embeddings for each sample within the 
+dataset. These embeddings are in fact the training data for the combiner module which will be explained 
+in Section 3.4. In our case, we use two of the most reliable and robust face representation learning 
+algorithms, namely MagFace [6] and FaceNet [52], for obtaining embeddings for each image. MagFace’s 
+learning procedure is for a universal embedding that is quality aware, meaning that the easier the sample 
+is for the recognition task, the closer its feature vector becomes to the center of the class. Furthermore, 
+FaceNet is an algorithm that directly learns a mapping from the samples to a compact Euclidean space 
+and the distances correlate to the similarity degree of a given pair of face images. In Phase 3, the 
+binarized images generated in Phase 2 are fed to the AE model in order to generate an embedding vector 
+1http://rasbt.github.io/mlxtend/  
+
+Faciallandmarkextraction
+1
+Face lancmark
+detector
+Encoder
+(MLXTEND)
+1
+-
+68facial
+1
+targetface
+-
+withposturep'
+landmarks
+512Dposefeatures
+Combiner
+1
+Facialrepresentationextraction
+1
+-
+1
+-
+512D augmented face
+Face
+Face
+-
+for identity i, emotion e,
+representation
+detector
+representation
+detector
+andposturep'
+(MTCNN)
+(FRL)
+1
+1
+sourceface
+with identity i,
+512Dfacial
+emotione,
+representation
+andposturep
+1
+1
+-
+1
+1
+a)alignandcrop
+b)inputspreparation
+c)posefeaturesextraction
+d)augmentedrepresentation7
+7
+representing posture features. Finally, in Phase 4, pose and face representation vectors are fed into the 
+combiner module to generate an augmented face representation vector.
+3.3.Facial Landmark Restoration using Autoencoders
+Autoencoders (AE)  are a particular type of neural networks whose main functionality is to encode the 
+input into a meaningfully compacted representation and decode this into the input space afterwards [63, 
+64]. Following this paradigm, in this paper, we have been inspired by  the work done by Meng et al. [65] 
+and decided to use an AE-based model for encoding our input space (binarized images of landmarks 
+explained in Section 3.2) into the latent space (embeddings), as shown in Figure 2. Given Si as a sample 
+of facial landmarks image, the output of F(Si) is a reconstructed image S’i, where F(Si) is. After the AE 
+model’s convergence, we can discard B (decoder) and take only A which has learned to encode the input 
+into an optimized and meaningful latent space representation denoted by Vi. It is worth mentioning that Vi 
+plays a vital role in our proposed method which is the latent representation of the posture of the face. 
+Figure 2 illustrates the proposed AE-based model and its architecture.
+Figure 2. A general architecture of an autoencoder-based model. FRA utilizes a typical convolutional 
+autoencoder with a bottleneck of size 512. This bottleneck vector is used in further steps.
+3.4.Combining Feature Vectors and Feature Extraction using Vision Transformers
+Vision Transformers (ViT) are deep learning models whose versatility in various fields such as natural 
+language processing, speech recognition and computer vision has made them a prominent choice for 
+researchers [66]. In comparison with the conventional CNNs, ViT models have achieved competitive 
+superior results in vision tasks like object detection [67], image recognition [68], image super-resolution 
+[69], and segmentation [70, 71].  
+At the core of ViT models, there is a mechanism of attention that has been probably one of the most 
+significant concepts in the domain of deep learning. Its inspiration is the biological attributes of human 
+beings in that, to recognize an object, we tend to focus on the most distinctive parts of that entity instead 
+of paying attention to all parts of it as a whole [72]. In terms of deep neural networks, this can be 
+interpreted as assigning importance scores for a given set of features where the higher scores are for more 
+relevant features and the lower ones for the features with less saliency [73]. As can be observed from 
+Figure 3, the model learns to have more focus on the parts which represent the target object in the image. 
+
+20
+5
+112x112x1
+56
+1@112x112
+112x112x1
+@
+4@
+Convolution
+Convoluton
++Max-Pool
+S
+Encoder (A)
+Decoder (B)8
+8
+Figure 3. The paradigm of combining two representation vectors using ViT. The combiner takes two 
+representation vectors with a size of 512 and combines them into a 32x32 matrix to be processed by a 
+vision-transformer component.
+Moreover, transformers [74] refer to a set of neural networks which use the mechanism of attention. 
+These models consist of multiple encoders and decoders whose architectures are identical to each other. 
+In these models, a multi-head self-attention (MSA) mechanism is used for encoding the input, followed 
+by decoders which include an extra attention layer in order to process the encoder’s output. Self-attention 
+is a function denoted in Equation (1). 
+  
+s.t.  
+(1)
+where , , and  are weight matrices used in linear transformations on inputs x to produce Q, K, and V. The 
+attention score is then calculated by  as the dot product of the query and each key, scaled by the 
+dimension dk of the key K. Put x = "x1, x2, x3, ... , xn" to calculate an answer based on a collection of 
+queries Q, keys K, and values V. In MSA, Q, K, and V are projected linearly and this is done for h 
+consecutive times with different learned weights. Then, by applying the self-attention mechanism on each 
+of the outputs in the previous step simultaneously, we obtain h outputs which are heads. Then, these 
+heads are concatenated to achieve the final output. The following demonstrates these computations in 
+mathematical terms. 
+ 
+s.t. 
+(2)
+MSAs, compared with CNNs, transform feature maps with huge data-specific kernels and this makes 
+them as expressive as the CNN-based architectures [75]. The key difference exists where convolutions 
+diversify feature maps whereas MSAs combine them. According to [76], the Fourier analysis of feature 
+maps demonstrates that convolutions boost high-frequency components whereas MSAs, on the other 
+hand, attenuate them. 
+Furthermore, finding elements that are more pertinent for the depiction of the altered posture is made 
+easier by the multi-head attention layer. In order to do this, the scaled dot product attention gives greater 
+weight to the characteristics of the input facial representation and encoded posture that is more pertinent 
+while providing less weight to the features that are less relevant [77]. The procedure chooses features 
+
+512Dpose
+Concatination and Reshape
+representation
+Normalization Layer
+512DAugmented
+Vision
+representation
+Transformer
+(ViT)
+512Dface
+representation9
+9
+from various input regions and aids in improving representation performance since there are several heads 
+in the attention layer.
+In this paper, we have opted for using a ViT-based architecture for extracting features. As stated before, 
+this policy ensures that the model is trained to attend to the most salient feature values within the identity 
+and posture-related feature vectors simultaneously.  Considering E of size  as the embedding obtained 
+from a pre-trained facial representation learning algorithm and Vi of shape  as the bottleneck vector 
+generated by the autoencoder part of FRA, we concatenate and reshape the produced feature vector to 
+make it of shape . 
+3.5.Generating Pose-Aware Face Embeddings
+After the ViT module in our proposed model, the output is normalized, making it into the range of 0-1, 
+after which it is fed to a fully connected layer. This layer helps us produce an output of a 1-d shape which 
+is also considered the embedding generated by our model. 
+3.6.Multi-Task Loss Function
+In the training procedure, we utilized a Multi-part Loss Function (MLF) as the learning objective. This 
+MLF comprises a Binary Cross-Entropy (BCE) loss function, which is used to train the autoencoder, so it 
+can reconstruct the posed style better. Since the activation function of the last layer of our autoencoder is 
+Sigmoid, it can lead to loss of saturation (plateau) [78]. This saturation could prevent gradient-based 
+learning algorithms from convergence. In order to avoid this issue, it would be better to have a logarithm 
+function in the objective function to undo the exponential function within the Sigmoid. This is why BCE is 
+preferred, because it uses a logarithm function, unlike Mean Squared Error (MSE).
+The second part of our loss function is a type of N-pair Loss [79]. N-pair loss generalizes triplet loss [52] 
+to include comparison with multiple negative samples. The objective of this function is to keep the 
+distance between the anchor and positive smaller than the distance between the anchor and negative 
+representations, as shown in Figure 4.
+Figure 4. Effect of the proposed loss function during the learning process. The N-pair loss allows the 
+model to distinguish between pose-variant representation vectors with the same identity and emotion, as 
+well as possible.
+The proposed multi-task loss function is defined as follows,
+
+10
+10
+(3)
+in which, yi and pi denote the reconstructed pose style and the original pose style, respectively, and also is 
+defined as
+(4)
+where m is a margin applied to impose the separability between genuine and imposter pairs, and f denotes 
+the proposed architecture. d is the euclidean distance applied on normalized features and it is given by 
+Equation (5).
+(5)
+In Equation (6), a and p denote the anchor (generated) representation and the positive (real) 
+representation, respectively. Additionally, , , and  denote the negative representation w.r.t pose, identity, 
+and emotion of the anchor face, respectively. Specifically, negative pose representations have the same 
+identity as the anchor, but with different poses. The same holds for negative emotion representation. But, 
+for negative identity representation, the representation of another person is chosen randomly, regardless 
+of what pose or emotion it has. The goal of the triplet loss is to achieve,
+(6)
+The optimal state for each single triplet loss is achieved when is equal to zero and  is greater than the 
+predefined margin.
+4.Results and Discussion
+In this section we first introduce the benchmark dataset that we have used for evaluating our proposed 
+method. Then, we elaborate the details of our implementation and introduce the metrics used in this 
+paper. Finally, we demonstrate our experimental results and discussion.
+4.1.Datasets
+With the object of benchmarking our results, we have used the KDEF dataset. It is a publicly available 
+dataset of 4900 face images, covering 140 unique identities. The images demonstrate face images with 
+varying pose and emotion styles. Some samples of these datasets are shown in Figure 5. 
+
+11
+11
+Figure 5. A few samples of the KDEF dataset. The KDEF dataset provides face images of 140 different 
+people in various postures and emotions.
+4.2.Implementation details
+We carried out our experiments on a machine with a Core i7-1165G7 @ 2.80GHz CPU with 64 
+Gigabytes of RAM and a GeForce RTX 2060 12 GB GPU. All models were implemented and trained 
+using the Pytorch framework. Table 2 shows the hyperparameter setting.
+Table 2. Details of the training procedure and the utilized FRLs. The hyperparameter settings are shown. 
+FRL arch.
+# epochs
+Init. learning rate
+Dropout 
+rate
+Triplet 
+margin
+ViT
+Embedding dim
+FC dim
+# Heads
+# Layers
+Patch 
+size
+MagFace
+255
+0.001
+0.4
+10.0
+256
+256
+4
+4
+8
+ArcFace
+320
+0.001
+0.4
+10.0
+CosFace
+157
+0.001
+0.05
+10.0
+Furthermore, with the object of fairly evaluating the proposed FRA, we divided KDEF datasets based 
+on identities with the following distributions:
+●
+We randomly selected all samples from 99 identities which nearly comprise 70.7 % of all 
+identities in KDEF as our training data.
+
+12
+12
+●
+We randomly selected all samples from 11 identities which nearly comprise 7.8 % of all identities 
+in KDEF as our validation data.
+●
+We randomly selected all samples from 30 identities which nearly comprise 21.5 % of all 
+identities in KDEF as our testing data.
+4.3.
+Experimental Results
+This section details our comprehensive experimental results. Table 3 shows the achieved 
+accuracy of the Support Vector Machine (SVM) [80] classifier on the embeddings generated in 
+three different experiments. These experiments are:
+(1) Pre-augmentation accuracy: In this experiment, the training happens on the embeddings extracted 
+using three different FRLs, namely MagFace, ArcFace, and CosFace, and the testing accuracy is 
+achieved on the testing partition of these embeddings (Train/Test split ratio is set 80/20). In this 
+experiment, we used no augmentation technique at all and this is done to find a baseline for the 
+quality of the original data in the chosen benchmark dataset.
+(2) Generated embeddings’ accuracy: In this experiment, we first augmented the original 
+embeddings to obtain the transformed embeddings. Then, we trained the SVM on the original data 
+and tested its performance on the generated embeddings by the proposed algorithm. This is done to 
+demonstrate how much the proposed model is able to sustain the identity, posture, and emotion-
+related features without any degradation. 
+(3) Post-augmentation accuracy: In this experiment, we have augmented the embeddings of the 
+training split using FRA, where the test split is the same as (1). Then, we trained the SVM classifier 
+on the training part and tested it on the testing one.
+Table3. Evaluation results of FRA. Pre-augmentation and post-augmentation accuracies show the 
+effectiveness of FRA. Generated embeddings’ accuracy denotes the sustainability of FRA.
+FRL
+Target
+(1) Pre-
+augmentation
+Accuracy (%)
+(2) Generated 
+embeddings’ 
+Accuracy 
+(%)  
+(3) Post-
+augmentation 
+Accuracy (%)
+MagFace
+Posture
+82.38
+98.12
+96.66
+Identity
+86.19
+93.61
+95.71
+Emotion
+44.76
+92.43
+99.04
+ArcFace 
+Posture
+89.12
+99.3
+97.9
+Identity
+86.61
+91.6
+96.65
+Emotion
+53.55
+92.98
+100
+CosFace
+Posture
+99.12
+99.91
+99.12
+Identity
+79.91
+88.11
+96.50
+Emotion
+54.14
+87.61
+97.37
+
+13
+13
+Based on Table 3, it is observed that our proposed algorithm improves the classification accuracy, not 
+only identity-wise but also in terms of emotion and posture. For instance, SVM outputs 86.19% accuracy 
+on the MagFace embeddings but after the augmentation, this score goes up to 95.71% in Post-
+augmentation. For the same data, the generated embeddings are much more representative which 
+improves the classification accuracy on the identity of the SVM outputs by 93.61%. This enhancement 
+can also be validated for ArcFace and CosFace since our algorithm increases the accuracy in all three 
+experiments. In addition, FRA can improve the accuracy of SVM embeddings remarkably. In addition to 
+improving the classification accuracy with respect to the identities of the embeddings, FRA improves that 
+pose and emotion-wise. Based on Table 5, the accuracy of the SVM classifier is increased from 86.19% 
+for MagFace embeddings to 95.71% after augmentation. This improvement for ArcFace and Cosface is 
+from 86.61% to 96.65% and from 79.91% to 97.37%, respectively.
+Furthermore, our generated embeddings should ensure the fact that they are linearly separable. This 
+means that the embeddings can be classified using a linear classifier such as SVM with a linear kernel. In 
+our experiments we used an SVM classifier with a linear kernel and based on Table 5, we can deduce that 
+FRA is able to improve the accuracy in Phase 2 and Phase 3 by a large margin, effectively enhancing the 
+performance of SVM with a linear kernel.
+Moreover, in order to show the independence of FRA from any FR algorithms, we adopted three such 
+approaches, namely MagFace, ArcFace, and CosFace. Based on Table 5, after augmenting the 
+embeddings generated by each of these algorithms, the classification accuracy of the SVM classifier is 
+increased significantly and this proves the fact that FRA is not dependent on any specific FR algorithm as 
+its requirements. 
+In addition, for our algorithms’ performance to be verified thoroughly, the reconstructed binary images 
+which are created by the AE part of the proposed approach are presented. These images are illustrated in 
+Figure 6, showing the original image, and the AE’s output when dealing with MagFace embeddings, 
+ArcFace embeddings, and CosFace embeddings. Also, the training and validation loss in the training 
+procedure of our proposed pipeline is shown in Figure 7. 
+
+14
+14
+Original landmarks
+AE-generated landmarks 
+for Magface FRL
+AE-generated landmarks 
+for Arcface FRL
+AE-generated landmarks 
+for Cosface FRL
+Figure 6. Some instances of reconstructed binarized facial landmark images. These reconstructed images 
+denote how good the AE is performing for each FRL model. The CosFace model reconstructs the most 
+landmarks more precisely.
+
+15
+15
+Total Loss Magface
+BCE Loss Magface
+Npair Loss Magface
+Total Loss Arcface
+BCE Loss Arcface
+Npair Loss Arcface
+Total Loss Cosface
+BCE Loss Cosface
+Npair Loss Cosface
+Figure 7. Total loss, BCE loss and Npair loss curves achieved by various FRL methods.
+The curves of Precision-Recall (PRC), Receiver Operating Characteristic (ROC), and confusion matrices 
+of all experiments for MagFace, ArcFace, and CosFace are shown in Figures 8-16.
+
+Train loss
+Validation loss
+20
+15
+Loss Error
+10
+5
+0
+0
+50 
+100
+150
+200
+250
+Epoch0.22
+Train BCE losS
+ValidationBCEloss
+0.20
+0.18
+0.16
+rror
+E
+0.14
+Loss
+0.12
+0.10
+0.08
+0.06
+0
+50
+100
+150
+200
+250
+EpochTrain Npair loss
+Validation Npair loss
+20
+15
+Loss Error
+10
+5
+0
+0
+50 
+100
+150
+200
+250
+EpochTrain loss
+Validation loss
+20
+15
+Error
+Loss
+10
+5
+0
+0
+50
+100
+150
+200
+250
+300
+EpochTrain BCE losS
+Validation BCE loss
+0.7
+0.6
+0.5
+rror
+E
+0.4
+LosS
+0.3
+0.2 
+0.1
+0
+50
+100
+150
+200
+250
+300
+EpochTrain Npair loss
+Validation Npair loss
+20
+15
+Loss Error
+10
+5
+0
+0
+50
+100
+150
+200
+250
+300
+EpochTrain loss
+14 
+Validation loss
+12 
+10
+Error
+8
+Loss
+6
+4
+2
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+EpochTrain BCE losS
+Validation BCE loss
+0.18
+0.16
+0.14
+Loss Error
+0.12
+0.10
+0.08
+0.06
+0.04
+0
+20
+40
+60
+80
+100
+120
+140
+160
+EpochTrain Npair loss
+14
+Validation Npair loss
+12 
+10
+Error
+8
+Loss
+6
+4
+2
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Epoch16
+16
+a)
+Pre-augmentation PRC (pose)
+b)
+Pre-augmentation PRC (id)
+c)
+Pre-augmentation PRC 
+(emotion)
+d)
+Post-augmentation PRC 
+(pose)
+e)
+Post-augmentation PRC (id)
+f)
+Post-augmentation PRC 
+(emotion)
+Figure 8. PRC curves for MagFace FRL. As the curves indicate, post-augmentation PRC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation PRC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+precision vs.recall curve (pose)
+1.0
+0.8
+0.6
+precision
+0.4
+0.2 
+class FL
+class HL
+0.0
+class HR
+class S
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+recallprecision vs. recall curve (id)
+1.0
+0.9
+class AF05
+class BF04
+0.8
+clasS AM31
+classBM13
+class BF11
+class AF01
+class AM15
+0.7 
+class AF03
+class BM16
+precision
+class AF10
+class AM22
+0.6
+class AM13
+class BMo5
+class BM23
+class AM34
+class BF12
+0.5
+class AF07
+class AM05
+class AF27
+class BF14
+class BM29
+0.4
+class AMo1
+class AF16
+class AF32
+class AF26
+0.3
+class BM07
+class AF29
+class BF16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (emo)
+1.0
+0.8
+precision
+0.6
+0.4
+class AF
+class AN
+class DI
+class HA
+0.2
+class NE
+class SA
+class SU
+0.0
+0.2 
+0.4 
+0.6
+0.8 
+1.0
+recallprecision vs.recall curve (pose
+class FL
+1.0
+class HL
+class HR
+0.8
+0.6
+precision
+0.4
+0.2 
+0.0
+0.0
+0.2 
+0.4 
+0.6 
+0.8
+1.0
+recallprecision vs.recall curve (id)
+1.00
+0.95
+class AF05
+0.90
+class BF04
+class AM31
+class BM13
+class BF11
+class AF01
+0.85
+class AM15
+class AF03
+precision
+class BM16
+class AF10
+0.80
+class AM22
+class AM13
+class BM05
+class BM23
+0.75
+class AM34
+class BF12
+class AF07
+class AM05
+0.70
+class AF27
+class BF14
+class BM29
+class AM01
+0.65
+class AF16
+class AF32
+class AF26
+class BM07
+0.60
+class AF29
+class BF16
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (emo)
+1.00
+0.99
+0.98
+precision
+0.97
+0.96
+0.95
+class AF
+class AN
+class DI
+class HA
+0.94
+class NE
+class SA
+class SU
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recall17
+17
+a)
+Pre-augmentation ROC (pose)
+b)
+Pre-augmentation ROC (id)
+c)
+Pre-augmentation ROC 
+(emotion)
+d)
+Post-augmentation ROC 
+(pose)
+e)
+Post-augmentation ROC (id)
+f)
+Post-augmentation ROC 
+(emotion)
+Figure 9. ROC curves for MagFace FRL. As the curves indicate, post-augmentation ROC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation ROC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+RoCcurve(pose)
+1.0
+0.8
+0.6
+truepositive rate
+0.4
+0.2
+class FL (AUC = 0.9609)
+class FR (AUC = 0.999)
+class HL (AUC = 0.9991)
+0.0
+class HR (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROC curve (id)
+1.0
+class AM23 (AUC = 1.0)
+0.8 -
+class AF05 (AUC = 1.0)
+class BM13 (AUC = 1.0)
+classAF13(AUC =0.9892)
+class BF28 (AUC = 1.0)
+class BM08 (AUC = 1.0)
+class AF31 (AUC = 1.0)
+0.6
+class BM35 (AUC = 1.0)
+true positive rate
+class BF33 (AUC = 1.0)
+class BM18 (AUC = 1.0)
+class BF13 (AUC = 1.0)
+class AF19 (AUC = 0.9918)
+class AM26 (AUC = 0.9904)
+class BM12 (AUC = 1.0)
+0.4
+class AF16 (AUC = 1.0)
+class AF23 (AUC = 1.0)
+class BM01 (AUC = 1.0)
+class BM03 (AUC = 0.9938)
+class BF10 (AUC = 1.0)
+class BM19 (AUC = 1.0)
+class AF04 (AUC = 1.0)
+0.2
+class BF32 (AUC = 1.0)
+class AM08 (AUC =0.9939)
+class BF01 (AUC = 1.0)
+class BM06 (AUC = 1.0)
+class AM14 (AUC = 1.0)
+class BM07 (AUC = 1.0)
+0.0
+class BF15 (AUC =0.9949)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROCcurve (emo)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+class AF (AUC =0.9948)
+classAN (AUC =0.9921)
+class DI (AUC = 0.9972)
+class HA (AUC = 0.9959)
+class NE (AUC = 0.9894)
+class SA (AUC = 0.9939)
+0.0
+classSU (AUC = 0.9916)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateRoC curve (pose)
+class FL (AUC = 0.459)
+1.0
+class FR (AUC = 0.4088)
+class HL (AUC = 0.0797)
+No skill
+0.8
+0.6
+true positive rate
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROC curve (id)
+1.0
+class AM23 (AUC = 1.0)
+0.8 -
+class AF05 (AUC = 1.0)
+class BM13 (AUC = 1.0)
+class AF13 (AUC = 0.9951)
+class BF28 (AUC = 1.0)
+class BM08 (AUC = 1.0)
+class AF31 (AUC = 1.0)
+0.6
+class BM35 (AUC = 1.0)
+true positive rate
+class BF33 (AUC = 1.0)
+class BM18 (AUC = 1.0)
+class BF13 (AUC = 1.0)
+class AF19 (AUC =0.9961)
+class AM26 (AUC = 0.9961)
+class BM12 (AUC = 1.0)
+0.4
+class AF16 (AUC = 1.0)
+class AF23 (AUC = 1.0)
+class BM01 (AUC = 1.0)
+class BM03 (AUC = 0.9951)
+class BF10 (AUC = 1.0)
+class BM19 (AUC = 1.0)
+class AF04 (AUC = 1.0)
+0.2
+class BF32 (AUC = 1.0)
+class AM08 (AUC = 0.9989)
+class BF01 (AUC = 1.0)
+class BM06 (AUC = 1.0)
+class AM14 (AUC = 1.0)
+class BM07 (AUC = 1.0)
+0.0
+class BF15 (AUC =0.999)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateRoCcurve(emo)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+class AF (AUC = 1.0)
+class AN (AUC = 1.0)
+class DI (AUC = 1.0)
+class HA (AUC = 1.0)
+class NE (AUC = 1.0)
+class SA (AUC = 0.9996)
+0.0
+class SU (AUC = 1.0)
+No skill
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rate18
+18
+a) Pre-augmentation Confusion (pose)
+b) Pre-augmentation Confusion (id)
+c) Pre-augmentation Confusion 
+(emotion)
+d) Post-augmentation Confusion (pose)
+e) Post-augmentation Confusion (id)
+f) Post-augmentation Confusion 
+(emotion)
+Figure 10. Confusion matrices for MagFace FRL. As the confusion matrices indicate, post-augmentation matrices 
+(d, e, and f) have significant improvements in comparison to pre-augmentation matrices (a, b, and c) for pose, 
+identity, and emotion, respectively.
+
+Confusionmatrix(pose
+10000
+8000
+6000
+4000
+2000
+FL
+HL
+HR
+SConfusionmatrix(id)
+AF05
+BF04
+AM31
+BM13
+BF11
+1000
+AF01
+AM15
+AF03
+BM16
+AF10
+800
+AM22
+AM13
+BM05
+BM23
+600
+AM34 -
+BF12
+AF07
+AM05
+AF27
+400
+BF14 
+BM29
+AM01
+AF16
+AF32
+200
+AF26 
+BM07
+AF29
+BF16
+AF05
+BF04
+AM31
+BM13
+BF11
+AF01
+AM15
+AF03
+BM16
+AF10
+AM22
+AM13
+BM05
+BM23
+AM34
+BF12
+AF07
+AM05
+AF27
+BF14
+BM29
+AM01
+AF16
+AF32
+AF26
+BM07
+AF29
+BF16Confusionmatrix(emo)
+4
+4000
+3500
+3
+3000
+2500
+2000
+1500
+1000
+ SA
+500
+SU
+0
+AF
+AN
+DI
+HA
+NE
+SA
+SUConfusionmatrix(pose)
+60
+50
+40
+30
+20
+S
+10
+0
+HL
+HR
+SConfusionmatrix(id)
+10
+AF05
+BF04
+AM31
+BM13
+BF11
+AF01
+8
+AM15
+AF03
+BM16
+AF10
+AM22
+AM13
+6
+BM05
+BM23
+AM34 
+BF12
+AF07
+4
+AM05
+AF27
+BF14 
+BM29
+AM01
+AF16
+2
+AF32
+AF26
+BM07
+AF29
+BF16
+AF05
+BF04
+AM31
+BM13
+BF11
+AF01
+AM15
+AF03
+BM16
+AF10
+AM22
+AM13
+BM05
+BM23
+AM34
+BF12
+AF07
+AM05
+AF27
+BF14
+BM29
+AM01
+AF16
+AF32
+AF26
+BM07
+AF29
+BF16Confusionmatrix(emo)
+-35
+4
+30
+3
+ 25
+- 20
+15
+10
+ SA
+- 5
+SU
+0
+AF
+AN
+DI
+HA
+NE
+SA
+sU19
+19
+a)
+Pre-augmentation PRC (pose)
+b)
+Pre-augmentation PRC (id)
+c)
+Pre-augmentation PRC 
+(emotion)
+d)
+Post-augmentation PRC 
+(pose)
+e)
+Post-augmentation PRC (id)
+f)
+Post-augmentation PRC 
+(emotion)
+Figure 11. PRC curves for ArcFace FRL. As the curves indicate, post-augmentation PRC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation PRC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+precision vs.recall curve (pose
+1.0
+0.8
+0.6
+class FL
+precision
+class FR
+class HL
+class HR
+class S
+0.4
+0.2 
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (id)
+1.0
+0.8
+class BM15
+class BF09
+class BM14
+class BF07
+class AM22
+class AM23
+0.6
+class BM18
+class BM26
+precision
+class AF14
+class BM11
+class AF10
+class BM16
+class AF33
+0.4
+class BF10
+class AF20
+class BF02
+class BM08
+class BF28
+class AM35
+class AM07
+0.2 
+class BF03
+class BM31
+class BF17
+class AM10
+class AF02
+class BM06
+0.0
+class BM23
+class BM13
+0.0
+0.2 
+0.4 
+0.6 
+0.8 
+1.0
+recallprecision vs.recall curve (emo)
+1.0
+0.8
+precision
+0.6
+0.4
+class AF
+class AN
+class DI
+class HA
+0.2
+class NE
+class SA
+class SU
+0.0
+0.2 
+0.4 
+0.6 
+0.8
+1.0
+recallprecision vs.recall curve (pose)
+class FL
+1.0
+class FR
+class HL
+0.8
+0.6
+precision
+0.4
+0.2 
+0.0
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (id)
+1.0
+class BM15
+0.9
+class BF09
+class BM14
+class BF07
+class AM22
+class AM23
+class BM18
+class BM26
+precision
+class AF14
+0.8
+class BM11
+class AF10
+class BM16
+class AF33
+class BF10
+class AF20
+class BF02
+class BM08
+0.7
+class BF28
+class AM35
+class AM07
+class BF03
+class BM31
+class BF17
+class AM10
+class AF02
+0.6
+class BM06
+class BM23
+class BM13
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (emo)
+1.2
+1.0
+0.8
+precision
+0.6
+0.4
+class AF
+0.2 
+class AN
+class DI
+class HA
+class NE
+class SA
+class SU
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recall20
+20
+a)
+Pre-augmentation ROC (pose)
+b)
+Pre-augmentation ROC (id)
+c)
+Pre-augmentation ROC 
+(emotion)
+d)
+Post-augmentation ROC 
+(pose)
+e)
+Post-augmentation ROC (id)
+f)
+Post-augmentation ROC 
+(emotion)
+Figure 12. ROC curves for ArcFace FRL. As the curves indicate, post-augmentation ROC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation ROC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+RoCcurve(pose)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+class FL (AUC = 0.9978)
+class FR (AUC = 0.2198)
+class HL (AUC = 0.9953)
+class HR (AUC = 0.9998)
+0.0
+classS (AUC=0.9981)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROC curve (id)
+1.0
+class BM15 (AUC = 1.0)
+0.8 -
+class BF09 (AUC = 1.0)
+class BM14 (AUC = 1.0)
+class BF07 (AUC = 1.0)
+class AM22 (AUC = 1.0)
+class AM23 (AUC = 0.9782)
+class BM18 (AUC = 1.0)
+0.6
+class BM26 (AUC = 1.0)
+true positive rate
+class AF14 (AUC = 1.0)
+class BM11 (AUC = 1.0)
+class AF10 (AUC = 0.9838)
+class BM16 (AUC = 1.0)
+class AF33 (AUC = 1.0)
+class BF10 (AUC = 0.9841)
+0.4
+class AF20 (AUC = 1.0)
+class BF02 (AUC = 0.9916)
+class BM08 (AUC = 1.0)
+class BF28 (AUC = 1.0)
+class AM35 (AUC = 1.0)
+class AM07 (AUC = 1.0)
+class BF03 (AUC = 1.0)
+0.2
+class BM31 (AUC = 1.0)
+class BF17 (AUC = 1.0)
+class AM10 (AUC = 1.0)
+class AF02 (AUC = 0.9875)
+class BM06 (AUC = 1.0)
+classBM23(AUC=0.9827)
+0.0
+class BM13 (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROCcurve (emo)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+classAF(AUC =0.9885)
+clasS AN (AUC = 0.9983)
+class DI (AUC = 0.9975)
+class HA (AUC = 0.9998)
+class NE (AUC =0.9901)
+class SA (AUC = 0.9947)
+0.0
+class SU (AUC = 0.9968)
+No skill
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateRoCcurve(pose)
+class FL (AUC = 0.1322)
+1.0
+class FR (AUC = 0.7495)
+class HL (AUC = 0.3596)
+No skill
+0.8
+0.6
+true positive rate
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+false positiverateROC curve (id)
+1.0
+class BM15 (AUC = 1.0)
+0.8 -
+class BF09 (AUC = 1.0)
+class BM14 (AUC = 1.0)
+class BF07 (AUC = 1.0)
+class AM22 (AUC = 1.0)
+class AM23 (AUC =0.9952)
+class BM18 (AUC = 1.0)
+0.6
+class BM26 (AUC = 1.0)
+true positive rate
+class AF14 (AUC = 1.0)
+class BM11 (AUC = 1.0)
+class AF10 (AUC = 0.9987)
+class BM16 (AUC = 1.0)
+class AF33 (AUC = 1.0)
+class BF10 (AUC = 0.9979)
+0.4
+class AF20 (AUC = 1.0)
+class BF02 (AUC = 1.0)
+class BM08 (AUC = 1.0)
+class BF28 (AUC = 1.0)
+class AM35 (AUC = 1.0)
+class AM07 (AUC = 1.0)
+class BF03 (AUC = 1.0)
+0.2
+claSs BM31 (AUC = 1.0)
+class BF17 (AUC = 1.0)
+class AM10 (AUC = 1.0)
+class AF02 (AUC = 1.0)
+class BM06 (AUC = 1.0)
+class BM23 (AUC = 0.994)
+0.0
+class BM13 (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateRoCcurve(emo)
+1.0
+0.8
+0.6
+truepositive rate
+0.4
+0.2
+class AF (AUC = 1.0)
+class AN (AUC = 1.0)
+class DI (AUC = 1.0)
+class HA (AUC = 1.0)
+class NE (AUC = 1.0)
+class SA (AUC = 1.0)
+0.0
+class SU (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rate21
+21
+a)
+Pre-augmentation Confusion 
+(pose)
+b)
+Pre-augmentation Confusion 
+(id)
+c)
+Pre-augmentation Confusion 
+(emotion)
+d)
+Post-augmentation Confusion 
+(pose)
+e)
+Post-augmentation Confusion 
+(id)
+f)
+Post-augmentation Confusion 
+(emotion)
+Figure 13. Confusion matrices for ArcFace FRL. As the confusion matrices indicate, post-augmentation matrices (d, 
+e, and f) have significant improvements in comparison to pre-augmentation matrices (a, b, and c) for pose, identity, 
+and emotion, respectively.
+
+Confusionmatrix(pose)
+10000
+L
+8000
+R
+6000
+主
+4000
+2000
+S
+FL
+FR
+HL
+HR
+sConfusionmatrix(id)
+BM15
+BF09
+BM14
+BF07
+AM22
+1000
+AM23
+BM18
+BM26
+AF14 
+800
+BM11
+AF10
+BM16
+AF33
+BF10
+600
+AF20 
+BF02
+BM08
+BF28
+AM35
+400
+AM07
+BF03
+BM31
+BF17
+AM10
+200
+AF02
+BM06
+BM23
+BM13
+BM15
+BF09
+BM14
+BF07
+AM22
+AM23
+BM18
+BM26
+AF14
+BM11
+AF10
+BM16
+AF33
+BF10
+AF20
+BF02
+BM08
+BF28
+AM35
+AM07
+BF03
+BM31
+BF17
+AM10
+AF02
+BM06
+BM23
+BM13Confusionmatrix(emo)
+4
+4000
+3
+3500
+3000
+2500
+2000
+NE
+1500
+1000
+ SA
+500
+SU
+0
+AF
+AN
+DI
+HA
+NE
+SA
+sUConfusion matrix(pose)
+80
+ 70
+60
+ 50
+40
+-30
+20
+S
+10
+0
+HL
+HR
+SConfusionmatrix(id)
+BM15
+BF09
+BM14
+10
+BF07
+AM22
+AM23
+BM18
+BM26
+AF14
+BM11
+AF10
+BM16
+AF33
+6
+BF10
+AF20 
+BF02
+BM08
+BF28
+AM35
+AM07
+BF03
+BM31
+BF17
+2
+AM10
+AF02
+BM06
+BM23
+BM13
+BM15
+BF09
+BM14
+BF07
+AM22
+AM23
+BM18
+BM26
+AF14
+BM11
+AF10
+BM16
+AF33
+BF10
+AF20
+BF02
+BM08
+BF28
+AM35
+AM07
+BF03
+BM31
+BF17
+AM10
+AF02
+BM06
+BM23
+BM13Confusionmatrix(emo)
+-35
+4
+30
+3
+25
+20
+15
+10
+5
+0
+AF
+AN
+DI
+HA
+NE
+SA
+sU22
+22
+a)
+Pre-augmentation PRC (pose)
+b)
+Pre-augmentation PRC (id)
+c)
+Pre-augmentation PRC 
+(emotion)
+d)
+Post-augmentation PRC 
+(pose)
+e)
+Post-augmentation PRC (id)
+f)
+Post-augmentation PRC 
+(emotion)
+Figure 14. PRC curves for CosFace FRL. As the curves indicate, post-augmentation PRC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation PRC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+precision vs.recall curve (pose)
+1.0
+0.8
+0.6
+class FL
+precision
+class FR
+class HL
+class HR
+class S
+0.4
+0.2 
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (id)
+1.0
+0.8 
+class BF25
+class AM30
+classBM13
+class AM33
+class BM21
+class BF14
+class BM19
+class BM17
+0.6
+precision
+class BF15
+class BF06
+class AM18
+class AF35
+class BF13
+class BM22
+class BM23
+0.4
+class BM25
+class AF08
+class AF26
+class BM33
+class AM21
+class AMo1
+class AF14
+class AF32
+0.2
+class BM28
+class AM10
+class BM29
+class BM05
+class AM06
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (emo)
+1.0
+0.9
+0.8
+0.7
+precision
+0.6
+0.5
+0.4 
+class AF
+0.3
+class AN
+class DI
+class HA
+class NE
+0.2
+class SA
+class SU
+0.0
+0.2 
+0.4 
+0.6 
+0.8 
+1.0
+recallprecision vs.recall curve (pose
+1.2
+class FL
+class FR
+class HL
+class HR
+class S
+1.0
+0.8
+precision
+0.6
+0.4
+0.2 
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (id)
+1.0
+class E
+BF25
+class AM30
+class E
+BM13
+class AM33
+0.9
+class BM21
+class BF14
+class BM19
+class BM17
+class BF15
+class BF06
+0.8
+class AM18
+class AF35
+precision
+class BF13
+class BM22
+class BM23
+class BM25
+class AF08
+0.7
+class AF26
+class BM33
+class AM21
+class AMol
+class AF14
+class AF32
+class BM28
+0.6
+class AM10
+class BM29
+class BM05
+class AM06
+0.5
+0.0
+0.2 
+0.4 
+0.6
+0.8
+1.0
+recallprecision vs.recall curve (emo)
+1.0
+0.9
+0.8
+0.7
+recision
+ 0.6
+0.5
+0.4
+class AF
+class AN
+class DI
+class HA
+0.3
+class NE
+class SA
+class SU
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+recall23
+23
+a)
+Pre-augmentation ROC (pose)
+b)
+Pre-augmentation ROC (id)
+c)
+Pre-augmentation ROC 
+(emotion)
+d) Post-augmentation ROC (pose)
+e) Post-augmentation ROC (id)
+f) Post-augmentation ROC (emotion)
+Figure 15. ROC curves for CosFace FRL. As the curves indicate, post-augmentation ROC curves (d, e, and f) have 
+significant improvements in comparison to pre-augmentation ROC curves (a, b, and c) for pose, identity, and 
+emotion, respectively.
+
+RoC curve (pose)
+1.0
+0.8 -
+0.6
+true positive rate
+0.4
+0.2
+class FL (AUC = 0.5661)
+class FR (AUC = 0.7062)
+class HL (AUC = 0.9998)
+class HR (AUC =0.9997)
+0.0
+class S (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROC curve (id)
+1.0
+class BF25 (AUC = 0.9997)
+0.8
+class AM30 (AUC = 1.0)
+class BM13 (AUC = 0.9996)
+class AM33 (AUC = 0.9969)
+class BM21 (AUC = 0.9795)
+class BF14 (AUC = 0.9894)
+class BM19 (AUC = 0.9983)
+0.6
+class BM17 (AUC = 0.9998)
+truepositive rate
+classBF15(AUC =0.9907)
+class BF06 (AUC = 0.994)
+class AM18 (AUC = 0.9926)
+class AF35 (AUC =0.9998)
+class BF13 (AUC = 0.9998)
+class BM22 (AUC =0.9999)
+0.4
+class BM23 (AUC =0.9999)
+class BM25 (AUC =0.9997)
+class AF08 (AUC =0.9986)
+class AF26 (AUC =0.9952)
+class BM33 (AUC = 0.9912)
+class AM21 (AUC = 0.9817)
+class AM01 (AUC = 0.9994)
+0.2
+class AF14 (AUC =0.9908)
+class AF32 (AUC = 0.9996)
+class BM28 (AUC = 0.9996)
+class AM10 (AUC =0.9999)
+class BM29(AUC =0.9985)
+class BM05 (AUC =1.0)
+0.0
+class AM06 (AUC = 0.9988)
+No skill
+0.0 
+0.2 
+0.4
+0.6
+0.8
+1.0
+false positive rateROCcurve (emo)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+claSSAF (AUC =0.9749)
+class AN (AUC = 0.9835)
+class DI (AUC = 0.9897)
+class HA (AUC = 0.9994)
+class NE (AUC =0.9949)
+class SA (AUC = 0.9814)
+0.0
+class SU (AUC = 0.9931)
+No skill
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROCcurve(pose)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2 
+class FL (AUC = 1.0)
+class FR (AUC = 1.0)
+class HL (AUC = 1.0)
+class HR (AUC = 1.0)
+0.0
+class S (AUC = 1.0)
+No skill
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0
+false positive rateROC curve (id)
+1.0
+class BF25 (AUC = 1.0)
+0.8 -
+class AM30 (AUC = 1.0)
+class BM13 (AUC = 1.0)
+class AM33 (AUC = 1.0)
+class BM21 (AUC = 1.0)
+class BF14 (AUC = 0.9932)
+class BM19 (AUC = 1.0)
+0.6
+class BM17 (AUC = 1.0)
+true positive rate
+class BF15 (AUC = 1.0)
+class BF06 (AUC = 1.0)
+class AM18 (AUC = 1.0)
+class AF35 (AUC = 1.0)
+class BF13 (AUC = 1.0)
+class BM22 (AUC = 1.0)
+0.4
+class BM23 (AUC = 1.0)
+class BM25 (AUC = 1.0)
+class AF08 (AUC = 1.0)
+class AF26 (AUC = 1.0)
+class BM33 (AUC = 1.0)
+class AM21 (AUC = 1.0)
+class AM01 (AUC = 1.0)
+0.2
+clasS AF14 (AUC = 0.9943)
+class AF32 (AUC = 1.0)
+class BM28 (AUC = 1.0)
+class AM10 (AUC = 1.0)
+class BM29 (AUC = 1.0)
+class BM05 (AUC = 1.0)
+0.0
+class AM06 (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rateROCcurve (emo)
+1.0
+0.8
+0.6
+true positive rate
+0.4
+0.2
+classAF (AUC =0.9853)
+class AN (AUC = 0.9996)
+class DI (AUC = 0.9965)
+class HA (AUC = 1.0)
+class NE (AUC =0.9932)
+class SA (AUC = 0.9998)
+0.0
+class SU (AUC = 1.0)
+No skill
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+false positive rate24
+24
+a)
+Pre-augmentation Confusion 
+(pose)
+b)
+Pre-augmentation Confusion 
+(id)
+c)
+Pre-augmentation Confusion 
+(emotion)
+d)
+Post-augmentation Confusion 
+(pose)
+e)
+Post-augmentation Confusion 
+(id)
+f)
+Post-augmentation Confusion 
+(emotion)
+Figure 16. Confusion matrices for CosFace FRL. As the confusion matrices indicate, post-augmentation matrices 
+(d, e, and f) have significant improvements in comparison to pre-augmentation matrices (a, b, and c) for the pose, 
+identity, and emotion, respectively.
+4.4. Discussion
+FR has long been a popular field of study among specialists and academics in the field of biometric 
+recognition and it has the advantages of being non-contact, amiable, and simple to accept. Although 
+remarkable performance has been shown by some state-of-the-art approaches presented in the literature, 
+in real-world scenarios, there still exists the demand to improve such algorithms. For better handling such 
+uncontrolled contexts especially when we face a lack of data, DA techniques are introduced to increase 
+the number of training samples by applying different manipulations. Classical techniques for image 
+transformations such as rotation, skewing, flipping, blurring, etc., and also GAN-based ones which utilize 
+deep generative models and disentangled features to create more realistically transformed face images 
+have well been studied for DA in the domain of FR. However, these techniques have their drawbacks. 
+Classical techniques mostly manipulate face images in a way that distorts their alignment causing FR 
+algorithms’ performance when generating distinct representative embeddings dramatically decrease. To 
+prove this, we have experimented with four different transformations, namely, horizontal flip, skewing, 
+blurring, and notifying. We augmented samples in the KDEF dataset and increased the training dataset 
+size to be 4 times more than the original dataset and used different FR algorithms for generating 
+
+Confusion matrix (pose)
+10000
+8000
+R
+6000
+4000
+2000
+S
+FL
+FR
+HL
+HRConfusionmatrix(id)
+BF25
+AM30
+1200
+BM13
+AM33
+BM21
+BF14
+1000
+BM19
+BM17
+BF15
+BF06
+AM18
+800
+AF35
+BF13
+BM22
+BM23
+600
+BM25
+AF08
+AF26
+BM33
+400
+AM21
+AM01
+AF14
+AF32
+BM28
+200
+AM10
+BM29
+BM05
+AM06
+BF25
+AM30
+BM13
+AM33
+BM21
+BF14
+BM19
+BM17
+BF15
+BF06
+AM18
+AF35
+BF13
+BM22
+M23
+BM25
+AF08
+AF26
+BM33
+AM21
+AM01
+AF14
+AF32
+BM28
+AM10
+BM29
+BM05
+AM06Confusionmatrix(emo)
+4000
+4
+3500
+3
+3000
+2500
+2000
+NE
+1500
+1000
+ SA
+500
+SU
+0
+AF
+AN
+DI
+HA
+NE
+SA
+SUConfusion matrix (pose)
+- 80
+- 70
+60
+50
+-40
+-30
+- 20
+10
+0
+FL
+FR
+HL
+HR
+SConfusionmatrix(id)
+BF25
+AM30
+BM13
+10
+AM33
+BM21
+BF14 
+BM19
+BM17
+BF15
+BF06
+AM18
+AF35
+BF13
+6
+BM22
+BM23
+BM25
+AF08
+AF26
+BM33
+AM21
+AM01
+AF14
+AF32
+2
+BM28
+AM10
+BM29
+BM05
+AM06
+BF25
+AM30
+BM13
+AM33
+BM21
+BF14
+BM19
+BM17
+BF15
+BF06
+AM18
+AF35
+BF13
+BM22
+BM23
+BM25
+AF08
+AF26
+BM33
+AM21
+AM01
+AF14
+AF32
+BM28
+AM10
+BM29
+BM05
+AM06Confusionmatrix(emo)
+35
+4
+30
+3
+- 25
+20
+15
+10
+- 5
+0
+AF
+AN
+DI
+HA
+NE
+SA
+SU25
+25
+embeddings. Then we classified the embeddings using SVM with respect to their identities. Table 6 
+details the results achieved by this experiment. 
+ 
+Table 4. Evaluation of MagFace, ArcFace, and CosFace using traditional augmentation techniques on the 
+KDEF dataset. Post-augmentation accuracy scores achieved denote the ineffectiveness of these techniques 
+for FR tasks.
+FR Algorithm
+Accuracy Score
+Pre-augmentation
+Post-augmentation
+MagFace
+20.49%
+18.54%
+ArcFace
+20.12%
+17.01%
+CosFace
+18.19%
+11.33%
+Table 4 shows that augmenting the face images using the classical approaches does not result in any 
+improvement and they, in fact, degrade the quality of embeddings. For instance, for the MagFace 
+algorithm, the accuracy obtained by SVM is decreased by 2% after applying DA.
+Moreover, we conducted another experiment using three state-of-the-art generative-based algorithms 
+namely, CPM [41], AttGAN [81], and PSGAN [40]. Following the previous experiments, we augmented 
+the face images and obtained the classification accuracy before and after augmentation. Table 7 shows the 
+results achieved by this experiment.
+Table 5. Evaluation of augmentation techniques using GANs on the KDEF dataset. In the best case, the 
+post-augmentation accuracy has increased a little and in some cases, it has caused a degradation in 
+accuracy.
+Algorithm
+Accuracy Score
+Pre-augmentation
+Post-augmentation
+CPM [40]
+20.49 %
+17.82 %
+AttGAN
+20.49 %
+26.70 %
+PSGAN [39]
+20.49 %
+23.40 %
+Based on Table 5, it can be claimed that these generative models do not contribute to classification 
+accuracy improvement. Therefore, in order to address this issue, in this paper, we propose a new 
+algorithm, named FRA, which effectively augments the training data for FR algorithms. FRA functions 
+with original embeddings and manipulates them in a way to be representative of the same identity of the 
+embedding and also a differed postural information existent in these representational embeddings. Results 
+achieved by our extensive experiments indicate the efficacy of FRA in augmenting samples in the FR 
+domain.
+5.Conclusion
+
+26
+26
+Since data scarcity is a common problem in deep learning-based solutions, it can be very challenging to 
+build up FR systems that are robust enough to recognize face images with extreme diversity. In this paper, 
+we proposed a novel method that augments the face data in latent space. The proposed method utilizes 
+two major components, one of which is an autoencoder and the other is a ViT-based model. The former is 
+used to encode the binarized input images consisting of sparse facial landmarks into a latent space. The 
+latter is used for extracting features from the combined embeddings coming from a pre-trained FRL 
+algorithm and the autoencoder part of our model. Lastly, the output of the proposed model is an 
+embedding representing the main identity with the same emotion but with a different posture. This way, 
+we managed to improve the classification accuracy by 9.52, 10.04, and 16.60, in comparison with the 
+based models of MagFace, ArcFace, and CosFace, respectively.
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+

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+Towards Edge-Cloud Architectures for Personal
+Protective Equipment Detection
+Jarosław Legierski
+jaroslaw.legierski@orange.com
+Orange Innovation, Orange
+Polska S.A.
+Warsaw, Poland
+Kajetan Rachwał
+Piotr Sowinski
+kajetan.rachwal@ibspan.waw.pl
+piotr.sowinski@ibspan.waw.pl
+Systems Research Institute Polish
+Academy of Sciences
+Poland
+Warsaw Univesity of Technology
+Warsaw, Poland
+Wojciech Niewolski
+wojciech.niewolski@orange.com
+Orange Innovation, Orange
+Polska S.A.
+Poland
+Warsaw Univesity of Technology
+Warsaw, Poland
+Przemysław Ratuszek
+Zbigniew Kopertowski
+przemyslaw.ratuszek@orange.com
+zbigniew.kopertowski@orange.com
+Orange Innovation, Orange
+Polska S.A.
+Warsaw, Poland
+Marcin Paprzycki
+Maria Ganzha
+marcin.paprzycki@ibspan.waw.pl
+maria.ganzha@ibspan.waw.pl
+Systems Research Institute Polish
+Academy of Sciences
+Warsaw, Poland
+Abstract
+Detecting Personal Protective Equipment in images and video
+streams is a relevant problem in ensuring the safety of con-
+struction workers. In this contribution, an architecture en-
+abling live image recognition of such equipment is proposed.
+The solution is deployable in two settings – edge-cloud and
+edge-only. The system was tested on an active construction
+site, as a part of a larger scenario, within the scope of the
+ASSIST-IoT H2020 project. To determine the feasibility of
+the edge-only variant, a model for counting people wearing
+safety helmets was developed using the YOLOX method. It
+was found that an edge-only deployment is possible for this
+use case, given the hardware infrastructure available on site.
+In the preliminary evaluation, several important observations
+were made, that are crucial to the further development and de-
+ployment of the system. Future work will include an in-depth
+investigation of performance aspects of the two architecture
+variants.
+Keywords: edge-cloud continuum architectures, PPE detec-
+tion, image recognition, worker safety
+1
+Introduction
+Nowadays, the demand for intelligent video analytics is grow-
+ing across a wide spectrum of application areas [17]. The
+key part of such systems is usually an image recognition (IR)
+component. However, as of today, the IR subsystem is, most
+commonly, deployed in the cloud. This approach offers mul-
+tiple benefits, such as availability of large and scalable com-
+putational resources, reliable APIs, and shifting the burden
+of system maintenance to the cloud service provider. How-
+ever, this comes at a cost. Sending data to the cloud raises
+both security and privacy concerns. Moreover, communicat-
+ing with the cloud always induces network latency, which
+may be significant in time-critical applications. To address
+issues brought about by cloud-centric solutions, edge com-
+puting has been proposed. Here, the core of the approach is
+processing the data as close to the source as possible. This
+allows for latency reduction, and helps ensure the security
+and privacy of data, which remains within the local network.
+However, edge computing has its own set of issues. Typically,
+the computational resources, which are available at the edge
+are considerably smaller. A possible solution to addressing
+the downsides of both these options is a combined approach –
+an edge-cloud continuum, where data is partially processed
+on the edge and partially in the cloud. However, this raises
+the obvious question: at which point(s), within the continuum,
+individual parts of the system should be deployed.
+Here, this question is considered within a real-world sce-
+nario of monitoring the entrance to an active construction
+site. Specifically, the system is tasked with ensuring that (1)
+no unauthorized people enter the worksite, and (2) every-
+body is wearing appropriate Personal Protective Equipment
+(PPE), i.e. helmets and safety vests. The scenario is evalu-
+ated as part of the ASSIST-IoT project, on a construction site
+in Warsaw, Poland, managed by the construction company
+Mostostal Warszawa. Here, the edge versus cloud discussion
+becomes particularly relevant. On the one hand, the privacy
+of workers is of paramount importance, while latencies must
+be minimized, to ensure a quick reaction, which hints at an
+arXiv:2301.01501v1  [cs.CV]  4 Jan 2023
+
+Legierski and Rachwał et al.
+edge deployment. On the other hand, given the limited hard-
+ware resources available on the edge, and the extremely harsh
+conditions of the construction site, a cloud deployment seems
+attractive.
+Given the possible benefits of both solutions, in this contri-
+bution, a solution is proposed for an edge-cloud continuum
+video analytics architecture. The architecture can be deployed
+in two variants (edge-only, and edge-cloud), described in the
+Architecture section. Moreover, to determine the viability of
+the solution, an initial experimental study was performed.
+Here, an IR model was developed and integrated with the
+edge-only variant of the architecture. Next, it was tasked with
+detecting when personnel wearing PPE entered and exited the
+work site.
+2
+Background
+To provide a context for this study, the state of the art of (1)
+IR system architectures and (2) machine learning models for
+PPE detection is summarized.
+System architectures. The most obvious benefit of deploy-
+ing IR systems on the edge is the decreased latency. This was
+demonstrated in [20], where facial recognition models were
+deployed on the edge. The authors found that deploying the
+models on the edge resulted in significantly better response
+speeds, as compared to a cloud deployment. In other stud-
+ies [8, 9], the viability of deploying deep convolutional neural
+networks (CNNs) in the edge-only scenario was investigated.
+CNNs are characterized by high resource utilization, and thus
+are typically deployed in the cloud. The studies found that
+deploying CNNs is viable on mobile devices, when parts of
+the computation can be offloaded to other edge devices. Edge
+deployment allowed to achieve a consistently low latency
+of 2.24 ms while using CNNs to perform real-time object
+tracking in augmented reality [8]. Both studies showed that
+the edge deployment distributing the workload increased the
+inference capabilities of the system, as the models could not
+be run on the disconnected mobile devices alone.
+One study [5] investigated an edge-cloud architecture, where
+data preprocessing servers were deployed close to the data
+source. The preprocessed data was then sent to a cloud-based
+deep-learning platform. This resulted in decreasing network
+latency and traffic. It also increased the security and privacy
+of the raw data.
+On the other hand, edge deployments are more limited
+in terms of the available hardware. Low computational re-
+sources naturally limit the size of models and inference speed.
+A study compared different implementations (based on Ten-
+sorFlow, TensorRT, and TFLite) of the same video processing
+model [6], and found them to differ in their resource utiliza-
+tion. The choice of implementation influenced the energy
+consumption of the model, as well as its inference speed.
+Interestingly, the slowest implementation (TFLite) was the
+most energy efficient. It was also found that TFLite managed
+to remain on par with the other implementations in terms of
+speed, when processing low-resolution video. In the case of
+high-resolution video, more resource-intensive models were
+needed to maintain the speed, suggesting that a cloud deploy-
+ment could be more beneficial in low-resource settings. Nev-
+ertheless, some resource-intensive models can be deployed
+on the edge, if resources available there are sufficient. The
+deployment proposed in a different study required all nodes
+to be equipped with a GPU [9]. This allowed the authors to
+use CNNs on the edge. A similar result was reported in [13],
+were IR models deployed on a Raspberry Pi 4B, equipped
+with a camera, and an Intel Neural Compute Stick 2 (a USB
+device for deep learning inference on the edge) were studied.
+These devices were chosen for their low power consumption
+and good computing capabilities. Overall, a model tasked
+with detecting PPE in the form of helmets and safety vests
+achieved precision on the order of 99.5%.
+Models for PPE detection. Effective video analytics-based
+methods for detecting the presence of protective helmets,
+worn by workers, due to its health and safety importance, is
+currently a hot research topic.
+The usage of existing, unmodified machine learning models
+for detecting protective head covers does not provide suffi-
+cient detection accuracy, as proven in a recent study [19].
+In said article, several versions of the popular YOLO algo-
+rithm [1] were compared. It was shown that the most effective
+version of YOLO for helmet detection is the v4. After improv-
+ing the loss function, it achieved more than 93% accuracy
+during tests. A similar study [10] focused on improving the
+YOLOv5 algorithm. The system achieved results close to 97%
+accuracy, thanks to the improvement of the structure of the
+neural network. Another study [18], also investigated improv-
+ing YOLOv5. However, instead of the algorithm itself, work
+was focused on processing of input data by applying filters
+on the input image. This allowed to improve the accuracy to
+above 95%. Yet another study [15] presented an approach for
+improving the detection speed and accuracy by designing a
+multi-level pyramidal feature fusion network based on the
+ConCaNet attention mechanism. Here, YOLOv3 was applied
+and a dataset with 6000 images was used. The results demon-
+strate the effectiveness of this approach, which managed to
+reduce the number of necessary parameters.
+Helmet detection can also be done using the SSD-MobileNet
+algorithm [4], which is based on yet another variant of CNN.
+An analysis of this method, reported in [7], tested its effec-
+tiveness and managed to reach 80% accuracy during tests.
+In a wider comparison of algorithm types [11], the authors
+proposed a helmet detection method based on a dynamically
+changing neural network – SHDDM (Safety Helmet Detec-
+tion Dynamic Model). The developed model analyzes the
+human posture and defines the area where the helmet should
+be located, to eliminate the detection of the helmet outside
+the head area and thus reduce the false positive rate. There are
+
+Towards Edge-Cloud Architectures for Personal Protective Equipment Detection
+also other approaches to helmet detection, such as methods
+based on color and shape used to to locate the face, and the
+proper wearing of a helmet [16]. Another solution used low-
+resolution images, captured from a video stream, using the
+Local Binary Pattern (LBP) and gray-level co-occurrence ma-
+trix (GLCM) methods along with a back-propagation neural
+network [14].
+Another study [2] investigated the usefulness of artificially
+created images in the training of CNNs for PPE detection. The
+paper presented the results achieved with YOLOv3, trained on
+artificial images generated by the Rockstar Advanced Game
+Engine (RAGE) from the Grand Theft Auto V video game.
+This approach achieved a mean average precision (mAP) of
+only 55.11% on a test dataset consisting of real-world images.
+The mAP for synthetic images was much higher at 87.24%.
+It should be noted that the poor results for the real-world
+images are most likely caused by the RAGE engine being
+unable to generate a sufficient amount of head, welding mask,
+ear protection, and chest object variations.
+As can be seen, there are many possibilities for detecting
+protective helmets. Here, the SHDDM is particularly note-
+worthy, as it has an important feature of checking whether the
+helmet is worn properly, and not only detecting its presence.
+This, in turn, is particularly relevant in real-world applica-
+tions.
+3
+Proposed Architecture
+The proposed video analytics system can be deployed in
+two architecture variants: edge-cloud (Fig. 1) and edge-only
+(Fig. 2). As outlined above, there are reasons to believe that
+both variants may be appropriate for the considered scenario.
+Both architectures share a common core deployed on the edge,
+consisting of: a camera, the Image Processor (IP) component,
+and the OSH (Occupational Safety and Health) manager’s
+mobile device.
+The camera (in the reported experiments the Dahua IPC-
+HFW5449T-ASE-LED was used) provides a live RTSP video
+stream, which is directed to the Image Processor. The IP is
+a service written in Python, which can optionally perform
+preliminary image analysis. Using configurable methods such
+as motion detection and brightness thresholding, the IP is able
+to discard image frames that do not contain moving people,
+reducing network traffic to components involved in actual im-
+age analysis. It is also responsible for communicating with the
+rest of the system, designed in accordance with the ASSIST-
+IoT reference architecture [3]. IP communicates with the rest
+of the system publishing alerts to an MQTT topic. This de-
+sign allows other components and devices in the ASSIST-IoT
+deployment to be notified in a streaming manner of any OSH
+violations, such as workers not wearing protective helmets.
+In the first version of the architecture – the edge-cloud
+deployment – the IP is configured to use the cloud-based
+AWS Rekognition platform, with its PPE detection service.
+ASSIST-IoT deployment 
+AWS
+Rekognition
+Video stream
+OSH alert
+Image Processor
+Camera
+OSH manager's
+mobile device
+Edge
+Filtered 
+video stream
+Inference 
+results
+AWS Cloud
+Figure 1. Edge-cloud deployment
+Video stream
+OSH alert
+Image Processor
+Camera
+OSH manager's
+mobile device
+Edge
+Orange AI&ML Platform
+Video stream
+Processing pipeline
+AI-X
+AI-2
+AI-1
+...
+Result adaptation
+Inference 
+results
+External platform connector
+ASSIST-IoT deployment 
+Figure 2. Edge deployment
+In the edge-only variant, the video analysis is performed by
+the Orange AI&ML Platform, which is deployed on a server
+on the construction site. This edge deployment allows for
+maintaining lower network latency, and ensures the privacy
+of worker data. The AI&ML Platform’s services are written
+as Python runnable modules that provide their own APIs and
+GUIs. The services can reuse the APIs and GUIs provided
+by the platform, or build them from scratch. A service col-
+lects frames from a video source, processes them in an ML
+pipeline specific to the service, and adapts or interprets the
+results. The inference results from the Platform are forwarded
+to external services, with the use of provided connectors. As
+the Orange AI&ML Platform operates on the edge, all video
+
+Legierski and Rachwał et al.
+processing takes place on the client’s site, ensuring full secu-
+rity of customer data (video) and compliance with appropriate
+regulations, such as GDPR.
+4
+Methodology
+As part of this study, a preliminary version of the edge-only
+variant of the architecture was deployed on an active construc-
+tion site. Using the Orange AI&ML Platform, a model was
+trained to count people wearing helmets entering and exiting a
+specific area. The system counts people in helmets in defined
+recognition areas (bounding boxes), crossing the yellow and
+green lines visible in Figs. 3 and 4. People entering the con-
+struction site are counted after crossing the green line, while
+people leaving are counted after crossing the yellow line. The
+machine learning pipeline consists of a YOLOX object detec-
+tion model, trained for detecting heads in helmets, and a Deep-
+SORT [12] multi-object tracking algorithm. The YOLOX
+model was trained using a dataset provided by the Northeast-
+ern University of China (https://public.roboflow.com/object-
+detection/hard-hat-workers).
+The system’s results were compared to those obtained
+from an algorithm built into the Dahua camera. It should
+be noted that the camera counted all people entering and leav-
+ing, including those without protective helmets. However, this
+should not impact the results much, as the safety regulations
+on this particular site forbid entering it without a helmet and
+the rule is strictly enforced before workers reach the counting
+location.
+The measurements were performed in two series – each
+using a different bounding box definition. A single series
+spanned the length of one workday on the construction site.
+The number of entering and leaving people was counted in
+hourly intervals (between 5 AM and 7 PM).
+5
+Results
+The Tables 1 and 2 present the results of the performed exper-
+iments. The Table 1 contains measurements made on 22nd
+November 2022, with the bounding box set as presented in
+Fig. 3. The average difference between the number of people
+entering, as measured by the camera and the model was equal
+to −6.21, with the standard deviation of σ = 5.08, whereas
+for people exiting it was 1.35 and σ = 2.73 respectively. The
+correlation between entrances detected by the camera and the
+model deployed on the AI&ML platform, expressed by the
+Pearson coefficient is 0.988, whereas for exits 0.995. The cor-
+relations were found to be statistically significant (p ≤ 0.05).
+Table 2 contains measurements from 24th November 2022
+(for modified detection areas, depicted in Fig. 4). On that day,
+the average difference for entering was −4.93 with σ = 4.25
+and for exiting 3.92 with σ = 4.92. For these measurements
+the Pearson coefficient for people entering is equal to 0.993
+and exiting 0.989. The correlations were found to be statisti-
+cally significant (p ≤ 0.05).
+Figure 3. Bounding boxes location on November 22 (before
+modification)
+Figure 4. Bounding boxes location on November 24 (after
+modification)
+The tables also present differences in the number of people
+detected by the camera and the AI&ML platform and the sum
+of these differences calculated for both movement directions:
+entries and exits.
+During the experiments, several unexpected events took
+place, which had a significant impact on the reported results.
+Workers were observed acting in an unexpected manner –
+lingering or walking around the detection area (Fig. 5). It
+was also noticed that sometimes the workers put on their
+helmets after having passed the detection area (Fig. 6). These
+behaviors present a challenge to the future system, as they
+significantly affect its accuracy.
+
+Inferen
+POlnferen
+me
+POTowards Edge-Cloud Architectures for Personal Protective Equipment Detection
+Hour
+05:00
+06:00
+07:00
+08:00
+09:00
+10:00
+11:00
+12:00
+13:00
+14:00
+15:00
+16:00
+17:00
+18:00
+Total
+Dahua In
+12
+65
+84
+47
+26
+84
+50
+51
+28
+70
+28
+8
+9
+0
+562
+Dahua Out
+2
+15
+21
+35
+81
+44
+61
+31
+63
+32
+59
+66
+26
+8
+544
+AI&ML In
+11
+78
+87
+52
+28
+96
+60
+58
+43
+75
+33
+18
+10
+0
+649
+AI&ML Out
+2
+13
+23
+33
+73
+44
+63
+31
+58
+31
+59
+62
+25
+8
+525
+Diff. In
+1
+-13
+-3
+-5
+-2
+-12
+-10
+-7
+-15
+-5
+-5
+-10
+-1
+0
+-87
+Diff. Out
+0
+2
+-2
+2
+8
+0
+-2
+0
+5
+1
+0
+4
+1
+0
+19
+Table 1. Entries and exits to the construction site, 22 November 2022.
+Hour
+05:00
+06:00
+07:00
+08:00
+09:00
+10:00
+11:00
+12:00
+13:00
+14:00
+15:00
+16:00
+17:00
+18:00
+Total
+Dahua In
+4
+57
+113
+62
+34
+73
+75
+65
+56
+93
+27
+10
+9
+0
+678
+Dahua Out
+0
+10
+29
+57
+80
+53
+74
+41
+82
+52
+49
+84
+20
+8
+639
+AI&ML In
+3
+61
+113
+68
+43
+76
+79
+73
+69
+98
+33
+21
+10
+0
+747
+AI&ML Out
+0
+11
+26
+46
+64
+48
+72
+38
+73
+49
+47
+82
+22
+6
+584
+Diff. In
+1
+-4
+0
+-6
+-9
+-3
+-4
+-8
+-13
+-5
+-6
+-11
+-1
+0
+-69
+Diff. Out
+0
+-1
+3
+11
+16
+5
+2
+3
+9
+3
+2
+2
+-2
+2
+55
+Table 2. Entries and exits to the construction site, 24 November 2022.
+6
+Concluding remarks
+The tested model demonstrated relatively good performance
+in the investigated scenario. Its accuracy when tasked with
+counting people wearing protective helmets was found to be
+sufficient, and was validated against a different system. A
+number of discrepancies between the counts of the model and
+the camera can be attributed to unexpected situations (Figs. 5
+and 6) and the fact that the Dahua camera did not differentiate
+people wearing and not wearing helmets. The high correla-
+tion coefficient between the camera and the Orange AI&ML
+Platform’s model allows to conclude that the two solutions
+perform comparably well.
+It should be noted that there were changes in the correla-
+tion between the days of experiments. These differences are
+explained by the changes to the bounding box. This is one of
+the parameters that have to be investigated further.
+Both variants of the proposed architecture can be used
+in the investigated scenario of PPE detection on a construc-
+tion site. The feasibility of using an edge-deployment was
+confirmed – the server’s computational capabilities were suf-
+ficient to maintain satisfactory inference accuracy. Therefore,
+it can be concluded that the construction site is equipped with
+sufficient hardware to warrant further experiments with the
+deployment.
+In the future, the two proposed architecture variants will be
+compared in terms of network latencies, resource utilization,
+and their accuracy. The presented model will also be tested
+further, which will include manually annotating the videos to
+obtain a ground truth for comparison. This will allow for de-
+termining the actual accuracy of the developed model. Further
+optimization of bounding box locations is also planned.
+Acknowledgments
+Work supported by ASSIST-IoT project funded from the Eu-
+ropean Union’s H2020 RIA program under grant 957258.
+References
+[1] Alexey Bochkovskiy, Chien-Yao Wang, and Hong-Yuan Mark Liao.
+2020. YOLOv4: Optimal speed and accuracy of object detection. arXiv
+preprint arXiv:2004.10934 (2020).
+[2] Marco di Benedetto, Enrico Meloni, Giuseppe Amato, Fabrizio Falchi,
+and Claudio Gennaro. 2019. Learning Safety Equipment Detection
+using Virtual Worlds. In 2019 International Conference on Content-
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+
+RC

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+page_content='pl  Systems Research Institute Polish  Academy of Sciences  Warsaw, Poland  Abstract  Detecting Personal Protective Equipment in images and video  streams is a relevant problem in ensuring the safety of con-  struction workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In this contribution, an architecture en-  abling live image recognition of such equipment is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The solution is deployable in two settings – edge-cloud and  edge-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The system was tested on an active construction  site, as a part of a larger scenario, within the scope of the  ASSIST-IoT H2020 project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' To determine the feasibility of  the edge-only variant, a model for counting people wearing  safety helmets was developed using the YOLOX method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It  was found that an edge-only deployment is possible for this  use case, given the hardware infrastructure available on site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  In the preliminary evaluation, several important observations  were made, that are crucial to the further development and de-  ployment of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Future work will include an in-depth  investigation of performance aspects of the two architecture  variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Keywords: edge-cloud continuum architectures, PPE detec-  tion, image recognition, worker safety  1  Introduction  Nowadays, the demand for intelligent video analytics is grow-  ing across a wide spectrum of application areas [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The  key part of such systems is usually an image recognition (IR)  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' However, as of today, the IR subsystem is, most  commonly, deployed in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This approach offers mul-  tiple benefits, such as availability of large and scalable com-  putational resources, reliable APIs, and shifting the burden  of system maintenance to the cloud service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' How-  ever, this comes at a cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Sending data to the cloud raises  both security and privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Moreover, communicat-  ing with the cloud always induces network latency, which  may be significant in time-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' To address  issues brought about by cloud-centric solutions, edge com-  puting has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Here, the core of the approach is  processing the data as close to the source as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This  allows for latency reduction, and helps ensure the security  and privacy of data, which remains within the local network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  However, edge computing has its own set of issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Typically,  the computational resources, which are available at the edge  are considerably smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A possible solution to addressing  the downsides of both these options is a combined approach –  an edge-cloud continuum, where data is partially processed  on the edge and partially in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' However, this raises  the obvious question: at which point(s), within the continuum,  individual parts of the system should be deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Here, this question is considered within a real-world sce-  nario of monitoring the entrance to an active construction  site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Specifically, the system is tasked with ensuring that (1)  no unauthorized people enter the worksite, and (2) every-  body is wearing appropriate Personal Protective Equipment  (PPE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' helmets and safety vests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The scenario is evalu-  ated as part of the ASSIST-IoT project, on a construction site  in Warsaw, Poland, managed by the construction company  Mostostal Warszawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Here, the edge versus cloud discussion  becomes particularly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' On the one hand, the privacy  of workers is of paramount importance, while latencies must  be minimized, to ensure a quick reaction, which hints at an  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='01501v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='CV]  4 Jan 2023  Legierski and Rachwał et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  edge deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' On the other hand, given the limited hard-  ware resources available on the edge, and the extremely harsh  conditions of the construction site, a cloud deployment seems  attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Given the possible benefits of both solutions, in this contri-  bution, a solution is proposed for an edge-cloud continuum  video analytics architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The architecture can be deployed  in two variants (edge-only, and edge-cloud), described in the  Architecture section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Moreover, to determine the viability of  the solution, an initial experimental study was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Here, an IR model was developed and integrated with the  edge-only variant of the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Next, it was tasked with  detecting when personnel wearing PPE entered and exited the  work site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  2  Background  To provide a context for this study, the state of the art of (1)  IR system architectures and (2) machine learning models for  PPE detection is summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  System architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The most obvious benefit of deploy-  ing IR systems on the edge is the decreased latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This was  demonstrated in [20], where facial recognition models were  deployed on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The authors found that deploying the  models on the edge resulted in significantly better response  speeds, as compared to a cloud deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In other stud-  ies [8, 9], the viability of deploying deep convolutional neural  networks (CNNs) in the edge-only scenario was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  CNNs are characterized by high resource utilization, and thus  are typically deployed in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The studies found that  deploying CNNs is viable on mobile devices, when parts of  the computation can be offloaded to other edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Edge  deployment allowed to achieve a consistently low latency  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='24 ms while using CNNs to perform real-time object  tracking in augmented reality [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Both studies showed that  the edge deployment distributing the workload increased the  inference capabilities of the system, as the models could not  be run on the disconnected mobile devices alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  One study [5] investigated an edge-cloud architecture, where  data preprocessing servers were deployed close to the data  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The preprocessed data was then sent to a cloud-based  deep-learning platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This resulted in decreasing network  latency and traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It also increased the security and privacy  of the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  On the other hand, edge deployments are more limited  in terms of the available hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Low computational re-  sources naturally limit the size of models and inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  A study compared different implementations (based on Ten-  sorFlow, TensorRT, and TFLite) of the same video processing  model [6], and found them to differ in their resource utiliza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The choice of implementation influenced the energy  consumption of the model, as well as its inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Interestingly, the slowest implementation (TFLite) was the  most energy efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It was also found that TFLite managed  to remain on par with the other implementations in terms of  speed, when processing low-resolution video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In the case of  high-resolution video, more resource-intensive models were  needed to maintain the speed, suggesting that a cloud deploy-  ment could be more beneficial in low-resource settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Nev-  ertheless, some resource-intensive models can be deployed  on the edge, if resources available there are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The  deployment proposed in a different study required all nodes  to be equipped with a GPU [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This allowed the authors to  use CNNs on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A similar result was reported in [13],  were IR models deployed on a Raspberry Pi 4B, equipped  with a camera, and an Intel Neural Compute Stick 2 (a USB  device for deep learning inference on the edge) were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  These devices were chosen for their low power consumption  and good computing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Overall, a model tasked  with detecting PPE in the form of helmets and safety vests  achieved precision on the order of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Models for PPE detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Effective video analytics-based  methods for detecting the presence of protective helmets,  worn by workers, due to its health and safety importance, is  currently a hot research topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The usage of existing, unmodified machine learning models  for detecting protective head covers does not provide suffi-  cient detection accuracy, as proven in a recent study [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  In said article, several versions of the popular YOLO algo-  rithm [1] were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It was shown that the most effective  version of YOLO for helmet detection is the v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' After improv-  ing the loss function, it achieved more than 93% accuracy  during tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A similar study [10] focused on improving the  YOLOv5 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The system achieved results close to 97%  accuracy, thanks to the improvement of the structure of the  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Another study [18], also investigated improv-  ing YOLOv5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' However, instead of the algorithm itself, work  was focused on processing of input data by applying filters  on the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This allowed to improve the accuracy to  above 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Yet another study [15] presented an approach for  improving the detection speed and accuracy by designing a  multi-level pyramidal feature fusion network based on the  ConCaNet attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Here, YOLOv3 was applied  and a dataset with 6000 images was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The results demon-  strate the effectiveness of this approach, which managed to  reduce the number of necessary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Helmet detection can also be done using the SSD-MobileNet  algorithm [4], which is based on yet another variant of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  An analysis of this method, reported in [7], tested its effec-  tiveness and managed to reach 80% accuracy during tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  In a wider comparison of algorithm types [11], the authors  proposed a helmet detection method based on a dynamically  changing neural network – SHDDM (Safety Helmet Detec-  tion Dynamic Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The developed model analyzes the  human posture and defines the area where the helmet should  be located, to eliminate the detection of the helmet outside  the head area and thus reduce the false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' There are  Towards Edge-Cloud Architectures for Personal Protective Equipment Detection  also other approaches to helmet detection, such as methods  based on color and shape used to to locate the face, and the  proper wearing of a helmet [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Another solution used low-  resolution images, captured from a video stream, using the  Local Binary Pattern (LBP) and gray-level co-occurrence ma-  trix (GLCM) methods along with a back-propagation neural  network [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Another study [2] investigated the usefulness of artificially  created images in the training of CNNs for PPE detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The  paper presented the results achieved with YOLOv3, trained on  artificial images generated by the Rockstar Advanced Game  Engine (RAGE) from the Grand Theft Auto V video game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  This approach achieved a mean average precision (mAP) of  only 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='11% on a test dataset consisting of real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The mAP for synthetic images was much higher at 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='24%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  It should be noted that the poor results for the real-world  images are most likely caused by the RAGE engine being  unable to generate a sufficient amount of head, welding mask,  ear protection, and chest object variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  As can be seen, there are many possibilities for detecting  protective helmets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Here, the SHDDM is particularly note-  worthy, as it has an important feature of checking whether the  helmet is worn properly, and not only detecting its presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  This, in turn, is particularly relevant in real-world applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  3  Proposed Architecture  The proposed video analytics system can be deployed in  two architecture variants: edge-cloud (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 1) and edge-only  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' As outlined above, there are reasons to believe that  both variants may be appropriate for the considered scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Both architectures share a common core deployed on the edge,  consisting of: a camera, the Image Processor (IP) component,  and the OSH (Occupational Safety and Health) manager’s  mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The camera (in the reported experiments the Dahua IPC-  HFW5449T-ASE-LED was used) provides a live RTSP video  stream, which is directed to the Image Processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The IP is  a service written in Python, which can optionally perform  preliminary image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Using configurable methods such  as motion detection and brightness thresholding, the IP is able  to discard image frames that do not contain moving people,  reducing network traffic to components involved in actual im-  age analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It is also responsible for communicating with the  rest of the system, designed in accordance with the ASSIST-  IoT reference architecture [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' IP communicates with the rest  of the system publishing alerts to an MQTT topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This de-  sign allows other components and devices in the ASSIST-IoT  deployment to be notified in a streaming manner of any OSH  violations, such as workers not wearing protective helmets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  In the first version of the architecture – the edge-cloud  deployment – the IP is configured to use the cloud-based  AWS Rekognition platform, with its PPE detection service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content="  ASSIST-IoT deployment  AWS  Rekognition  Video stream  OSH alert  Image Processor  Camera  OSH manager's  mobile device  Edge  Filtered  video stream  Inference  results  AWS Cloud  Figure 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=" Edge-cloud deployment  Video stream  OSH alert  Image Processor  Camera  OSH manager's  mobile device  Edge  Orange AI&ML Platform  Video stream  Processing pipeline  AI-X  AI-2  AI-1  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Result adaptation  Inference  results  External platform connector  ASSIST-IoT deployment  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Edge deployment  In the edge-only variant, the video analysis is performed by  the Orange AI&ML Platform, which is deployed on a server  on the construction site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This edge deployment allows for  maintaining lower network latency, and ensures the privacy  of worker data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The AI&ML Platform’s services are written  as Python runnable modules that provide their own APIs and  GUIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The services can reuse the APIs and GUIs provided  by the platform, or build them from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A service col-  lects frames from a video source, processes them in an ML  pipeline specific to the service, and adapts or interprets the  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The inference results from the Platform are forwarded  to external services, with the use of provided connectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' As  the Orange AI&ML Platform operates on the edge, all video  Legierski and Rachwał et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  processing takes place on the client’s site, ensuring full secu-  rity of customer data (video) and compliance with appropriate  regulations, such as GDPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  4  Methodology  As part of this study, a preliminary version of the edge-only  variant of the architecture was deployed on an active construc-  tion site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Using the Orange AI&ML Platform, a model was  trained to count people wearing helmets entering and exiting a  specific area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The system counts people in helmets in defined  recognition areas (bounding boxes), crossing the yellow and  green lines visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' People entering the con-  struction site are counted after crossing the green line, while  people leaving are counted after crossing the yellow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The  machine learning pipeline consists of a YOLOX object detec-  tion model, trained for detecting heads in helmets, and a Deep-  SORT [12] multi-object tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The YOLOX  model was trained using a dataset provided by the Northeast-  ern University of China (https://public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='roboflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='com/object-  detection/hard-hat-workers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The system’s results were compared to those obtained  from an algorithm built into the Dahua camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It should  be noted that the camera counted all people entering and leav-  ing, including those without protective helmets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' However, this  should not impact the results much, as the safety regulations  on this particular site forbid entering it without a helmet and  the rule is strictly enforced before workers reach the counting  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The measurements were performed in two series – each  using a different bounding box definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A single series  spanned the length of one workday on the construction site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  The number of entering and leaving people was counted in  hourly intervals (between 5 AM and 7 PM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  5  Results  The Tables 1 and 2 present the results of the performed exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The Table 1 contains measurements made on 22nd  November 2022, with the bounding box set as presented in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The average difference between the number of people  entering, as measured by the camera and the model was equal  to −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='21, with the standard deviation of σ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='08, whereas  for people exiting it was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='35 and σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='73 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The  correlation between entrances detected by the camera and the  model deployed on the AI&ML platform, expressed by the  Pearson coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='988, whereas for exits 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The cor-  relations were found to be statistically significant (p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Table 2 contains measurements from 24th November 2022  (for modified detection areas, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' On that day,  the average difference for entering was −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='93 with σ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='25  and for exiting 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='92 with σ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' For these measurements  the Pearson coefficient for people entering is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='993  and exiting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The correlations were found to be statisti-  cally significant (p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Bounding boxes location on November 22 (before  modification)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Bounding boxes location on November 24 (after  modification)  The tables also present differences in the number of people  detected by the camera and the AI&ML platform and the sum  of these differences calculated for both movement directions:  entries and exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  During the experiments, several unexpected events took  place, which had a significant impact on the reported results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Workers were observed acting in an unexpected manner –  lingering or walking around the detection area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' It  was also noticed that sometimes the workers put on their  helmets after having passed the detection area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' These  behaviors present a challenge to the future system, as they  significantly affect its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Inferen  POlnferen  me  POTowards Edge-Cloud Architectures for Personal Protective Equipment Detection  Hour  05:00  06:00  07:00  08:00  09:00  10:00  11:00  12:00  13:00  14:00  15:00  16:00  17:00  18:00  Total  Dahua In  12  65  84  47  26  84  50  51  28  70  28  8  9  0  562  Dahua Out  2  15  21  35  81  44  61  31  63  32  59  66  26  8  544  AI&ML In  11  78  87  52  28  96  60  58  43  75  33  18  10  0  649  AI&ML Out  2  13  23  33  73  44  63  31  58  31  59  62  25  8  525  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In  1  13  3  5  2  12  10  7  15  5  5  10  1  0  87  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Out  0  2  2  2  8  0  2  0  5  1  0  4  1  0  19  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Entries and exits to the construction site, 22 November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Hour  05:00  06:00  07:00  08:00  09:00  10:00  11:00  12:00  13:00  14:00  15:00  16:00  17:00  18:00  Total  Dahua In  4  57  113  62  34  73  75  65  56  93  27  10  9  0  678  Dahua Out  0  10  29  57  80  53  74  41  82  52  49  84  20  8  639  AI&ML In  3  61  113  68  43  76  79  73  69  98  33  21  10  0  747  AI&ML Out  0  11  26  46  64  48  72  38  73  49  47  82  22  6  584  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In  1  4  0  6  9  3  4  8  13  5  6  11  1  0  69  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Out  0  1  3  11  16  5  2  3  9  3  2  2  2  2  55  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Entries and exits to the construction site, 24 November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  6  Concluding remarks  The tested model demonstrated relatively good performance  in the investigated scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Its accuracy when tasked with  counting people wearing protective helmets was found to be  sufficient, and was validated against a different system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' A  number of discrepancies between the counts of the model and  the camera can be attributed to unexpected situations (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' 5  and 6) and the fact that the Dahua camera did not differentiate  people wearing and not wearing helmets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The high correla-  tion coefficient between the camera and the Orange AI&ML  Platform’s model allows to conclude that the two solutions  perform comparably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  It should be noted that there were changes in the correla-  tion between the days of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' These differences are  explained by the changes to the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This is one of  the parameters that have to be investigated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Both variants of the proposed architecture can be used  in the investigated scenario of PPE detection on a construc-  tion site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The feasibility of using an edge-deployment was  confirmed – the server’s computational capabilities were suf-  ficient to maintain satisfactory inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Therefore,  it can be concluded that the construction site is equipped with  sufficient hardware to warrant further experiments with the  deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  In the future, the two proposed architecture variants will be  compared in terms of network latencies, resource utilization,  and their accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' The presented model will also be tested  further, which will include manually annotating the videos to  obtain a ground truth for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' This will allow for de-  termining the actual accuracy of the developed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Further  optimization of bounding box locations is also planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Acknowledgments  Work supported by ASSIST-IoT project funded from the Eu-  ropean Union’s H2020 RIA program under grant 957258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  References  [1] Alexey Bochkovskiy, Chien-Yao Wang, and Hong-Yuan Mark Liao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' YOLOv4: Optimal speed and accuracy of object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' ASSIST-IoT: A reference architecture for next generation  Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='1109/ICNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='8117585  [6] Anis Koubaa, Adel Ammar, Anas Kanhouch, and Yasser AlHabashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Cloud Versus Edge Deployment Strategies of Real-Time Face  Recognition Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' IEEE Transactions on Network Science and  Engineering 9, 1 (2022), 143–160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Deep Learning-Based Safety Helmet Detection in Engineering  Management Based on Convolutional Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Edge Assisted  Real-Time Object Detection for Mobile Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' In The  25th Annual International Conference on Mobile Computing and  Legierski and Rachwał et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Unexpected worker behavior – staying in the detec-  tion area for longer  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content=' Unexpected worker behavior – putting the helmet  on behind the detection line  Networking (Los Cabos, Mexico) (MobiCom ’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='1145/3213344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' Safety Helmet Wearing  Recognition Based on Improved YOLOv5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='1109/ICTech55460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='  Safety Helmet Detection Dynamic Model Based on the Critical Area  Attention Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
+page_content='9661778  [14] JIANG Xinhua, XUE Heru, ZHANG Lina, and ZHOU Yanqing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AzT4oBgHgl3EQfif2Z/content/2301.01501v1.pdf'}
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EdA0T4oBgHgl3EQfA_-G/content/tmp_files/2301.01970v1.pdf.txt

@@ -0,0 +1,2269 @@
+CAT: LoCalization and IdentificAtion Cascade Detection Transformer
+for Open-World Object Detection
+Shuailei Ma 1* Yuefeng Wang1† Jiaqi Fan1 Ying Wei1‡
+Thomas H. Li3 Hongli Liu2 Fanbing Lv2
+1Northeast University, 2Changsha Hisense Intelligent System Research Institute Co., Ltd.
+3Information Technology R&D Innovation Center of Peking University,
+Abstract
+Open-world object detection (OWOD), as a more gen-
+eral and challenging goal, requires the model trained from
+data on known objects to detect both known and unknown
+objects and incrementally learn to identify these unknown
+objects.
+The existing works which employ standard de-
+tection framework and fixed pseudo-labelling mechanism
+(PLM) have the following problems: (𝑖) The inclusion of de-
+tecting unknown objects substantially reduces the model’s
+ability to detect known ones. (𝑖𝑖) The PLM does not ade-
+quately utilize the priori knowledge of inputs. (𝑖𝑖𝑖) The fixed
+selection manner of PLM cannot guarantee that the model
+is trained in the right direction. We observe that humans
+subconsciously prefer to focus on all foreground objects and
+then identify each one in detail, rather than localize and
+identify a single object simultaneously, for alleviating the
+confusion. This motivates us to propose a novel solution
+called CAT: LoCalization and IdentificAtion Cascade De-
+tection Transformer which decouples the detection process
+via the shared decoder in the cascade decoding way. In the
+meanwhile, we propose the self-adaptive pseudo-labelling
+mechanism which combines the model-driven with input-
+driven PLM and self-adaptively generates robust pseudo-
+labels for unknown objects, significantly improving the abil-
+ity of CAT to retrieve unknown objects. Comprehensive ex-
+periments on two benchmark datasets, 𝑖.𝑒., MS-COCO and
+PASCAL VOC, show that our model outperforms the state-
+of-the-art in terms of all metrics in the task of OWOD, in-
+cremental object detection (IOD) and open-set detection.
+1. Introduction
+Open-world object detection (OWOD) is a more prac-
+tical detection problem in computer vision, making artifi-
+*First author. Email: xiaomabufei@gmail.com
+†Code url: https://github.com/xiaomabufei/CAT
+‡Corresponding author. Email: weiying@ise.neu.edu.cn
+Bear
+Frog
+Flower
+Squirrel
+Unknown
+Cat
+BeeUnknown
+Unknown
+Unknown
+Unknown
+Figure 1. When faced with new scenes in open world, humans sub-
+consciously focus on all foreground objects and then identify them
+in detail in order to alleviate the confusion between the known and
+unknown objects and get a clear view. Motivated by this, our CAT
+utilizes the shared decoder to decouple the localization and iden-
+tification process in the cascade decoding way, where the former
+decoding process is used for localization and the latter for identi-
+fication.
+cial intelligence (AI) smarter to face more difficulties in real
+scenes. Within the OWOD paradigm, the model’s life-span
+is pushed by iterative learning process. At each episode, the
+model trained only by known objects needs to detect known
+objects while simultaneously localizing unknown objects
+and identifying them into the unknown class. Human an-
+notators then label a few of these tagged unknown classes
+of interest gradually. The model given these newly-added
+annotations will continue to incrementally update its knowl-
+edge without retraining from scratch.
+Recently, the work [17] proposed an open-world ob-
+ject detector, ORE, based on the two-stage Faster R-CNN
+[33] pipeline. ORE utilizes an auto-labelling step to obtain
+pseudo-unknowns for training model to detect unknown ob-
+jects and learns an energy-based binary classifier to distin-
+guish the unknown class from known classes. However,
+its success largely relies on a held-out validation set which
+1
+arXiv:2301.01970v1  [cs.CV]  5 Jan 2023
+
+is leveraged to estimate the distribution of unknown ob-
+jects in the energy-based classifier. To alleviate the prob-
+lems in ORE, OW-DETR [13] proposes to use the detection
+transformer [3, 38] for OWOD in a justifiable way and di-
+rectly leverages the framework of DDETR [38]. In addi-
+tion, OW-DETR proposes an attention-driven PLM which
+selects pseudo labels for unknown objects according to the
+attention scores.
+For the existing works, we find the following hindering
+problems. (𝑖) Owing to the inclusion of detecting unknown
+objects, the model’s ability to detect known objects substan-
+tially drops. To alleviate the confusion between known and
+unknown objects, humans prefer to dismantle the process of
+open-world object detection rather than parallelly localize
+and identify open-world objects like most standard detec-
+tion models. (𝑖𝑖) To the best of our knowledge, in the exist-
+ing OWOD PLM, models leverage the learning process for
+known objects to guide the generation of pseudo labels for
+unknown objects, without leveraging the prior conditions of
+the inputs (𝑡𝑒𝑥𝑡𝑢𝑟𝑒,𝑙𝑖𝑔ℎ𝑡 𝑓 𝑙𝑜𝑤,𝑒𝑡𝑐). As a result, the model
+cannot learn knowledge beyond the data annotation. (𝑖𝑖𝑖)
+The fixed selection manner of PLM cannot guarantee that
+the model learns to detect unknown objects in the right di-
+rection, due to the uncertain quality of the pseudo labels.
+The models may be worse for detecting unknown objects.
+When faced with a new scene, humans prefer focusing
+on all foreground objects and then analyse them in detail,
+as shown in Figure.1. Motivated by this and the aforemen-
+tioned observations, we propose a novel LoCalization and
+IdentificAtion Cascade Detection Transformer. CAT com-
+prises three dedicated components namely, self-adaptive
+pseudo-labelling mechanism, shared transformer de-
+coder and cascade decoupled decoding structure. The
+self-adaptive PLM maintains the ability of CAT to ex-
+plore the knowledge beyond the known objects and self-
+adaptively adjusts the pseudo-label generation according to
+the model training process. Via the cascade decoupled de-
+coding structure, the shared transformer decoder decouples
+the localization and identification process for alleviating the
+influence of detecting unknown objects on the detection of
+known objects, where the former decoding process is used
+for localization and the latter for identification. In the mean-
+while, we observe the structure substantially improves the
+model’s ability for incremental object detection according
+to the experiments. In addition, we explore the decoupled
+structures for detection transformer. Our contributions can
+be summarized fourfold:
+• We propose a novel localization and identification cas-
+cade detection transformer (CAT), which decouples
+the localization and identification process of detection
+and alleviates the influence of detecting unknown ob-
+jects on the detection of known ones.
+• We introduce a novel pseudo-labelling mechanism
+which self-adaptively combines the model-driven and
+input-driven pseudo-labelling during the training pro-
+cess for generating robust pseudo-labels and exploring
+knowledge beyond known objects.
+• We explore the decoupled decoding methods of the de-
+tection transformer, 𝑖.𝑒., the fully decoupled decoding
+structure and the cascade decoupled decoding struc-
+ture.
+• Our extensive experiments on two popular bench-
+marks demonstrate the effectiveness of the proposed
+CAT. CAT outperforms the recently introduced ORE
+and OW-DETR for OWOD, IOD and open-set detec-
+tion. For OWOD, CAT achieves absolute gains ranging
+from 11.8% to 18.3% in terms of unknown recall over
+OW-DETR.
+2. Problem Formulation
+At time 𝑡, let K𝑡 = {1,2,...,𝐶} denote the set of known
+object classes and U𝑡 = {𝐶 + 1,...} denote the unknown
+classes which might be encountered at the test time. The
+known object categories K𝑡 are labeled in the dataset
+D𝑡 = {J 𝑡,L𝑡} where J 𝑡 denotes the input images and
+L𝑡 denotes the corresponding labels at time 𝑡. The train-
+ing image set consists of 𝑀 images J 𝑡 = {𝑖1,𝑖2,...,𝑖𝑀 }
+and corresponding labels L𝑡 = {ℓ1,ℓ2,...,ℓ𝑀 }. Each ℓ𝑖 =
+{T1,T2,...,T𝑁 } denotes a set of 𝑁 object instances with
+their class labels 𝑐𝑛 ⊂ K𝑡 and locations, 𝑥𝑛, 𝑦𝑛,𝑤𝑛, ℎ𝑛
+denote the bounding box center coordinates, width and
+height respectively. The Open-World Object Detection re-
+moves the artificial assumptions and restrictions in tradi-
+tional object detection and makes object detection tasks
+more aligned with real life. It requires the trained model
+M𝑡 not only to detect the previously encountered known
+classes 𝐶 but also to identify an unseen class instance as
+belonging to the unknown class. In addition, it requires the
+object detector to be capable of incremental update for new
+knowledge and this cycle continues over the detector’s lifes-
+pan. In incremental updating phase, the unknown instances
+identified by M𝑡 are annotated manually, and along with
+their corresponding training examples, update D𝑡 to D𝑡+1
+and K𝑡 to K𝑡+1 = {1,2,...,𝐶,...,𝐶 +n}, the model adds the
+𝑛 new classes to known classes and updates itself to M𝑡+1
+without retraining from scratch on the whole dataset D𝑡+1.
+3. Proposed method
+This section elaborates the proposed CAT in details. In
+Sec.3.1, the overall architecture of CAT is described in de-
+tail. A novel self-adaptive adjustment strategy for pseudo-
+labelling is proposed in Sec.3.2. We explore to decouple
+the decoding process of the detection transformer and pro-
+pose the localization and identification cascade decoupled
+2
+
+Figure 2. Overall Architecture of proposed CAT framework. The proposed CAT consists of a multi-scale feature extractor, the shared trans-
+former decoder, the regression prediction branch, and the self-adaptive pseudo-labelling. The multi-scale feature extractor comprises the
+mainstream feature extraction backbone and a deformable transformer encoder, for extracting multi-scale features. The shared transformer
+decoder is a deformable transformer decoder and decouples the localization and identification process in the cascade decoding way. The
+regression prediction branch contains the bounding box regression branch 𝐹𝑟𝑒𝑔, novelty objectness branch 𝐹𝑜𝑏 𝑗, and novelty classification
+branch 𝐹𝑐𝑙𝑠. While the novelty classification and objectness branches are single-layer feed-forward networks (FFN) and the regression
+branch is a 3-layer FFN.
+decoding structure in Sec.3.3. In Sec.3.4, we illustrate the
+end-to-end training strategy of CAT.
+3.1. Overall Architecture
+As shown in Figure.2, for a given image J ∈ R𝐻×𝑊 ×3,
+CAT uses a hierarchical feature extraction backbone to
+extract multi-scale features Z𝑖 ∈ R
+H
+4×𝑖2 ×
+𝑤
+4×2𝑖 ×2𝑖𝐶𝑠,𝑖 = 1,2,3.
+The feature maps 𝑍𝑖 are projected from dimension 𝐶𝑠
+to dimension 𝐶𝑑 by using 1×1 convolution and concate-
+nated to 𝑁𝑠 vectors with 𝐶𝑑 dimensions after flattening
+out.Afterwards, along with supplement positional encod-
+ing 𝑃𝑛 ∈ R𝑁𝑠×𝐶𝑑, the multi-scale features are sent into the
+deformable transformer encoder to encode semantic fea-
+tures. The encoded semantic features 𝑀 ∈ R𝑁𝑠×𝑐𝑑 are ac-
+quired and sent into the shared decoder together with a
+set of 𝑁 learnable location queries and positional embed-
+dings 𝑃𝑚 ∈ R𝑁𝑠×𝐶𝑑. Aided by interleaved cross-attention
+and self-attention modules, the shared decoder transforms
+the location queries Q location ∈ R𝑁 ×𝐷 to a set of N loca-
+tion query embeddings E location ∈ R𝑁 ×𝐷. The Elocation are
+then input to the regression branch to locate N foreground
+bounding boxes containing the known classes and unknown
+classes. Meanwhile, the E location are used as class queries
+and sent into the shared decoder together with the 𝑀 and
+𝑃𝑚 again. The shared decoder transforms the class queries
+to 𝑁 class query embeddings Eclass that are corresponding to
+the location query embeddings. The Eclass are then sent into
+the objectness and novelty classification branch to predict
+the objectness and category respectively. After selecting the
+unique queries that best match the known instances by a bi-
+partite matching loss, the remaining queries are utilized to
+select the unknown category instances and generate pseudo
+labels by self-adaptive pseudo-labelling mechanism.
+3.2. Self-Adaptive Pseudo-labelling
+Pseudo labels play an important role in guiding mod-
+els to detect unknown object instances, determining the up-
+per learning limitation of the model. The existing meth-
+ods [13,17] only use model-driven pseudo-labelling and do
+not take full advantage of the inputs’ priori knowledge (light
+flow, textures, 𝑒𝑡𝑐).
+The model-driven pseudo-labelling
+[13] makes the model’s learning get caught up in the knowl-
+edge of known objects, for the reason that the only source
+of knowledge for the model is known object instances.
+In addition, their fixed selection manner cannot guarantee
+the right learning direction for unknown objects. We pro-
+pose to combine model-driven with input-driven pseudo-
+labelling [31, 36, 39] for expanding the knowledge sources
+of the model. In the meanwhile, the pseudo-labels selec-
+tion scheme should not be fixed, but be adapted as train-
+ing and able to adjust itself when facing the unexpected
+problems. In this paper, a novel pseudo-labelling mech-
+anism is proposed for self-adaptively combining model-
+driven and input-driven pseudo-labelling according to the
+situation faced by the model, where the attention-driven
+pseudo-labelling [13] is used as the model-driven pseudo-
+3
+
+Shared Decoder
+Multi-Scale Feature
+Decoder
+Decoder
+Decoder
+Freg
+Layer 1
+Layer 2
+Layer N
+Deformable
+Transformer
+Encoder
+Fobj
+Decoder
+Decoder
+Decoder
+Layer 1
+Layer 2
+Layer N
+Fcls
+Shared Decoder
+human
+cup
+unknown
+Positional Encoding
+Self-Adaptive Pseudo-Labelling
+Pseudo labels
+Model-driven
+Positional Embeddings
+Pseudo-labelling
+Self-Adaptive
+Location Queries
+Adjustment Strategy
+Location Embeddings
+Input-driven
+Class Queries
+Pseudo-labelling
+Class Embeddingslabelling and selective search [36] is selected as the input-
+driven pseudo-labelling. In self-adaptive pseudo-labelling
+mechanism, the model-driven pseudo-labelling generates
+pseudo-labels’ candidate boxes 𝑃𝑚 and the corresponding
+confidence 𝑠𝑜, and the input-driven pseudo-labelling gen-
+erates pseudo-label candidate boxes 𝑃𝐼 . The object confi-
+dence of generated pseudo labels is formulated as follows:
+S𝑖 = (𝑛𝑜𝑟𝑚 (𝑠𝑜))W𝑚 ·
+�
+max
+1≤ 𝑗 ≤|P𝐼 |
+�
+IOU
+�
+𝑃𝐼
+𝑗, 𝑃𝑚
+𝑖
+��� W𝐼
+, (1)
+where IOU(·) (Intersection-over-union [34]) is the most
+commonly used metric for comparing the similarity be-
+tween two arbitrary shapes, 𝑖 denotes the index of the
+pseudo labels. W𝑚 and W𝐼 are the self-adaptive weights,
+which are controlled by the 𝑀𝑒𝑎𝑠��𝑟𝑒𝑟, 𝑆𝑒𝑛𝑠𝑜𝑟 and
+𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑟, as formulated below:
+W𝑡 = 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑟(W𝑡−1, 𝑆𝑒𝑛𝑠𝑜𝑟(𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑟(𝐿𝑚))), (2)
+where 𝐿𝑚 represents the loss memory which is stored and
+updated in real time during model training. The formulation
+is illustrated in Equation.3:
+𝐿𝑚 = DEQUE(𝑙𝑜𝑠𝑠𝑡−1,𝑙𝑜𝑠𝑠𝑡−2,··· ,𝑙𝑜𝑠𝑠𝑡−𝑛),
+(3)
+where 𝑡 is the current iteration. Considering the sensitivity
+of the model and the uneven quality of the data, we leverage
+𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑟 to obtain the trend of the losses Δ𝑙 for replacing
+the single loss. The formula is as follows:
+𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑟(𝐿𝑚) =
+�𝑛
+𝑖=1 𝛼𝑖 · 𝑙𝑜𝑠𝑠𝑡−𝑖
+�𝑁
+𝑗=𝑛+1 𝛽 𝑗 · 𝑙𝑜𝑠𝑠𝑡− 𝑗
+,
+𝑛 < 𝑁 < 𝑇,
+(4)
+where 𝛼 and 𝛽 denote the weighted average weights and
+�𝑛
+𝑖=1 𝛼𝑖 = �𝑁
+𝑗=𝑛+1 𝛽 𝑗 = 𝛼𝑖−𝛼𝑖−1
+𝛼𝑖+1−𝛼𝑖 = 𝛽𝑗−𝛽𝑗−1
+𝛽𝑗+1−𝛽𝑗 = 1. In the 𝑆𝑒𝑛𝑠𝑜𝑟,
+the variable of the weight Δ𝑤 is acquired as follows:
+𝑆𝑒𝑛𝑠𝑜𝑟(Δ𝑙) =
+� 𝜋𝑛𝑚𝑎 · 𝑆𝑖𝑔𝑚𝑜𝑖𝑑(Δ𝑙 −1),Δ𝑙 > 1,
+−𝜋𝑝𝑚𝑎 ·Δ𝑙,Δ𝑙 ≤ 1,
+(5)
+where 𝜋𝑝𝑚𝑎 and 𝜋𝑛𝑚𝑎 represents the positive and negative
+momentum amplitude, respectively. In the 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑟, we
+use Equation.6 to update the self-adaptive weight via a in-
+cremental way [5,14,17], for memory storage and enhanc-
+ing the robustness (more explanations in Appendix A.1).
+���
+���
+W 𝑡
+𝑚 = W 𝑡−1
+𝑚
++Δ𝑤 ×W 𝑡−1
+𝑚
+,
+W 𝑡
+𝐼 = W 𝑡−1
+𝐼
+−Δ𝑤 ×W 𝑡−1
+𝐼
+,
+W 𝑡
+𝑚,W 𝑡
+𝐼 = 𝑛𝑜𝑟𝑚 �W 𝑡
+𝑚,W 𝑡
+𝐼
+� ,
+(6)
+where 𝑛𝑜𝑟𝑚(·) is the normalization operation. The update
+strategy for the weights during training is shown in Algo-
+rithm.1.
+Algorithm 1 COMPUTINGADAPTIVEWEIGHTS
+Input: Loss Memory: 𝐿𝑚; Current Interation: 𝑡; Positive
+Momentum Amplitude: 𝜋𝑝𝑚𝑎; Negative Momentum
+Amplitude: 𝜋𝑛𝑚𝑎; 𝑇𝑠𝑡𝑎𝑟𝑡: Start iteration; 𝑇𝑏: Weight
+updating cycle; Loss← Compute using Equation.11
+Output: self-adaptive weights 𝑊𝑚𝑡 and 𝑊𝐼 𝑡
+1: while 𝑡𝑟𝑎𝑖𝑛 do
+2:
+if 𝑡 ≤ 𝑇𝑠𝑡𝑎𝑟𝑡 then
+3:
+Initialise 𝑊𝑚0 ← 0.8 and 𝑊𝐼 0 ← 0.2
+4:
+Initialise 𝐿𝑚 using Equation.3
+5:
+else
+6:
+Update 𝐿𝑚 using Equation.3
+7:
+if 𝑡%𝑇𝑏 == 0 then
+8:
+Compute Δ𝑙 using 𝐿𝑚 and Equation.4
+9:
+Compute Δ𝑤 using Δ𝑙 and Equation.5
+10:
+Update W𝑚𝑡 and W𝐼 𝑡 using Equation.6
+11:
+end if
+12:
+end if
+13: end while
+3.3. Exploration of Decoupled Decoding Structure
+Detection transformer [2, 3, 7, 21, 27, 38] leverages the
+object queries to detect object instances, where each ob-
+ject query represents an object instance.
+In the decod-
+ing stage, the object queries are updated to query embed-
+dings by connecting object queries with semantic informa-
+tion from the encoded semantic features. The generated
+query embeddings couple the location and category infor-
+mation for both object localization and identification pro-
+cess simultaneously. For open-world object detection, the
+model requires to detect the known objects, localize the un-
+known objects and identify them as the unknown class. For
+the parallel decoding structure, we observe that the inclu-
+sion of detecting unknown reduces the model’s ability to
+detect known objects.
+Inspired by how humans subcon-
+sciously confront new scenarios, we propose to decouple
+the decoding process of DETR for mitigating the impact of
+unknown object detection on detecting known objects. In
+this paper, we explore two decoupled decoding ways, 𝑖.𝑒.,
+the fully decoupled decoding structure and the cascade de-
+coupled decoding structure.
+3.3.1
+Fully Decoupled Decoding Structure
+For decoupling the location and category information, an
+intuitive way is to carry out the localization and identifi-
+cation process independently. Motivated by this, the fully
+decoupled decoding structure (FD) is proposed. In the fully
+decoupled decoding structure, location and class queries are
+two sets of mutually independent queries sent to the shared
+decoder. This operation of FD is shown in Figure 3 (a),
+4
+
+Figure 3. (a) The fully decoupled decoding structure has two independent decoding processes for localization and identification. (b) In
+the cascade decoupled decoding structure, the location embeddings are used as class queries for knowledge retention. (c) For the coupled
+decoding structure, the same query is put into the decoder for localization and identification.
+which is formulated as follows:
+ELocation = F𝑠 (F𝑒(∅(J), 𝑃𝑛), 𝑃𝑚,Q Location,R) ,
+(7)
+EClass = F𝑠 (F𝑒(∅(J), 𝑃𝑛), 𝑃𝑚,Q Class ,R) ,
+(8)
+where F𝑠(·) denotes the shared decoder. F𝑒(·) is the en-
+coder and ∅(·) is the backbone. 𝑃𝑛 and 𝑃𝑚 stands for the
+positional encoding and embeddings, respectively. R repre-
+sents the reference points and J denotes the input image.
+Q Class stands for the class queries.
+3.3.2
+Cascade Decoupled Decoding Structure
+Inspired by how people react to new scenarios, a cascade
+decoupled decoding structure is proposed to decode the en-
+coded features in a cascade way so that the localization pro-
+cess is not restricted by the category information, while the
+identification process can get help from the location knowl-
+edge in the cascade structure. The operation of localization
+and identification cascade decoding structure is expressed
+as follows:
+ELocation = F𝑠 (F𝑒(∅(J), 𝑃𝑛), 𝑃𝑚,Q Location,R) ,
+(9)
+EClass = F𝑠 (F𝑒(∅(J), 𝑃𝑛), 𝑃𝑚,ELocation ,R) .
+(10)
+As shown in Figure.3 (b), the location embeddings are used
+as class queries to generate class embeddings.
+3.4. Training and Inference
+Our CAT is trained end-to-end using the following joint
+loss formulation:
+𝐿 = 𝐿𝑙𝑜𝑐𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 + 𝐿𝑖𝑑𝑒𝑛𝑡𝑖 𝑓 𝑖𝑐𝑎𝑡𝑖𝑜𝑛 + 𝐿𝑜𝑏 𝑗𝑒𝑐𝑡𝑛𝑒𝑠𝑠,
+(11)
+where 𝐿𝑙𝑜𝑐𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛, 𝐿𝑖𝑑𝑒𝑛𝑡𝑖 𝑓 𝑖𝑐𝑎𝑡𝑖𝑜𝑛 and 𝐿𝑜𝑏 𝑗𝑒𝑐𝑡𝑛𝑒𝑠𝑠 de-
+notes the loss terms for foreground localization, novelty
+identification and object scoring, respectively. When a set
+of new categories are introduced at each episode, we em-
+ploy an exemplar replay based finetuning to alleviate catas-
+trophic forgetting of learned classes and then finetune the
+model using a balanced set of exemplars stored for each
+known class. The bounding boxes and categories predic-
+tions of the known and 𝑡𝑜𝑝-k unknown objects are simulta-
+neous used during evaluation.
+4. Experiments
+4.1. Datasets and Metrics
+The experiments are implemented on two mainstream
+splits of MS-COCO [23] and Pascal VOC [10] dataset.
+We group the classes into a set of nonoverlapping tasks
+�
+𝑇1,...,𝑇𝑡,...
+�
+. The class in task 𝑇 𝑐 only appears in tasks
+where 𝑡 ≥ 𝑐. In task 𝑇 𝑐, classes encountered in {𝑇 𝑐 : 𝑐 ≤ 𝑡}
+and {𝑇 𝑐 : 𝑐 > 𝑡} are considered as known and unknown
+classes, respectively.
+OWOD SPLIT [17] spilts the 80 classes of MS-COCO into
+4 tasks and selects training set for each task from the MS-
+COCO and Pascal VOC training set images. Pascal VOC
+testing and MS-COCO validation set are used for evalua-
+tion. See more details in Appendix A.2.
+MS-COCO SPLIT [13] mitigates data leakage across tasks
+in [17] and is more challenging. The training and testing
+data are selected from MS-COCO.
+Metrics: Following the most commonly used evaluation
+metric for object detection, we use mean average preci-
+sion (mAP) to evaluate the known objects.
+Inspired by
+[1,9,13,17,25], U-Recall, Wilderness Impact (WI, see de-
+tailed in Appendix A.3) and Absolute Open-Set Error (A-
+OSE) are used as main metric for unknown objects. U-
+Recall measures the ability of the model to retrieve un-
+5
+
+Location Queries
+Location Queries
+Encoded Semantic Features
+Location
+Encoded Semantic Features
+Location
+I Semantic Features
+Object Queries
+Embeddings
+Shared
+Embeddings
+Shared
+Location
+Decoder
+Decoder
+Embeddings
+Class Queries
+Decoder
+Class Queries
+Shared
+Class Embeddings
+Shared
+Decoder
+Decoder
+Class Embeddings
+Class Embeddings
+(a) Fully Decoupled Decoding
+(b) Cascade Decoupled Decoding
+(c) Coupled DecodingTable 1. State-of-the-art comparison on OWOD split. The comparison is shown in terms of U-Recall, WI, A-OSE and known class mAP.
+U-Recall measures the ability of the model to retrieve unknown object instances for OWOD problem. Both WI and A-OSE implicitly
+quantify the effevtiveness of the model in handling unknown objects. For a fair comparison, we compare with the recently introduced
+OW-DETR [13] and ORE [17] not employing EBUI (the results are reproduced by the same GPUs as our model). The CAT achieves
+improved all metrics over the existing works across all tasks, demonstrating our model’s effectiveness for OWOD problem. U-Recall, WI
+and A-OSE cannot be computed in Task 4 due to the absence of unknown test annotations, for the reason that all 80 classes are known.
+Task IDs →
+Task 1
+Task 2
+Task 3
+Task 4
+U-Recall
+WI
+A-OSE
+mAP(↑)
+U-Recall
+WI
+A-OSE
+mAP(↑)
+U-Recall
+WI
+A-OSE
+mAP(↑)
+mAP(↑)
+(↑)
+(↓)
+(↓)
+Current
+known
+(↑)
+(↓)
+(↓)
+Previously
+known
+Current
+known
+Both
+(↑)
+(↓)
+(↓)
+Previously
+known
+Current
+known
+Both
+Previously
+known
+Current
+known
+Both
+Faster-RCNN [33]
+-
+0.0699
+13396
+56.4
+-
+0.0371
+12291
+3.7
+26.7
+15.2
+-
+0.0213
+9174
+2.5
+15.2
+6.7
+0.8
+14.5
+4.2
+Faster-RCNN
++ Finetuning
+Not applicable in Task 1
+-
+0.0375
+12497
+51.0
+25.0
+38.0
+-
+0.0279
+9622
+38.2
+13.6
+30.0
+29.7
+13.0
+25.6
+DDETR [38]
+-
+0.0608
+33270
+60.3
+-
+0.0368
+18115
+4.5
+31.3
+17.9
+-
+0.0197
+9392
+3.3
+22.5
+8.5
+2.5
+16.4
+6.0
+DDETR
++ Finetuning
+Not applicable in Task 1
+-
+0.0337
+17834
+54.5
+34.4
+44.8
+-
+0.0195
+10095
+40.0
+17.8
+33.3
+32.5
+20.0
+29.4
+Cascade
+-
+0.0476
+42083
+60.5
+-
+0.0308
+21928
+5.0
+33.7
+19.2
+-
+0.0189
+12189
+4.2
+24.9
+10.2
+3.6
+18.2
+7.6
+Cascade
++ Finetuning
+Not applicable in Task 1
+-
+0.0296
+20587
+55.4
+35.0
+46.0
+-
+0.0184
+12854
+42.4
+19.2
+35.2
+34.6
+21.8
+31.6
+ORE-EBUI [17]
+4.9
+0.0621
+10459
+56.0
+2.9
+0.0282
+10445
+52.7
+26.0
+39.4
+3.9
+0.0211
+7990
+38.2
+12.7
+29.7
+29.6
+12.4
+25.3
+OW-DETR [13]
+7.1
+0.0590
+10248
+58.9
+6.8
+0.0279
+8540
+52.9
+29.1
+41.0
+7.8
+0.0191
+6840
+38.1
+14.7
+30.3
+30.8
+13.3
+26.4
+Ours:CAT
+21.8
+0.0581
+7070
+59.9
+18.6
+0.0263
+5902
+54.0
+33.6
+43.8
+23.9
+0.0177
+5189
+42.1
+19.8
+34.7
+35.1
+17.1
+30.6
+(+14.7)
+(-0.0009)
+(-3178)
+(+1.0)
+(+11.8)
+(-0.0016)
+(-2638)
+(+1.1)
+(+4.5)
+(+2.8)
+(+16.1)
+(-0.0014)
+(-1651)
+(+4.0)
+(+5.1)
+(+4.4)
+(+4.3)
+(+3.8)
+(+4.2)
+Table 2. State-of-the-art comparison on MS-COCO split. The
+comparison is shown in terms of U-Recall and mAP. Although
+the MS-COCO split is more challenging, our model gets a more
+significant improvement on this in comparison to ORE and OW-
+DETR. The significant metric improvements demonstrate that our
+CAT has the ability to retrieve new knowledge beyond the range
+of closed set and would not be limited by category knowledge of
+existing objects. See Sec.4.3 for more details.
+Task IDs ↓
+Metrics
+ORE
+OW-DETR
+Our:CAT
+U-Recall(↑)
+1.5
+5.7
+24.0 (+18.3)
+Task1
+mAP(↑)
+Current known
+61.4
+71.5
+74.2 (+2.7)
+U-Recall(↑)
+3.9
+6.2
+23.0 (+16.8)
+Previously known
+56.5
+62.8
+67.6 (+4.8)
+Current known
+26.1
+27.5
+35.5 (+8.0)
+Task2
+mAP(↑)
+Both
+40.6
+43.8
+50.7 (+6.9)
+U-Recall(↑)
+3.6
+6.9
+24.6 (+17.7)
+Previously known
+38.7
+45.2
+51.2 (+6.0)
+Current known
+23.7
+24.9
+32.6 (+7.7)
+Task3
+mAP(↑)
+Both
+33.7
+38.5
+45.0 (+6.5)
+Previously known
+33.6
+38.2
+45.4 (+7.2)
+Current known
+26.3
+28.1
+35.1 (+7.0)
+Task4
+mAP(↑)
+Both
+31.8
+33.1
+42.8 (+9.7)
+known object instances for OWOD problem. Both WI and
+A-OSE implicitly quantify the effevtiveness of the model in
+handling unknown objects.
+4.2. Implementation Details
+The multi-scale feature extractor consists of a Resnet-
+50 [16] pretrained on ImageNet [8] in a self-supervised [4]
+manner and a deformable transformer encoder whose num-
+ber of layer is set to 6. For the shared decoder, we use a
+deformable transformer decoder and the numbder of layer
+is set to 6, too. We set the number of queries 𝑀 = 100, the
+dimension of the embeddings 𝐷 = 256 and the number of
+pseudo-labels 𝑘 = 5. During inference, 𝑡𝑜𝑝-50 high scor-
+ing detections are used for evaluation for per image. More
+details are described in the Appendix A.4.
+4.3. Comparison With State-of-the-art Methods
+For a fair comparison, we compare CAT with ORE [17]
+without the energy-based unknown identifier (EBUI) that
+relies on held-out validation data with weak unknown object
+supervision and OW-DETR [13] to demonstrate the effec-
+tiveness of our method for OWOD problem. We present the
+comparison in terms of known class mAP, unknown class
+recall, WI, and A-OSE, where U-Recall, WI and A-OSE
+cannot be computed in Task 4 due to the absence of un-
+known test annotations, for the reason that all 80 classes are
+known. Furthermore, we demonstrate the effectiveness of
+our model for incremental object detection in comparison
+to [13,17,30,35].
+OWOD SPLIT: The results compared with the state-of-
+the-art methods on OWOD split for OWOD problem are
+shown in Table.1. The performance of proposed standard
+cascade detection transformer is also reported to be com-
+pared with Faster R-CNN [33] and the standard Deformable
+DETR [38] frameworks, for demonstrating the power of
+localization identification cascade structure.
+These three
+can only identify known objects, and so U-Recall cannot
+be computed for them. Benefiting from the self-adaptive
+pseudo-labelling, the ability of CAT to detect unknown ob-
+jects goes substantially beyond the existing models. Com-
+pared with OW-DETR’s U-Recall of 7.1, 6.8 and 7.8 on
+Task 1, 2 and 3, our CAT achieves 21.8, 18.6 and 23.9 in the
+6
+
+corresponding tasks, achieving significant absolute gains up
+to 16.1%. In terms of WI and A-OSE, CAT also exceeds
+them in all tasks. The ability to detect known objects and al-
+leviate catastrophic forgetting of previous knowledge gains
+an improved performance with significant gains, achieving
+significant absolute gains up to 4.4%. This demonstrates the
+significant performance of the cascade decoding structure.
+In addition, we report qualitative results in Figure.4, along
+with failure case analysis. See more detailed qualitative re-
+sults in Appendix B.2.
+MS-COCO SPLIT: We report the results on MS-COCO
+split in Table.2.
+MS-COCO split mitigates data leakage
+across tasks and assign more data to each Task, while CAT
+receives a more significant boost compared with OWOD
+split. Compared with OW-DETR’s U-Recall of 5.7, 6.2 and
+6.9 on Task 1, 2 and 3, our CAT achieves 24.0, 23.0 and
+24.6 in the corresponding tasks, achieving significant abso-
+lute gains up to 18.3%. Furthermore, the performance on
+detecting known objects achieves significant absolute gains
+up to 9.7%. This demonstrates that our CAT has the more
+powerful ability to retrieve new knowledge and detect the
+known objects when faced with more difficult tasks.
+Figure 4. Predictions from CAT after being trained on Task 1.
+The results show that the model not only detects other categories
+in the total category that have not yet been learned, such as ‘key-
+board’, ‘kite’ and ‘dining table’, but also accurately detects cate-
+gories outside the total category, such as ‘calendar’, ‘table lamp’
+and ‘rubbish bins’. The approach misclassifies two of the ‘bird’
+as ‘aeroplane’ and ‘unknown’, showing the limitation of CAT. See
+more detailed qualitative results and analysis in Appendix B.2.
+Incremental Object Detection: To intuitively present our
+CAT’s ability for detecting object instances, we compare it
+to [13,17,30,35] on the incremental object detection (IOD)
+task. We evaluate the experiments on three standard set-
+tings, where a group of classes (10, 5 and last class) are in-
+troduced incrementally to a detector trained on the remain-
+ing classes (10, 15 and 19), based on PASCAL VOC 2007
+dataset [10]. As the results shown in Table.3, CAT outper-
+forms the existing method in a great migration on all three
+settings, indicating the power of localization and identifica-
+tion cascade detection transformer for IOD.
+Table 3. State-of-the-art comparison for incremental object detec-
+tion for three different settings on PASCAL VOC dataset. The
+comparison is shown in terms of overall mAP. Our CAT achieves
+significant performance in comparison to existing works on all the
+three settings. See more details in Sec.4.3 and Appedix B.1.
+Method
+10+10 settings
+15+5 settings
+19+1 settings
+ILOD [35]
+63.2
+65.8
+68.2
+Faster ILOD [30]
+62.1
+67.9
+68.5
+ORE [17]
+64.5
+68.5
+68.8
+OW-DETR [13]
+65.7
+69.4
+70.2
+Ours: CAT
+67.7 (+2.0)
+72.2 (+2.8)
+73.8 (+3.6)
+4.4. Ablation Study
+We conduct abundant ablative experiments to verify the
+effectiveness of CAT’s components on the OWOD split
+[17].
+Cascade Decoupled Decoding Structure: We compare
+between OW-DETR, fully decoupled decoding structure
+and CAT in Table.4. The results illustrate that the decoupled
+decoding structure improves the performance of detecting
+known objects and does mitigate the influence of unknown
+objects on the detection of known objects to some extent.
+Because it reduces the difficulty of parameter learning and
+mitigates the risk of confusion for known and unknown ob-
+jects by disassembling the localization and identification
+process of detection. Compared with the fully decoupled
+decoding structure, the cascade decoupled decoding struc-
+ture is able to allow the identification process to draw on
+location information while the localization process is not
+limited by category knowledge and outperforms it.
+Self-Adaptive Pseudo-labelling: As shown in Figure.5 (a)
+and (b), we performed a number of ablation experiments
+on Task 1 for different update cycles, positive and nega-
+tive momentum amplitudes. The results demonstrate that
+the self-adaptive pseudo-labelling makes the training pro-
+cess of CAT robust, as we analyzed earlier. Especially for
+the pink line, even if there are unexpected situations in the
+training process, CAT can still self-adjust and develop in
+a good direction. In addition, we compare the attention-
+driven (AD) and self-adaptive (SA) paeudo-labelling mech-
+anism in Table.5 and Figure.5 (c). The results demonstrate
+that our self-adaptive pseudo-labelling mechanism signifi-
+cantly improves the model’s ability to retrieve unknown ob-
+jects. During training, CAT requires double decoding pro-
+cesses so that it is affected by generated pseudo-labels twice
+as often as OW-DETR. Thus, for the high quality pseudo-
+7
+
+person:76%
+unknown:29%
+unkn0wn:35%
+tvmonitor:93%
+unknown:29%
+aeroplane:79%
+2
+bird:45%
+unkn0wn:29%
+unkn0wn:31%
+unkn0wn:26%
+unknunkn0wh:26%
+unknown:27%
+unkn0wn:26%
+unkn0wn:28%
+person:94%
+chair:82%
+chair:91%
+pottedplant:81%
+unknown:25%
+unkn0wn:30%Figure 5. (a) and (b) illustrate performance comparison between different update cycles, positive and negative momentum amplitude on
+A-OSE and U-Recall. Where the cycle is set to 150 and 300, the positive momentum amplitude is set to 25%, 33% and 50%, the negative
+momentum amplitude is set to 50%, respectively. The lighter coloured lines are the real data and the corresponding darker coloured lines
+are the data after smoothing. (c) shows performance comparison between AD and SA. See detail in Sec.4.4.
+Table 4. Performance comparison between different decoupled de-
+coding structures and OW-DETR. ‘FD’ refers to the fully decou-
+pled decoding structure. See more details in Sec.4.4.
+Task IDs ↓
+Metrics
+OW-DETR
+FD
+CAT
+Task1
+mAP(↑)
+Current known
+59.3
+57.9
+59.9
+Previously known
+53.0
+49.5
+54.0
+Current known
+29.4
+29.4
+33.6
+Task2
+mAP(↑)
+Both
+41.3
+39.4
+43.8
+Previously known
+38.1
+41.2
+42.1
+Current known
+15.0
+18.5
+19.8
+Task3
+mAP(↑)
+Both
+30.5
+33.5
+34.7
+Previously known
+30.6
+33.3
+35.1
+Current known
+14.0
+15.8
+17.1
+Task4
+mAP(↑)
+Both
+26.8
+28.9
+30.6
+labels, CAT could learn better to detect unknown objects
+than OW-DETR. For the low quality pseudo-labels, CAT
+would also be affected to a greater extent. The results in Ta-
+ble.5 further demonstrate this investigation and the robust-
+ness of our pseudo-labelling mechanism to generate pseudo
+labels.
+Open-set Detection Comparison: To further demonstrate
+CAT’s ability to handle unknown instances in open-set data,
+we follow the same evaluation protocol as [13, 17, 26] and
+report the performance in Table.6. CAT achieves promising
+performance in comparison to the existing methods.
+5. Relation to Prior Works
+The issue of standard object detection [3,6,12,15,22,24,
+29,32,33,38,40] has been raised for several years, number-
+ous works have investigated this problem and push the field
+to certain heights. Whereas the strong assumption that the
+label space of object categories to be encountered during the
+life-span of the model is the same as during its training re-
+sults that these methods cannot meet real-world needs. The
+success of [11,18–20,28,33] demonstrates the feasibility of
+Table 5. Performance comparison AD and SA pseudo-labelling
+mechanism.
+The results demonstrate that SA substantially en-
+hances the model’s ability to retrieve unknown objects
+Method
+Task IDs ↓
+AD
+SA
+U-Recall
+WI
+A-OSE
+
+
+7.1
+0.0590
+10248
+Task1
+
+
+19.8
+0.0578
+8360
+
+
+6.8
+0.0279
+8540
+Task2
+
+
+16.8
+0.0268
+6452
+
+
+7.8
+0.0191
+6840
+OW-DETR
+Task3
+
+
+21.8
+0.0175
+5310
+
+
+5.4
+0.0533
+41474
+Task1
+
+
+21.8
+0.0581
+7070
+
+
+4.9
+0.0271
+20410
+Task2
+
+
+18.6
+0.0263
+5902
+
+
+6.0
+0.0186
+11078
+CAT
+Task3
+
+
+23.9
+0.0177
+5189
+Table 6. Performance comparison on open-set object detection
+task. Our CAT achieves significant performance in comparison to
+existing works. See more details in Sec.4.4.
+Evaluated on →
+VOC
+WR1
+Standard Faster R-CNN [35]
+81.8
+77.1
+Standard RetinaNet
+79.2
+73.8
+Dropout Sampling [26]
+78.1
+71.1
+ORE [17]
+81.3
+78.2
+OW-DETR [13]
+82.1
+78.6
+Ours: CAT
+83.2 (+1.1)
+79.5 (+0.9)
+foreground localization based on the position and appear-
+ance of objects. ORE [17] and OW-DETR [13] leverage
+the models of standard object detection and pseudo labels
+to detect objects in open world. In this paper, we propose a
+novel transformer [37] based framework, CAT, for OWOD.
+CAT decouples the localization and identification process
+and connects them in a cascade approach. In CAT, the fore-
+ground localization process is not limited by the category
+8
+
+24
+24
+3e+4
+20
+20
+2.6e+4
+16
+2.2e+4
+I
+12
+1.8e+4
+Tp=150, Tpma=25%,Tnma=50%
+Tp=150, Tpma=33%,Tnma=50%
+1.4e+4
+Tp=150, Tpma=50%,Tnma=50%
+Tp=300, Tpma=25%,Tnma=50%
+CAT + Self_Adaptive
+le+z
+CAT + Attention_Driven
+Tp=300, Tpma=33%,Tnma=50%
+OW-DETR + Self_Adaptive
+be+s
+Tp=300, Tpma=50%,Tnma=50%
+OW-DETR + Attention_Drive
+5
+20
+25
+30
+35
+40
+45
+0
+15
+20
+25
+30
+35
+40
+45
+50
+10
+15
+20
+25
+30
+35
+40
+45
+50of known objects, whereas the process of foreground iden-
+tification can use information from the localization process.
+Along with self-adaptive pseudo-labelling, CAT can gain
+information beyond the data annotation and maintain a sta-
+ble learning process according to self-regulation.
+6. Conclusions
+In this paper, we analyze the drawbacks of the paral-
+lel decoding structure for open-world object detection and
+explore the decoupled decoding structures of the detection
+transformer. Motivated by the subconscious reactions of
+humans when facing new scenes, we propose a novel lo-
+calization and identification cascade detection transformer
+(CAT), which decouples the localization and identification
+process via the cascade decoding structure. The cascade
+decoding structure alleviates the influence of detecting un-
+known objects on the detection of known objects.
+With
+the self-adaptive pseudo-labelling mechanism, CAT gains
+knowledge beyond the data annotation, generates pseudo
+labels with robustness and maintains a stable training pro-
+cess via self-adjustment. The extensive experiments on two
+popular benchmarks, 𝑖.𝑒., PASCAL VOC and MS COCO
+demonstrate that CAT consistently outperforms the existing
+works for all task settings on all splits and achieves state-of-
+the-art performance in the incremental object detection and
+open-set detection.
+Acknowledgment
+This work is supported by National Natural Science
+Foundation of China (grant No.61871106 and No.6137015
+2), Key R&D projects of Liaoning Province, China (grant
+No.2020JH2/10100029), and the Open Project Program Fo-
+undation of the Key Laboratory of Opto-Electronics Infor-
+mation Processing, Chinese Academy of Sciences (OEIP-
+O-202002).
+A. Additional Experiments Material
+A.1. Theory For Self-Adaptive Pseudo-labelling
+For 0 < 𝑤2 < 𝑤1 < 1, we find the potential relationship
+as follows:
+� 𝑥𝑤1 > 𝑥𝑤2,𝑖 𝑓 𝑥 > 1
+𝑥𝑤1 < 𝑥𝑤2,𝑖 𝑓 𝑥 < 1
+(12)
+Thus, for 𝑥𝑤1 ·𝑦𝑤2 and 𝑤1 > 𝑤2, if 𝑥 > 1 and 𝑦 > 1, 𝑥 weights
+more and 𝑦 weights more if 𝑥 < 1 and 𝑦 < 1.
+For the self-adaptive pseudo-labelling, we first normal-
+ize 𝑠𝑜 to the range 0 to 1. Considering that the model it-
+self has little knowledge in the early stages of model train-
+ing, the model-driven pseudo-labelling should weight less
+than the input-deiven pseudo-labelling. As the training time
+of the model increasing, the knowledge base of the model
+grows and the weight of the model-driven pseudo-labelling
+gets bigger. Combining this with the patterns above, we set
+W𝑚0 to 0.8, W𝐼 0 to 0.2 and update them as follows:
+���
+���
+W 𝑡
+𝑚 = W 𝑡−1
+𝑚
++Δ𝑤 ×W 𝑡−1
+𝑚
+,
+W 𝑡
+𝐼 = W 𝑡−1
+𝐼
+−Δ𝑤 ×W 𝑡−1
+𝐼
+,
+W 𝑡
+𝑚,W 𝑡
+𝐼 = 𝑛𝑜𝑟𝑚 �W 𝑡
+𝑚,W 𝑡
+𝐼
+� ,
+(13)
+A.2. Additional Illustration For Data Split
+As shown in Table.7, the OWOD split proposed in ORE
+groups all VOC classes and data as 𝑇𝑎𝑠𝑘 1. The remaining
+60 classes of MS-COCO are grouped into three successive
+tasks (𝑇𝑎𝑠𝑘 2, 3, 4) with semantic drifts. However, it leads
+data leakage across tasks since different classes which be-
+long to a super-categories are introduced in different tasks.
+The MS-COCO split proposed in OW-DETR is a stricter
+split, where all the classes of a super-categories are intro-
+duced at a time in a task.
+Table 7. The table shows task composition in the OWOD and MS-
+COCO split for Open-world evaluation protocol. The semantics of
+each task and the number of images and instances(objects) across
+splits are shown.
+Task ID
+Task 1
+Task 2
+Task 3
+Task 4
+OWOD split
+Semantic split
+VOC
+Classes
+Outdoor, Accessories,
+Appliances, Truck
+Sports,
+Food
+Electronic, Indoor,
+Kitchen, Furniture
+# training images
+16551
+45520
+39402
+40260
+# test images
+4952
+1914
+1642
+1738
+# train instances
+47223
+113741
+114452
+138996
+# test instances
+14976
+4966
+4826
+6039
+MS-COCO split
+Semantic split
+Animals,Person,
+Vehicles
+Appliances, Accessories,
+Outdoor, Furniture
+Sports,
+Food
+Electronic, Indoor,
+Kitchen
+# training images
+89490
+55870
+39402
+38903
+# test images
+3793
+2351
+1642
+1691
+# train instances
+421243
+163512
+114452
+160794
+# test instances
+17786
+7159
+4826
+7010
+A.3. WI, A-OSE and U-Recall Metrics
+In this paper, we mainly illustrate the state-of-the-art
+comparison in terms of wilderness impact (WI), absolute
+open-set error (A-OSE), unknown recall (U-Recall) and
+mean average precision (mAP). WI measures the model’s
+confusion in predicting an unknown instance as known
+class. The calculation formula is as follows:
+WI =
+𝑃K
+𝑃K∪U
+−1
+(14)
+Where 𝑃K is the prediction on known classes and 𝑃K∪U
+is the prediction on known and unknown classes. A-OSE
+devotes the total number of unknown instances detected as
+known classes. Both WI and A-OSE indicate the degree of
+confusion in predicting the known classes in the presence
+9
+
+of unknown instances. Furthermore, U-Recall directly mea-
+sures the model’s ability to retrieve the unknown instances.
+A.4. Additional Implementation Details
+For selective search, we use the 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑒𝑎𝑟𝑐ℎ
+function in Selectivesearch library and the scale, sigma,
+min size of parameter is set to 500, 0.9 and 200, respec-
+tively. In addition, we eliminate candidate boxes with less
+than 2000 pixel points. The multi-scale feature maps ex-
+tracted from the backbone are projected to feature maps
+with 256-channels using 1 × 1 convolution filters and used
+as multi-scale input to deformable transformer encoder. The
+PyTorch library and eight NVIDIA RTX 3090 GPUs are
+used to train our CAT framework with a batch size of 3 im-
+ages per GPUs. In each task, the CAT framework is trained
+for 50 epochs and finetuned for 20 epochs during the in-
+cremental learning step. We train our CAT using the Adam
+optimizer with a base learning rate of 2 × 10−4, 𝛽1 = 0.9,
+𝛽2 = 0.999, and weight decay of 10−4. For finetuning dur-
+ing incremental step, the learning rate is reduced by a factor
+of 10 and trained using a set of 50 stored exemplars per
+known class.
+B. Additional Results
+B.1. Incremental Object Detection
+Table.8 shows a detailed comparison of CAT with exist-
+ing approaches on PASCAL VOC. Evaluation is performed
+on three standard settings, where a group of classes (10, 5
+and last class) are introduced incrementally to a detector
+trained on the remaining classes (10,15 and 19). Our CAT
+performs favorably against existing approaches on all three
+settings, illustrating the power of localization identification
+cascade detection transformer for incremental objection de-
+tection.
+B.2. Additional Qualitative Results
+Figure.6 illustrates the visualization results comparison
+between OW-DETR and our CAT. We use OW-DETR and
+CAT which are both trained on Task 1, the known classes
+are ‘aeroplane’, ‘bicycle’, ‘bird’, ‘boat’, ‘bottle’, ‘bus’,
+‘car’, ‘cat’, ‘chair’, ‘cow’, ‘diningtable’, ‘dog’, ‘horse’,
+‘motorbike’, ‘person’, ‘pottedplant’, ‘sheep’, ‘sofa’, ‘train’
+and ‘tvmonitor’. The results show that our CAT substan-
+tially outperforms OW-DETR in terms of the ability to ex-
+plore unknown objects and the accuracy of detection due to
+the clever cascade decoupled decoding structure and self-
+adaptive pseudo-labelling. As shown in the first row, OW-
+DETR identifies the background and known objects as un-
+knowns and the real unknown object (carton) as the back-
+ground, and our model accurately identifies the carton as
+the unknown object. In the second row, OW-DETR iden-
+tifies the two calendars as the chair and the background,
+respectively, and the keyboard as the background, and our
+CAT accurately identifies them as unknown objects. The
+third row shows that OW-DETR fails to detect the true
+unknown object (frame) and identifies two known objects
+(sofa) as one. Our model accurately identifies the frame as
+an unknown object and also accurately identifies the two
+sofas.
+Figure.7 describes the visualization results comparison
+between CAT and Oracle. We visualize the detection results
+of our model for known and unknown objects, as well as the
+ground truth on the tasks corresponding to the weights, in-
+cluding the labels of known and unknown categories, where
+the objects of unknown categories are the objects of other
+categories that have not yet appeared in the total categories
+of the dataset.
+Our model can accurately detect known
+objects and unknown objects outside the total class of the
+dataset, such as the electric plug and sound switch in the
+first row, the camera in the second row and the kitten toy
+in the third row. It is also worth noting that although our
+model detects the audio, it does not identify it as an un-
+known object, but as a remote, showing the limitations of
+our model.
+Figure.8 exhibits the visualization performance on in-
+cremental object detection. We visualize the detection re-
+sults of the weights corresponding to different tasks for the
+same scenario. The results show that our CAT can identify
+unknown kinds of objects as the unknown class and accu-
+rately identify their classes after incrementally learning the
+unknown classes, such as sports ball and tennis racket in
+the first row, surfboard in the second row and traffic light in
+the third row.
+C. Societal Impact and Limitations
+Open-world object detection makes artificial intelligence
+smarter to face more problems in real life. It takes object de-
+tection to a cognitive level, as the model requires more than
+simply remembering the objects learned, it requires deeper
+thinking about the scene.
+Although our results demonstrate significant improve-
+ments over ORE and OW-DETR in terms of WI, A-OSE,
+U-Recall and mAP, the performances are still on the lower
+side due to the challenging nature of the open-world de-
+tection problem.
+In this paper, we are mainly commit-
+ted to enhance the model’s ability to explore unknown
+classes. However, the confidence level of our model for
+the detection of unknown objects still needs to be im-
+proved, and this is what we will strive for in the fu-
+ture.
+References
+[1] Ankan Bansal, Karan Sikka, Gaurav Sharma, Rama Chel-
+lappa, and Ajay Divakaran. Zero-shot object detection. In
+10
+
+Figure 6. Visualization results comparison between OW-DETR and our CAT. We use OW-DETR and CAT which are both trained on Task
+1, the known classes are ‘aeroplane’, ‘bicycle’, ‘bird’, ‘boat’, ‘bottle’, ‘bus’, ‘car’, ‘cat’, ‘chair’, ‘cow’, ‘diningtable’, ‘dog’, ‘horse’,
+‘motorbike’, ‘person’, ‘pottedplant’, ‘sheep’, ‘sofa’, ‘train’ and ‘tvmonitor’. The results show that our CAT substantially outperforms
+OW-DETR in terms of the ability to explore unknown objects and the accuracy of detection due to the clever cascade decoupled decoding
+structure and self-adaptive pseudo-labelling mechanism. As shown in the first row, OW-DETR identifies the background and known objects
+as unknowns and the real unknown object (carton) as the background, and our model accurately identifies the carton as the unknown
+object. In the second row, OW-DETR identifies the two calendars as the chair and the background, respectively, and the keyboard as the
+background, and our CAT accurately identifies them as unknown objects. The third row shows that OW-DETR not only does not detect the
+true unknown object (frame), but also identifies two known objects (sofa) as one. Our model accurately identifies the frame as an unknown
+object and also accurately identifies the two sofas.
+11
+
+OW-DETR
+Ours:CAT
+i0g:29%
+unkn0wn:41%
+d0g:57%
+d0g:66%
+dog:68%
+unknown:60
+unknown:25%
+unkn0wn:29%
+unknown:35%
+pers0n:85%
+person:76%
+unknown:29%
+chair:25%
+unkn0wn:27%
+unkn0wn:38%
+unkn0wn:26%
+:68
+person:
+unknown:34%Table 8. The detailed comparison of CAT with existing approaches on PASCAL VOC. Evaluation is performed on three standard settings,
+where a group of classes (10, 5 and last class) are introduced incrementally to a detector trained on the remaining classes (10,15 and 19).
+Our CAT performs favorably against existing approaches on all three settings, illustrating the power of localization identification cascade
+detection transformer for incremental objection detection.
+10 + 10 setting
+aero
+cycle
+bird
+boat
+bottle
+bus
+car
+cat
+chair
+cow
+table
+dog
+horse
+bike
+person
+plant
+sheep
+sofa
+train
+tv
+mAP
+ILOD
+69.9
+70.4
+69.4
+54.3
+48
+68.7
+78.9
+68.4
+45.5
+58.1
+59.7
+72.7
+73.5
+73.2
+66.3
+29.5
+63.4
+61.6
+69.3
+62.2
+63.2
+Faster ILOD
+72.8
+75.7
+71.2
+60.5
+61.7
+70.4
+83.3
+76.6
+53.1
+72.3
+36.7
+70.9
+66.8
+67.6
+66.1
+24.7
+63.1
+48.1
+57.1
+43.6
+62.1
+ORE - (CC + EBUI)
+53.3
+69.2
+62.4
+51.8
+52.9
+73.6
+83.7
+71.7
+42.8
+66.8
+46.8
+59.9
+65.5
+66.1
+68.6
+29.8
+55.1
+51.6
+65.3
+51.5
+59.4
+ORE - EBUI
+63.5
+70.9
+58.9
+42.9
+34.1
+76.2
+80.7
+76.3
+34.1
+66.1
+56.1
+70.4
+80.2
+72.3
+81.8
+42.7
+71.6
+68.1
+77
+67.7
+64.5
+OW - DETR
+75.4
+63.9
+57.9
+50.0
+52.0
+70.9
+79.5
+72.4
+44.3
+57.9
+59.7
+73.5
+77.7
+75.2
+76.2
+44.9
+68.8
+65.4
+79.3
+69.0
+65.7
+Ours: CAT
+76.5
+75.7
+67.0
+51.0
+62.4
+73.2
+82.3
+83.7
+42.7
+64.4
+56.8
+74.1
+75.8
+79.2
+78.1
+39.9
+65.1
+59.6
+78.4
+67.4
+67.7
+15 + 5 setting
+aero
+cycle
+bird
+boat
+bottle
+bus
+car
+cat
+chair
+cow
+table
+dog
+horse
+bike
+person
+plant
+sheep
+sofa
+train
+tv
+mAP
+ILOD
+70.5
+79.2
+68.8
+59.1
+53.2
+75.4
+79.4
+78.8
+46.6
+59.4
+59
+75.8
+71.8
+78.6
+69.6
+33.7
+61.5
+63.1
+71.7
+62.2
+65.8
+Faster ILOD
+66.5
+78.1
+71.8
+54.6
+61.4
+68.4
+82.6
+82.7
+52.1
+74.3
+63.1
+78.6
+80.5
+78.4
+80.4
+36.7
+61.7
+59.3
+67.9
+59.1
+67.9
+ORE - (CC + EBUI)
+65.1
+74.6
+57.9
+39.5
+36.7
+75.1
+80
+73.3
+37.1
+69.8
+48.8
+69
+77.5
+72.8
+76.5
+34.4
+62.6
+56.5
+80.3
+65.7
+62.6
+ORE - EBUI
+75.4
+81
+67.1
+51.9
+55.7
+77.2
+85.6
+81.7
+46.1
+76.2
+55.4
+76.7
+86.2
+78.5
+82.1
+32.8
+63.6
+54.7
+77.7
+64.6
+68.5
+OW - DETR
+78.0
+80.7
+79.4
+70.4
+58.8
+65.1
+84.0
+86.2
+56.5
+76.7
+62.4
+84.8
+85.0
+81.8
+81.0
+34.3
+48.2
+57.9
+62.0
+57.0
+69.4
+Ours: CAT
+75.3
+81.0
+84.4
+64.5
+56.6
+74.4
+84.1
+86.6
+53.0
+70.1
+72.4
+83.4
+85.5
+81.6
+81.0
+32.0
+58.6
+60.7
+81.6
+63.5
+72.2
+19 + 1 setting
+aero
+cycle
+bird
+boat
+bottle
+bus
+car
+cat
+chair
+cow
+table
+dog
+horse
+bike
+person
+plant
+sheep
+sofa
+train
+tv
+mAP
+ILOD
+69.4
+79.3
+69.5
+57.4
+45.4
+78.4
+79.1
+80.5
+45.7
+76.3
+64.8
+77.2
+80.8
+77.5
+70.1
+42.3
+67.5
+64.4
+76.7
+62.7
+68.2
+Faster ILOD
+64.2
+74.7
+73.2
+55.5
+53.7
+70.8
+82.9
+82.6
+51.6
+79.7
+58.7
+78.8
+81.8
+75.3
+77.4
+43.1
+73.8
+61.7
+69.8
+61.1
+68.5
+ORE - (CC + EBUI)
+60.7
+78.6
+61.8
+45
+43.2
+75.1
+82.5
+75.5
+42.4
+75.1
+56.7
+72.9
+80.8
+75.4
+77.7
+37.8
+72.3
+64.5
+70.7
+49.9
+64.9
+ORE - EBUI
+67.3
+76.8
+60
+48.4
+58.8
+81.1
+86.5
+75.8
+41.5
+79.6
+54.6
+72.8
+85.9
+81.7
+82.4
+44.8
+75.8
+68.2
+75.7
+60.1
+68.8
+OW - DETR
+82.2
+80.7
+73.9
+56.0
+58.6
+72.1
+82.4
+79.6
+48.0
+72.8
+64.2
+83.3
+83.1
+82.3
+78.6
+42.1
+65.5
+55.4
+82.9
+60.1
+70.2
+Ours: CAT
+86.0
+85.8
+78.8
+65.3
+61.3
+71.4
+84.8
+84.8
+52.9
+78.4
+71.6
+82.7
+83.8
+81.2
+80.7
+43.7
+75.9
+58.5
+85.2
+61.1
+73.8
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+Figure 7. Visualization results comparison between CAT and Oracle. We visualize the detection results of our model for known and
+unknown objects, as well as the ground truth on the tasks corresponding to the weights, including the labels of known categories and the
+labels of unknown categories, where the objects of unknown categories are the objects of other categories that have not yet appeared in the
+total categories of the dataset. Our model can accurately detect known objects and unknown objects outside the total class of the dataset,
+such as the electric plug and sound switch in the first row, the camera in the second row and the kitten toy in the third row. It is also worth
+noting that although our model detects the audio, it does not identify it as an unknown object, but as a remote, showing the limitations of
+our model.
+13
+
+Oracle
+Ours:CAT
+tvmonitor
+tvmonitor:79%
+remote:28%
+mote:35
+O
+unknown:3
+unknown:38%
+1kn0wn:34%
+unkn0wn:26%%
+keyboard
+keyboard:42
+mouse
+mouse
+person
+tvmonitor:88%
+unkn0wn:36%
+sofa:43%
+cat:80%
+cat
+unknown:26%
+unknown:43
+unknown:35%
+inknov
+unknown:25%Figure 8. Visualization performance on incremental object detection. We visualize the detection results of the weights corresponding to
+different tasks for the same scenario. The results show that our CAT can identify unknown kinds of objects as the unknown class and
+accurately identify their classes after incrementally learning the unknown classes, such as sports ball and tennis racket in the first row,
+surfboard in the second row and traffic light in the third row.
+[16] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
+Deep residual learning for image recognition. In Proceed-
+14
+
+Before Learning
+After Learning
+tennis racket:84%
+erson:89%
+unk
+sports
+ball:80%
+erson:87%
+surfb0ard:60%
+unkn0wn:30%
+raffic light:60%3%
+unknowi
+traffic
+1ght:78%
+ight:6
+car:88%
+car:68%
+unknown
+ar:58
+car
+23%
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+

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+arXiv:2301.00100v1  [math.AP]  31 Dec 2022
+COBORDISM INVARIANCE OF THE INDEX FOR
+REALIZATIONS OF ELLIPTIC OPERATORS REVISITED
+THOMAS KRAINER
+Abstract. We revisit an argument due to Lesch [11, 12] for proving the cobor-
+dism invariance of the index of Dirac operators on even-dimensional closed
+manifolds and combine this with recent work by the author [10] to show van-
+ishing results for the spectral flow for families of selfadjoint Fredholm realiza-
+tions of elliptic operators in case the family is induced on the boundary by an
+elliptic operator on a compact space. This work is motivated by studying the
+behavior of the index of realizations of elliptic operators under cobordisms of
+statified manifolds.
+1. Introduction
+One of the original proofs of the Atiyah-Singer Index Theorem is based on showing
+that the index of Dirac type operators is invariant under cobordisms, see Palais
+[17]. This proof is analytic in nature and rooted in the classical theory of elliptic
+boundary value problems. Other proof strategies for the index theorem such as the
+heat equation proof have generally been favored because these proofs require less
+sophisticated analytic techniques than the original cobordism proof.
+Higson [9] gave a proof of the cobordism invariance of the index by attaching an
+infinite half-cylinder to the boundary and extending the operator from the manifold
+with boundary to the manifold with cylindrical end. The Dirac type operator on
+the resulting odd-dimensional complete manifold is essentially selfadjoint, and the
+analytic arguments involved in Higson’s proof are considerably simpler compared
+to the original proof. Lesch [11], on the other hand, gave a proof by attaching a
+(generalized) cone to the boundary and extended the operator from the manifold
+with boundary to a cone operator; while conic manifolds are incomplete and thus
+dealing with domains of realizations of the resulting conic Dirac type operator is
+needed, Lesch’s approach is still much simpler from a functional analytic point of
+view than the original proof because the maximal and minimal domains of L2-
+based realizations in the conic case differ only by a finite-dimensional space – the
+price to pay is the more intricate analysis to deal with the singularity which at
+this juncture has been introduced artificially. Several other analytic proofs of the
+cobordism invariance of the index [3, 16], a K-theory proof [5], and generalizations
+[4, 8, 14, 19] have since been found.
+This note is motivated by recent advances in elliptic theory on stratified man-
+ifolds with incomplete iterated wedge metrics [1, 2, 6, 7, 15, 18] and gives an ap-
+plication of the spectral flow formula for indicial operators obtained in our recent
+paper [10]. Stratified cobordisms and the cobordism invariance of the index for the
+2020 Mathematics Subject Classification. Primary: 58J20; Secondary: 58J05, 58J32, 58J30.
+Key words and phrases. Manifolds with singularities, index theory, cobordism.
+1
+
+2
+THOMAS KRAINER
+signature operator have been considered in [1, 2], where especially in [2] the opera-
+tor is no longer essentially selfadjoint and suitable boundary conditions associated
+with the singular strata are considered; stratified cobordism and the invariance of
+the index are used in an essential way to establish the properties of the signature
+of a Cheeger space considered in that paper.
+From our point of view Lesch’s proof [11, 12] of the cobordism invariance of
+the index is very natural in the context of elliptic theory on stratified manifolds
+because, unlike in the classical smooth case, singular analysis and dealing with
+boundary conditions associated with singular strata already are essential features
+of the investigations here.
+In this note we will revisit and extend Lesch’s proof from the Dirac case to more
+general operators of any order, and what amounts to the vanishing of the index in
+the Dirac case (for null-cobordisms) will accordingly generalize to the vanishing of
+the spectral flow for indicial families. Our recent paper [10] on indicial operators,
+which are abstract functional analytic model operators associated to generalized
+conical singularities, is the basis for this. We will only be concerned with null-
+cobordisms and proving vanishing results here; more general notions of cobordisms
+and cobordism invariance follow upon reduction to this case. Without detailing the
+precise assumptions, the argument proceeds as follows:
+Let (M, g) be a Riemannian manifold, and let U = U(Y ) ⊂ M be an open subset
+that is isometric to (0, ε) × Y with product metric dx2 + gY for some ε > 0, where
+(Y, gY ) is another Riemannian manifold. The reader ought to think of both M and
+Y as the open interior of compact stratified manifolds M and Y equipped with
+incomplete iterated wedge metrics, where Y is a boundary hypersurface of M, and
+U(Y ) is a collar neighborhood. Let E → M be a Hermitian vector bundle such
+that E
+��
+U(Y ) ∼= π∗
+Y E isometrically, where E → Y is a Hermitian vector bundle, and
+πY : (0, ε) × Y → Y is the canonical projection. Let
+A : C∞
+c (M; E ) → C∞
+c (M; E )
+be an elliptic differential operator of order µ ≥ 1 that is symmetric with respect to
+the inner product induced by the Riemannian and Hermitian metrics, and suppose
+that A is in U(Y ) of the form
+A ∼= A∧ = x−1
+µ
+�
+j=0
+aj(y, Dy)(xDx)j : C∞
+c ((0, ε) × Y ; π∗
+Y E) → C∞
+c ((0, ε) × Y ; π∗
+Y E),
+where aj(y, Dy) ∈ Diffµ−j(Y ; E). Let
+p(σ) =
+µ
+�
+j=0
+aj(y, Dy)σj : C∞
+c (Y ; E) → C∞
+c (Y ; E), σ ∈ C,
+be the indicial family. Now suppose that
+Amin : Dmin(A) ⊂ L2(M; E ) → L2(M; E )
+(1.1)
+is some closed symmetric extension of A : C∞
+c (M; E ) ⊂ L2(M; E ) → L2(M; E ),
+and let Amax : Dmax(A) ⊂ L2(M; E ) → L2(M; E ) be the adjoint – we point out
+here that Amin is not necessarily the minimal extension of A from C∞
+c (M; E ), and
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+3
+therefore Amax is not the largest L2-based closed extension either, i.e. we only have
+Dmin(A) ⊃ {u ∈ L2(M; E ); ∃uk ∈ C∞
+c (M; E ), uk → u in L2(M; E ),
+and Auk ⊂ L2(M; E ) Cauchy},
+Dmax(A) ⊂ {u ∈ L2(M; E ); ∃v ∈ L2(M; E ) :
+⟨Aφ, u⟩L2(M;E ) = ⟨φ, v⟩L2(M;E ) ∀φ ∈ C∞
+c (M; E )},
+and these inclusions are generally proper. The reader ought to think of the operator
+A as an elliptic iterated incomplete wedge operator on M, and the domain Dmin(A)
+as determined by previously chosen boundary conditions for A associated with
+singular strata of M away from the boundary hypersurface Y ⊂ M.
+One of the main points now is that under suitable localization and compatibility
+assumptions these extensions of A should localize to U(Y ) and be fully captured
+by the extensions of the indicial operator
+A∧ : C∞
+c (R+; E1) ⊂ L2(R+ × Y ; π∗
+Y E) → L2(R+ × Y ; π∗
+Y E).
+(1.2)
+Here
+Hµ
+comp(Y ; E) ⊂ E1 ⊂ Hµ
+loc(Y ; E)
+is the common domain for the indicial family p(σ) : E1 ⊂ E0 → E0, σ ∈ C,
+where E0 = L2(Y ; E), giving rise to a holomorphic family of unbounded Fredholm
+operators that are selfadjoint for σ ∈ R.
+The reader ought to think of E1 as
+determined by certain lateral boundary conditions associated with the singular
+strata of Y , obtained via restriction to U(Y ) by the previously determined boundary
+conditions on M for A that gave rise to Dmin(A); the localization and compatibility
+assumptions are such that the boundary conditions previously chosen for A on M
+should be selfadjoint away from the boundary hypersurface Y . The upshot of all
+of this is that we obtain a unitary equivalence
+�
+Dmax(A)/Dmin(A), [·, ·]A
+� ∼=
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+of finite-dimensional indefinite inner product spaces by passing to representatives
+supported in U(Y ) ∼= (0, ε)×Y , thus allowing transitioning between M and R+×Y ;
+here
+[·, ·]A : Dmax(A) × Dmax(A) → C,
+[u, v]A = 1
+i
+�
+⟨Amaxu, v⟩L2 − ⟨u, Amaxv⟩L2
+�
+is the adjoint pairing, and likewise for [·, ·]A∧, while Dmin(A∧) is the domain of the
+closure A∧,min of (1.2), and Dmax(A∧) is the domain of the adjoint A∧,max = A∗
+∧,min.
+In particular, we have
+sgn
+�
+Dmax(A)/Dmin(A), [·, ·]A
+�
+= sgn
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+for the signatures of these spaces. On the one hand, using the spectral flow formula
+from [10], we have
+sgn
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+= SF[ p(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ],
+while on the other hand sgn
+�
+Dmax(A)/Dmin(A), [·, ·]A
+�
+= 0 if (1.1) is Fredholm or
+the embedding Dmax(A) ֒→ L2(M; E ) is compact, which combined leads to the
+desired conclusion that
+SF[ p(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ] = 0.
+
+4
+THOMAS KRAINER
+In the Dirac case the spectral flow of the indicial family is easily seen to equal
+the Fredholm index of the operator D : D(D) ⊂ L2(Y ; E−) → L2(Y ; E+) on the
+even-dimensional boundary Y , thus recovering cobordism invariance of the index
+in this context.
+The structure of this paper is as follows: In Section 2 we briefly review what is
+needed from extension theory of symmetric operators, in particular the criteria that
+ensure that
+�
+Dmax(A)/Dmin(A), [·, ·]A
+�
+is finite-dimensional with signature zero. In
+Section 3 we review results from our paper [10] on indicial operators in the form in
+which they are needed here; we also address in this section how indicial operators of
+first order that model the Dirac case fit into this framework in order to obtain the
+desired conclusions about the cobordism invariance of the index when specializing to
+such operators. In Section 4 we fill in the details of the outline above and prove the
+null-cobordism theorem (Theorem 4.2). Finally, in Appendix A, we discuss the null-
+cobordism theorem for smooth manifolds; assumptions appear much weaker here
+on the geometry and the participating objects because the analytic tools available
+in this case are rich enough to create the preconditions needed to apply the null-
+cobordism theorem rather than having to assume them from the outset.
+With
+the ongoing further development of singular analysis on stratified manifolds we
+anticipate similar reductions and simplifications for such cases in the future as well.
+2. Preliminaries from extension theory
+Let H be a separable complex Hilbert space, and suppose Amin : Dmin ⊂ H → H
+is closed, densely defined, and symmetric. Let Amax := A∗
+min : Dmax ⊂ H → H be
+the adjoint. We equip Dmax with the graph inner product
+⟨u, v⟩Amax = ⟨u, v⟩ + ⟨Amaxu, Amaxv⟩
+and associated graph norm. Then Dmin ⊂
+�
+Dmax, ∥ · ∥Amax
+�
+is a closed subspace,
+and
+Dmax = Dmin ⊕ ker(Amax + i) ⊕ ker(Amax − i)
+by von Neumann’s formulas. The dimensions
+n± = dim ker(Amax ∓ λi) ∈ N0 ∪ {∞},
+λ > 0,
+are the deficiency indices of the operator Amin and independent of λ > 0. The
+operators
+Amin ± iλ : Dmin ⊂ H → H,
+λ > 0,
+are injective and have closed range, and we have n± < ∞ if and only if Amin ± iλ
+is Fredholm, in which case n± = − ind(Amin ± iλ). The adjoint pairing
+[·, ·]A : Dmax × Dmax → C,
+[u, v]A = 1
+i
+�
+⟨Amaxu, v⟩ − ⟨u, Amaxv⟩
+�
+descends to a nondegenerate Hermitian sesquilinear form (indefinite inner product)
+[·, ·] : Dmax/Dmin × Dmax/Dmin → C.
+If dim Dmax/Dmin < ∞, i.e. if Amin has finite deficiency indices, the signature of
+the adjoint pairing is given by
+sgn
+�
+Dmax/Dmin, [·, ·]
+�
+= n+ − n−.
+The following criteria are standard and useful for verification that n+ = n− < ∞.
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+5
+Proposition 2.1. Suppose Amin : Dmin ⊂ H → H is Fredholm. Then Amin has
+finite and equal deficiency indices, and therefore
+sgn
+�
+Dmax/Dmin, [·, ·]
+�
+= 0.
+Proof. Because Amin : Dmin ⊂ H → H is Fredholm there exists ε > 0 such that
+Amin + iλ : Dmin ⊂ H → H is Fredholm for −ε < λ < ε, and consequently both
+n± < ∞ and Amin + iλ is Fredholm for all λ ∈ R. Now
+R ∋ λ �→ Amin + iλ : Dmin ⊂ H → H
+is a continuous Fredholm function and therefore has constant index. Thus
+n+ = − ind(Amin + i) = − ind(Amin − i) = n−.
+□
+Proposition 2.2. If the embedding
+�
+Dmax, ∥ · ∥Amax
+�
+֒→ H is compact then Amin
+has finite and equal deficiency indices.
+Proof. The norms ∥·∥Amax and ∥·∥H are equivalent on ker(Amax±i), and the identity
+map
+�
+ker(Amax ± i), ∥ · ∥Amax
+�
+→
+�
+ker(Amax ± i), ∥ · ∥H
+�
+is compact by assumption.
+Thus dim ker(Amax ± i) < ∞. Now Amin ± i : Dmin ⊂ H → H are both Fredholm,
+and because Dmin ֒→ H is compact we have ind(Amin − i) = ind(Amin + i). The
+proposition is proved.
+□
+3. Indicial operators
+We consider indicial operators of the form
+A∧ = x−1
+µ
+�
+j=0
+aj(xDx)j : C∞
+c (R+; E1) ⊂ L2(R+; E0) → L2(R+; E0),
+(3.1)
+where µ ∈ N and E0 and E1 are separable complex Hilbert spaces such that E1 ֒→
+E0 is continuous and dense, and the operators aj : E1 → E0 are continuous for
+j = 0, . . . , µ. Let
+p(σ) =
+µ
+�
+j=0
+ajσj : E1 → E0,
+σ ∈ C
+(3.2)
+be the indicial family associated with A∧. We make the following assumptions:
+(i) p(σ) : E1 ⊂ E0 → E0 is closed, densely defined, and Fredholm for σ ∈ C, and
+the map C ∋ σ �→ p(σ) ∈ L (E1, E0) is holomorphic.
+(ii) We have p(σ)∗ = p(σ) : E1 ⊂ E0 → E0 as unbounded operators in E0.
+(iii) For (λ, σ) ∈ R2 and |λ, σ| ≥ R ≫ 0 sufficiently large p(σ) + iλ : E1 → E0 is
+invertible with
+sup
+|λ,σ|≥R
+�
+(1 + λ2 + σ2µ)
+1
+2 ���
+p(σ) + iλ
+�−1��
+L (E0) +
+���
+p(σ) + iλ
+�−1��
+L (E0,E1)
+�
+< ∞,
+and for every k ∈ {1, . . ., µ} we have
+sup
+|λ,σ|≥R
+(1 + λ2 + σ2µ)
+k
+2µ ���
+∂k
+σp(σ)
+��
+p(σ) + iλ
+�−1��
+L (E0) < ∞.
+In [10] we systematically studied operators of the kind (3.1) under such assumptions.
+We summarize some of the findings below:
+
+6
+THOMAS KRAINER
+(1) The operator (3.1) is symmetric and densely defined in L2(R+; E0). Let
+A∧,min be its closure, and A∧,max = A∗
+∧,min be the adjoint. Then
+dim Dmax(A∧)/Dmin(A∧) < ∞,
+i.e., A∧ has finite deficiency indices.
+(2) The boundary spectrum
+specb(p) = {σ ∈ C; p(σ) : E1 → E0 is not invertible} ⊂ C
+is discrete, and every strip |ℑ(σ)| ≤ K, K > 0, contains only finitely many
+elements of specb(p). The elements of the boundary spectrum are generally
+referred to as indicial roots.
+(3) Fix an arbitrary cut-off function ω ∈ C∞
+c (R+) with ω ≡ 1 near x = 0. For
+each indicial root σ0 ∈ specb(p) let
+Eσ0(p) =
+�
+u = ω
+k
+�
+j=0
+ej logj(x)xiσ0; k ∈ N0 and ej ∈ E1,
+and p(σ)(Mu)(σ) is holomorphic at σ = σ0
+�
+,
+(3.3)
+where
+�
+Mu
+�
+(σ) =
+� ∞
+0
+x−iσu(x) dx
+x
+is the Mellin transform of u. This space is finite-dimensional for every σ0,
+and we have
+Dmax(A∧) = Dmin(A∧) ⊕
+�
+σ0���specb(p)
+− 1
+2 <ℑ(σ0)< 1
+2
+Eσ0(p).
+(3.4)
+(4) We have
+x
+1
+2 H (R+; E1) ∩ L2(R+; E0) ֒→ Dmin(A∧),
+and Dmin(A∧) = x
+1
+2 H (R+; E1) ∩ L2(R+; E0) if and only if p(σ) : E1 → E0
+is invertible for all ℑ(σ) = − 1
+2.
+The space H (R+; E1) is the completion of C∞
+c (R+; E1) with respect to
+the norm
+∥u∥2
+H =
+�
+R
+∥p(σ + iγ0)(Mu)(σ)∥2
+E0 dσ,
+where γ0 ∈ R is arbitrary such that p(σ + iγ0) : E1 → E0 is invertible for
+all σ ∈ R. We have
+H (R+; E1) ֒→ Hµ
+b (R+; E0) ∩ L2
+b(R+; E1),
+and in typical situations these spaces are equal; this is the case, for instance,
+if
+sup
+σ∈R
+∥p(σ + iγ0)(⟨σ⟩µ + iΛ)−1∥L (E0) < ∞,
+(3.5)
+where Λ : E1 ⊂ E0 → E0 is selfadjoint (e.g. for Λ = p(0)).
+(5) While not discussed in [10] it is not hard to see that, under the added
+assumption that the embedding E1 ֒→ E0 is compact, multiplication by a
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+7
+cut-off function ω ∈ C∞
+c (R+) with ω ≡ 1 near x = 0 induces a compact
+operator ω : xαH (R+; E1) → L2
+b(R+; E0) for every α > 01.
+Consequently, if additionally p(σ) : E1 → E0 is invertible for all ℑ(σ) =
+− 1
+2, we obtain a compact map ω : Dmax(A∧) → L2(R+; E0), and a bounded
+map 1 − ω : Dmax(A∧) → Dmin(A∧). The latter is based on the identity
+Dmin(A∧) = x
+1
+2 H (R+; E1) ∩ L2(R+; E0) and localization properties of the
+space H (R+; E1) (see [10, Proposition 7.6]).
+(6) The adjoint pairing
+[·, ·]A∧ : Dmax(A∧) × Dmax(A∧) → C,
+[u, v]A∧ = 1
+i
+�
+⟨A∧,maxu, v⟩L2(R+;E0) − ⟨u, A∧,maxv⟩L2(R+;E0)
+�
+induces a nondegenerate Hermitian sesquilinear form
+[·, ·] : Dmax(A∧)/Dmin(A∧) × Dmax(A∧)/Dmin(A∧) → C,
+and its signature is given by the spectral flow of the indicial family (3.2)
+along the real line:
+sgn
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]
+�
+= SF[ p(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ].
+(3.6)
+Note that p(σ) : E1 → E0 is invertible for |σ| ≥ T ≫ 0 large enough,
+σ ∈ R, and the spectral flow in (3.6) then refers to p(σ) on the interval
+−T ≤ σ ≤ T . Only crossings of real indicial roots contribute terms to the
+spectral flow.
+The focus in this paper is on the signature of the adjoint pairing, and by (3.6)
+only real indicial roots are relevant. In order to obtain simple expressions for the
+minimal domain and the maximal domain (3.4) of A∧ it is sometimes convenient
+to introduce a scaling parameter t > 0 to remove any small non-real indicial roots
+from the strip |ℑ(σ)| ≤ 1
+2. This leads to
+A∧,t = x−1
+µ
+�
+j=0
+ajtj(xDx)j : C∞
+c (R+; E1) ⊂ L2(R+; E0) → L2(R+; E0)
+with indicial family
+pt(σ) = p(tσ) : E1 ⊂ E0 → E0,
+σ ∈ C,
+and the standing assumptions on p(σ) imply that the analogous properties are also
+satisfied for pt(σ), and all estimates are locally uniform with respect to t > 0. In
+particular, the spectral flow
+SF[ pt(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ]
+is independent of t > 0 by homotopy invariance, and thus
+sgn
+�
+Dmax(A∧,t)/Dmin(A∧,t), [·, ·]
+�
+1The function a0(x, σ) = xαω(x)p(σ + iγ0)−1 is a Mellin symbol taking values in the com-
+pact operators E0 → E0, and we have sup{⟨log(x)⟩j⟨σ⟩µ+k∥(xDx)l∂k
+σa0(x, σ)∥L (E0); (x, σ) ∈
+R+ × R} < ∞ for all j, k, l ∈ N0.
+Thus the Mellin pseudodifferential operator opM(a0) :
+L2
+b(R+; E0) → L2
+b(R+; E0) is compact, which implies compactness of the multiplication opera-
+tor ω : xαH (R+; E1) → L2
+b(R+; E0) as asserted.
+
+8
+THOMAS KRAINER
+is independent of t > 0. For 0 < t ≤ t0 small enough, pt(σ) : E1 ⊂ E0 → E0 is
+invertible for all 0 < |ℑ(σ)| ≤ 1
+2. We then have
+Dmin(A∧,t) = x
+1
+2 H (R+; E1) ∩ L2(R+; E0),
+where the definition of H (R+; E1) is accordingly based on pt(σ), and
+Dmax(A∧,t) = Dmin(A∧,t) ⊕
+�
+σ0∈specb(pt)∩R
+Eσ0(pt).
+If (3.5) holds for p(σ) it is true for all pt(σ), and in this case the space
+H (R+; E1) = Hµ
+b (R+; E0) ∩ L2
+b(R+; E1)
+is independent of t > 0; thus the minimal domain
+Dmin(A∧) = x
+1
+2 Hµ
+b (R+; E0) ∩ x
+1
+2 L2
+b(R+; E1) ∩ L2(R+; E0)
+is independent of 0 < t ≤ t0.
+Operators of first order. Let D : D(D) ⊂ H1 → H2 be closed and densely
+defined, and let D∗ : D(D∗) ⊂ H2 → H1 be the adjoint. Write
+E0 =
+H1
+⊕
+H2
+and E1 =
+D(D)
+⊕
+D(D∗)
+֒→ E0.
+We assume that D (and therefore also D∗) is Fredholm, and that the embeddings
+for both domains D(D) ֒→ H1 and D(D∗) ֒→ H2 are compact. Consider then
+D∧ = x−1
+��1
+0
+0
+−1
+�
+(xDx)+
+� 0
+D∗
+D
+0
+��
+: C∞
+c (R+; E1) ⊂ L2(R+; E0) → L2(R+; E0)
+with indicial family
+D(σ) =
+�
+σ
+D∗
+D
+−σ
+�
+: E1 ⊂ E0 → E0,
+σ ∈ C.
+Now D(σ) satisfies the assumptions previously stated for indicial families with
+µ = 1, including (3.5) with Λ = D(0); see Lemma 3.8 for the required estimates.
+Therefore the conclusions summarized above hold for D∧, and by Lemma 3.9 we
+have
+sgn
+��
+Dmax(D∧)/Dmin(D∧), [·, ·]
+�
+= ind[D : D(D) ⊂ H1 → H2].
+(3.7)
+The only real indicial root is σ0 = 0, and after possibly introducing a sufficiently
+small scaling parameter t > 0 and replacing D∧ by
+D∧,t = x−1
+�
+t
+�
+1
+0
+0
+−1
+�
+(xDx) +
+�
+0
+D∗
+D
+0
+��
+we have
+Dmin(D∧,t) = x
+1
+2 H1
+b (R+; E0) ∩ x
+1
+2 L2
+b(R+; E1) ∩ L2(R+; E0),
+Dmax(D∧,t) = Dmin(D∧,t) ⊕ E0(Dt).
+In this case E0(Dt) = E0(D) is also independent of t > 0, and we have
+E0(D) =
+�
+u = ω
+� k
+k∗
+�
+; k ∈ ker(D), k∗ ∈ ker(D∗)
+�
+.
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+9
+This follows from (3.3) in view of
+D(σ)−1 = σ
+�
+1
+0
+0
+−1
+�
+[D(0)2 + σ2]−1 + D(0)[D(0)2 + σ2]−1
+=
+�ΠD
+0
+0
+−ΠD∗
+� 1
+σ + holomorphic
+near σ = 0, where ΠD : H1 → ker(D) and ΠD∗ : H2 → ker(D∗) are the orthogonal
+projections onto the kernels of D and D∗, respectively. For sufficiently small t > 0
+a brief calculation shows that the adjoint pairing is given by
+�
+ω
+�
+k1
+k∗
+1
+�
+, ω
+�
+k2
+k∗
+2
+��
+D∧,t = t
+�
+⟨k1, k2⟩H1 − ⟨k∗
+1, k∗
+2⟩H2
+�
+for kj ∈ ker(D) and k∗
+j ∈ ker(D∗), j = 1, 2, which provides a direct justification for
+(3.7) for D∧,t (for small t > 0) that does not rely on the spectral flow.
+Lemma 3.8. For (λ, σ) ∈ R2 write z = σ + iλ ∈ C and consider
+D(z) = D(σ) + iλ =
+� z
+D∗
+D
+−z
+�
+: E1 ⊂ E0 → E0.
+Then D(z) is invertible for all z ∈ C \ {0}, and
+sup
+|z|≥1
+{|z| · ∥D(z)−1∥L (E0) + ∥D(z)−1∥L (E0,E1)} < ∞.
+Proof. We have D(z)∗ = D(z), and
+D(z)∗D(z) = D(z)D(z)∗ =
+�|z|2 + D∗D
+0
+0
+|z|2 + DD∗
+�
+= D(0)2 + |z|2.
+This operator is invertible for z ∈ C \ {0}, and consequently D(z) is invertible with
+D(z)−1 = D(z)∗[D(z)D(z)∗]−1 = [zΠ1−zΠ2][D(0)2+|z|2]−1+D(0)[D(0)2+|z|2]−1,
+where Πj : E0 → Hj ⊂ E0 is the orthogonal projection, j = 1, 2.
+In view of
+D(0)[zΠ1 − zΠ2] = [zΠ2 − zΠ1]D(0) we have
+D(0)D(z)−1 = [zΠ2 − zΠ1]D(0)[D(0)2 + |z|2]−1 + D(0)2[D(0)2 + |z|2]−1.
+The Spectral Theorem implies
+sup
+|z|≥1
+{∥D(0)2[D(0)2+|z|2]−1∥+∥zD(0)[D(0)2+|z|2]−1∥+∥z2[D(0)2+|z|2]−1∥} < ∞,
+where ∥ · ∥ = ∥ · ∥L (E0). The lemma now follows.
+□
+Lemma 3.9. We have
+ind[D : D(D) ⊂ H1 → H2] = SF
+�
+D(σ) : E1 ⊂ E0 → E0, σ ∈ R
+�
+.
+Proof. Let K = ker(D(0)) = ker(D) ⊕ ker(D∗). Then
+D(σ) =
+�DK(σ)
+0
+0
+DK⊥(σ)
+�
+:
+K
+⊕
+K⊥ ∩ E1
+→
+K
+⊕
+K⊥
+,
+σ ∈ R.
+
+10
+THOMAS KRAINER
+Now DK(σ) : K → K, σ ̸= 0, has eigenvalues σ, −σ of multiplicities dim ker(D) and
+dim ker(D∗), respectively, and DK⊥(σ) is invertible for all σ ∈ R. Thus
+ind D = dim ker(D) − dim ker(D∗)
+= SF
+�
+DK(σ) : K → K, σ ∈ R
+�
+= SF
+�
+D(σ) : E1 ⊂ E0 → E0, σ ∈ R
+�
+.
+□
+4. The null-cobordism theorem
+We now revisit the setting discussed in the introduction to prove the null-cobordism
+theorem. We make the following product type assumptions on the geometry and
+the operator:
+Let (M, g) be a Riemannian manifold, and let U = U(Y ) ⊂ M be an open subset
+that is isometric to (0, ε) × Y with product metric dx2 + gY for some ε > 0, where
+(Y, gY ) is another Riemannian manifold. Let E → M be a Hermitian vector bundle
+such that E
+��
+U(Y ) ∼= π∗
+Y E isometrically, where E → Y is a Hermitian vector bundle,
+and πY : (0, ε) × Y → Y is the canonical projection. Let
+A : C∞
+c (M; E ) → C∞
+c (M; E )
+be an elliptic differential operator of order µ ≥ 1 that is symmetric with respect to
+the inner product induced by the Riemannian and Hermitian metrics, and suppose
+that A is in U(Y ) of the form
+A ∼= A∧ = x−1
+µ
+�
+j=0
+aj(y, Dy)(xDx)j : C∞
+c ((0, ε) × Y ; π∗
+Y E) → C∞
+c ((0, ε) × Y ; π∗
+Y E),
+where aj(y, Dy) ∈ Diffµ−j(Y ; E). Let
+p(σ) =
+µ
+�
+j=0
+aj(y, Dy)σj : C∞
+c (Y ; E) → C∞
+c (Y ; E), σ ∈ C,
+be the indicial family. We assume that p(σ) : E1 ⊂ E0 → E0 satisfies the assump-
+tions stated in Section 3 with E0 = L2(Y ; E) and some domain
+Hµ
+comp(Y ; E) ⊂ E1 ⊂ Hµ
+loc(Y ; E).
+We also assume that the embedding E1 ֒→ E0 is compact, and that p(σ) : E1 → E0
+is invertible for 0 < |ℑ(σ)| ≤ 1
+2; as explained in Section 3, the latter can generally
+be achieved by introducing a scaling parameter (which for geometric operators
+typically corresponds to scaling the metric). The closed extensions of the indicial
+operator
+A∧ : C∞
+c (R+; E1) ⊂ L2(R+ × Y ; π∗
+Y E) → L2(R+ × Y ; π∗
+Y E)
+are then described as explained in Section 3. Let
+Amin : Dmin(A) ⊂ L2(M; E ) → L2(M; E )
+be a closed symmetric extension of A : C∞
+c (M; E ) ⊂ L2(M; E ) → L2(M; E ), and
+let Amax : Dmax(A) ⊂ L2(M; E ) → L2(M; E ) be the adjoint; as discussed in the
+introduction, Amin is generally not the minimal extension of A from C∞
+c (M; E ),
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+11
+and thus Amax is not the largest L2-based closed extension. By elliptic regularity
+we have
+Hµ
+comp(M; E ) ⊂ Dmin(A) ⊂ Dmax(A) ⊂ Hµ
+loc(M; E ).
+By a cut-off function we mean any function ω ∈ C∞
+c ([0, ε)) such that ω ≡ 1 near
+x = 0, and we consider ω a function on M supported in U(Y ).
+We make the
+following localization and compatibility assumptions between A and A∧:
+• For every cut-off function ω, multiplication by 1 − ω gives a continuous
+operator Dmax(A) → Dmin(A). We also assume that 1 − ω : Dmin(A) →
+L2(M; E ) is compact.
+• For every cut-off function ω, multiplication by ω gives continuous operators
+Dmin(A) → Dmin(A∧) and Dmin(A∧) → Dmin(A).
+To make sense of the mappings above note that
+M ⊃ U(Y ) ∼= (0, ε) × Y ⊂ R+ × Y,
+which allows transitioning both ways between functions on M supported in U(Y )
+and functions on R+ × Y supported in (0, ε) × Y . We will use these transitions
+freely in what follows.
+Proposition 4.1. Let ω ∈ C∞
+c ([0, ε)) be any cut-off function. The map
+Dmax(A)/Dmin(A) ∋ u + Dmin(A) �−→ ωu + Dmin(A∧) ∈ Dmax(A∧)/Dmin(A∧)
+is well-defined, and induces a unitary equivalence between the indefinite inner prod-
+uct spaces
+�
+Dmax(A)/Dmin(A), [·, ·]A
+� ∼=
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+.
+Proof. We first prove that multiplication by ω gives a well-defined map
+Dmax(A) ∋ u �→ ωu ∈ Dmax(A∧).
+Note that with u also ωu ∈ Dmax(A) by our localization assumption. Now pick
+another cut-off function ˜ω ∈ C∞
+c ([0, ε)) such that ˜ω ≡ 1 in a neighborhood of
+supp(ω). Let φ ∈ Dmin(A∧) be arbitrary, and write φ = ˜ωφ + (1 − ˜ω)φ. Since
+Dmin(A∧) = x
+1
+2 H (R+; E1) ∩ L2(R+; E0)
+as a consequence of our assumptions we have that both ˜ωφ, (1 − ˜ω)φ ∈ Dmin(A∧),
+see Section 3. We also have ˜ωφ ∈ Dmin(A) by our localization and compatibility as-
+sumption with respect to the minimal domains. Using the locality of the differential
+operators A∧ and A we get
+⟨A∧φ, ωu⟩ = ⟨A∧(˜ωφ), ωu⟩ = ⟨A(˜ωφ), ωu⟩ = ⟨˜ωφ, Amax(ωu)⟩
+= ⟨φ, ˜ωAmax(ωu)⟩ = ⟨φ, Amax(ωu)⟩.
+As this is valid for all φ ∈ Dmin(A∧) we see that ωu ∈ Dmax(A∧) with A∧,max(ωu)
+given as the restriction of Amax(ωu) to U(Y ) and extended trivially to R+ × Y . As
+for u ∈ Dmin(A) we also have ωu ∈ Dmin(A∧) by assumption, we thus obtain that
+the map
+Dmax(A)/Dmin(A) ∋ u + Dmin(A) �−→ ωu + Dmin(A∧) ∈ Dmax(A∧)/Dmin(A∧)
+is well-defined.
+Conversely, multiplication by ω likewise gives a well-defined map
+Dmax(A∧) ∋ u �→ ωu ∈ Dmax(A).
+
+12
+THOMAS KRAINER
+Note that if u ∈ Dmax(A∧) then ωu ∈ Dmax(A∧) and (1 − ω)u ∈ Dmin(A∧) by
+Section 3.
+Now let ˜ω ∈ C∞
+c ([0, ε)) be such that ˜ω ≡ 1 in a neighborhood of
+supp(ω). Let φ ∈ Dmin(A) be arbitrary, and write φ = ˜ωφ + (1 − ˜ω)φ; by the
+localization and compatibility assumptions both terms are in Dmin(A), and we also
+have ˜ωφ ∈ Dmin(A∧). We get
+⟨Aφ, ωu⟩ = ⟨A(˜ωφ), ωu⟩ = ⟨A∧(˜ωφ), ωu⟩ = ⟨˜ωφ, A∧,max(ωu)⟩
+= ⟨φ, ˜ωA∧,max(ωu)⟩ = ⟨φ, A∧,max(ωu)⟩.
+This shows that ωu ∈ Dmax(A) with Amax(ωu) given by A∧,max(ωu) in U(Y ) and
+extended trivially to M. We thus obtain a map
+Dmax(A∧)/Dmin(A∧) ∋ u + Dmin(A∧) �−→ ωu + Dmin(A) ∈ Dmax(A)/Dmin(A),
+and both maps are inverses of each other.
+Finally, as for both A and A∧ each class in Dmax/Dmin has a representative
+supported in U(Y ), and by the standing product type assumptions both adjoint
+pairings agree on those representatives, the proposition follows.
+□
+Theorem 4.2 (Null-Cobordism Theorem). Under the stated product type, local-
+ization, and compatibility assumptions we have
+SF[ p(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ] = 0.
+If moreover
+p(σ) =
+�
+σ
+D∗
+D
+−σ
+�
+:
+D(D)
+⊕
+D(D∗)
+⊂ L2
+�
+Y ;
+E−
+⊕
+E+
+�
+→ L2
+�
+Y ;
+E−
+⊕
+E+
+�
+with an elliptic Fredholm operator of first order
+D : D(D) ⊂ L2(Y ; E−) → L2(Y ; E+),
+then ind[D : D(D) ⊂ L2(Y ; E−) → L2(Y ; E+)] = 0.
+Proof. By Proposition 4.1 we have a unitary equivalence between the indefinite
+inner product spaces
+�
+Dmax(A)/Dmin(A), [·, ·]A
+� ∼=
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+.
+Because
+sgn
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+= SF[ p(σ) : E1 ⊂ E0 → E0, −∞ < σ < ∞ ]
+by (3.6) it suffices to show that
+sgn
+�
+Dmax(A)/Dmin(A), [·, ·]A
+�
+= 0,
+and by Proposition 2.2 this will be the case if the embedding Dmax(A) ֒→ L2(M; E )
+is compact. Because A has finite deficiency indices we only need to prove that
+Dmin(A) ֒→ L2(M; E ) is compact. Now let ω, ˜ω ∈ C∞
+c ([0, ε)) be cut-off functions
+such that ˜ω ≡ 1 in a neighborhood of supp(ω). By assumption the multiplication
+operator
+1 − ω : Dmin(A) → L2(M; E )
+is compact, and
+˜ω : Dmin(A) → Dmin(A∧)
+is continuous. Now
+Dmin(A∧) = x
+1
+2 H (R+; E1) ∩ L2(R+; E0),
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+13
+and because E1 ֒→ E0 is compact, multiplication by ω is a compact operator
+ω : Dmin(A∧) → L2(R+; E0),
+see Section 3. Consequently, using the product type assumptions, the composition
+ω = ω˜ω : Dmin(A) → L2(M; E )
+is compact, which shows that the embedding ι = ω+(1−ω) : Dmin(A) → L2(M; E )
+is compact. Finally, the vanishing of the index in the special case of operators of
+first order follows from (3.7).
+□
+Appendix A. The null-cobordism theorem for closed manifolds
+In this appendix we discuss a version of the null-cobordism Theorem 4.2 for closed
+manifolds. Most of the previous assumptions no longer explicitly appear in this
+version, e.g., we do not assume product type geometry, and there isn’t an operator
+A on M at the outset, but symbolic assumptions instead. As mentioned in the
+introduction this is due to the richness of analytic tools available for this situation
+that allows to create the preconditions needed to apply Theorem 4.2 instead of
+having to assume them from the outset.
+Let Y be a closed, compact Riemannian manifold and E → Y be a Hermitian
+vector bundle, and consider a family
+p(σ) =
+µ
+�
+j=0
+aj(y, Dy)σj : C∞(Y ; E) → C∞(Y ; E), σ ∈ R,
+(A.1)
+where aj(y, Dy) ∈ Diffµ−j(Y ; E), and µ ≥ 1.
+We assume that the parameter-
+dependent principal symbol
+σσ(p)(y, η; σ) =
+µ
+�
+j=0
+σσ(aj)(y, η)σj : Ey → Ey
+(A.2)
+is invertible on
+�
+T ∗Y × R
+�
+\ 0, and that p(σ) = p(σ)∗ is (formally) selfadjoint. By
+elliptic and analytic Fredholm theory,
+R ∋ σ �→ p(σ) : Hµ(Y ; E) ⊂ L2(Y ; E) → L2(Y ; E)
+is a family of selfadjoint unbounded Fredholm operators acting in L2(Y ; E) that
+is invertible for all σ ∈ R except at finitely many points, and it makes sense to
+consider the spectral flow
+SF[p(σ)] := SF[p(σ) : Hµ(Y ; E) ⊂ L2(Y ; E) → L2(Y ; E), −∞ < σ < ∞] ∈ Z
+associated with p(σ).
+Lemma A.3. The spectral flow is an invariant of the principal symbol (A.2) in
+the sense that if pj(σ), j = 1, 2, are two elliptic selfadjoint families of order µ ≥ 1
+of the form (A.1) with σσ(p1)(y, η; σ) = σσ(p2)(y, η; σ) then SF[p1(σ)] = SF[p2(σ)].
+Proof. Let R > 0 be such that
+p1(σ) + s[p2(σ) − p1(σ)] : Hµ(Y ; E) ⊂ L2(Y ; E) → L2(Y ; E)
+is invertible for |σ| ≥ R > 0 and all 0 ≤ s ≤ 1. Consequently, this family is a ho-
+motopy of selfadjoint Fredholm functions on [−R, R], invertible at both endpoints,
+and by the homotopy invariance of the spectral flow for such families we see that
+SF[p1(σ)] = SF[p2(σ)].
+□
+
+14
+THOMAS KRAINER
+Suppose there exists a compact Riemannian manifold M with ∂M = Y . Utilizing
+the geodesic flow from the boundary in the direction of the inner normal vector field
+shows that there exists ε > 0 and a collar neighborhood map U(Y ) ∼= [0, ε) × Y
+near the boundary such that the metric in U(Y ) takes the form dx2 + gY (x) with
+a smooth family of metrics gY (x) on Y , 0 ≤ x < ε, and such that gY (0) = gY is
+the given metric on Y . Moreover, by choosing ε > 0 small enough, there exists a
+defining function for ∂M on M that in U(Y ) is represented by projection onto the
+coordinate in [0, ε). We’ll also denote this global defining function by x : M → R+.
+In particular,
+T ∗M
+��
+Y = T ∗Y ⊕ span{dx
+��
+Y }
+subject to these choices, and we can split variables (y, η; σ) ∈ T ∗M
+��
+Y accordingly.
+Theorem A.4 (Null-Cobordism Theorem). Let M be a compact Riemannian man-
+ifold M with ∂M = Y , and let E → M be a Hermitian vector bundle with E
+��
+Y = E.
+Let T ∗M
+��
+Y ∼= T ∗Y × R subject to the choices described above, and suppose there ex-
+ists a symmetric, elliptic, differential principal symbol a ∈ C∞(T ∗M \0; End(π∗E ))
+of order µ such that
+a(y, η; σ) = σσ(p)(y, η; σ) for (y, η; σ) ∈
+�
+T ∗M \ 0
+���
+Y ,
+where π : T ∗M → M is the canonical projection. Then SF[p(σ)] = 0.
+With the family p(σ) from (A.1) we associate the indicial operator
+A∧ = x−1
+µ
+�
+j=0
+aj(y, Dy)(xDx)j : C∞
+c (R+×Y ; E) ⊂ L2(R+×Y ; E) → L2(R+×Y ; E).
+(A.5)
+Here we also write E for its pull-back to R+ ×Y with respect to the projection onto
+Y , and equip R+ × Y with the product metric dx2 + gY . Then A∧ is symmetric
+and densely defined. Let Dmin(A∧) be the domain of the closure, and Dmax(A∧)
+be the domain of the adjoint.
+Proof of Theorem A.4. In the previously fixed collar neighborhood U(Y ) ∼= [0, ε)×
+Y we utilize standard deformations of the Riemannian metric on M, the Hermitian
+metric on E , and the principal symbol a to reduce to a product type structure near
+the boundary, as follows:
+Pick an isomorphism E
+��
+U(Y ) ∼= π∗
+Y E that is the identity over Y , where πY :
+[0, ε) × Y → Y is the projection map. With respect to the pull-back of the given
+Hermitian metric on E to π∗
+Y E, the metric on E
+��
+U(Y ) under this isomorphism is
+then represented by h(x, y) ∈ C∞([0, ε) × Y ; End(π∗
+Y E)) such that h = h∗ > 0 and
+h(0, y) = Id. Choose C∞-functions φ, ψ : [0, ε) → R with
+φ ≡ 0 on 0 ≤ x ≤ ε
+3, 0 < φ < 2ε
+3 on ε
+3 < x < 2ε
+3 , and φ ≡ x on 2ε
+3 ≤ x < ε;
+ψ ≡ x on 0 ≤ x ≤ ε
+3, ψ > 0 on ε
+3 < x < 2ε
+3 , and ψ ≡ 1 on 2ε
+3 ≤ x < ε.
+We then deform the Riemannian metric on U(Y ) and Hermitian metric on E
+��
+U(Y )
+to
+˜g = dx2 + gY (φ(x)) and ˜h(x, y) = h(φ(x), y) ∈ C∞([0, ε) × Y ; End(π∗
+Y E)),
+respectively, which both connect seamlessly with the Riemannian metric on M
+outside U(Y ), and the Hermitian metric on E . We also change the principal symbol
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+15
+in
+◦U(Y ) to
+˜a(x, y, η; σ) = ψ(x)−1a(φ(x), y, η; ψ(x)σ) : Ey → Ey
+(A.6)
+for (x, y, η; σ) ∈ T ∗�
+(0, ε)×Y
+�
+\0 with the obvious identifications of variables, which
+again connects seamlessly outside the collar neighborhood. The new homogeneous
+principal symbol ˜a ∈ C∞(T ∗
+◦M \ 0, End(π∗E )) is symmetric with respect to the
+new metric on E , and elliptic over
+◦
+M. In
+◦
+U(Y ) we have
+˜a(x, y, η; σ) = x−1 σσ(p)(y, η; xσ) : Ey → Ey for 0 < x < ε
+3
+by construction, which aligns with the principal symbol of A∧ from (A.5). Let
+now A ∈ Diffµ(
+◦M; E ) be symmetric C∞
+c (
+◦M; E ) → C∞
+c (
+◦M; E ) with respect to the
+L2-inner product associated with the modified metrics on M and E , respectively,
+such that the principal symbol σσ(A) = ˜a on T ∗
+◦M \ 0, and such that in
+◦U(Y ) we
+have A = A∧ on C∞
+c ((0, ε
+4) × Y ; E). Then
+A = x−1P : C∞
+c (
+◦
+M; E ) ⊂ L2(M; E ) = x− 1
+2 L2
+b(M; E ) → x− 1
+2 L2
+b(M; E )
+is symmetric, and P ∈ Diffµ
+b (M; E ) is b-elliptic (see [13]). Moreover, by construction
+p(σ) is the indicial family of the operator P.
+By analytic Fredholm theory p(σ) : Hµ(Y ; E) → L2(Y ; E) is invertible for σ ∈ C
+except for the discrete set specb(p). In the sequel it will be convenient to assume
+that specb(p) ∩ {σ ∈ C; 0 < |ℑ(σ)| ≤
+1
+2} = ∅. As explained in Section 3, this
+can be achieved by replacing p(σ) by p(tσ) for sufficiently small t > 0 if necessary,
+which does not impact the spectral flow. Moreover, the assumptions of the theorem
+pertaining to the principal symbol of p(σ) also hold for p(tσ); to see this pick a
+C∞-function χ : [0, ε) → R with
+χ ≡ t on 0 ≤ x ≤ ε
+3, χ > 0 on ε
+3 < x < 2ε
+3 , and χ ≡ 1 on 2ε
+3 ≤ x < ε,
+and alter the principal symbol (A.6) in
+◦U(Y ) to
+˜a(x, y, η; σ) = ψ(x)−1a(φ(x), y, η; ψ(x)χ(x)σ) : Ey → Ey
+for (x, y, η; σ) ∈ T ∗�
+(0, ε) × Y
+�
+\ 0. We may thus proceed without loss of generality
+under the assumption that specb(p) ∩ {σ ∈ C; 0 < |ℑ(σ)| ≤ 1
+2} = ∅. In view of
+Section 3 for A∧ and by invoking elliptic regularity for A we then get
+Dmin(A∧) = x
+1
+2 Hµ
+b (R+; L2(Y ; E)) ∩ x
+1
+2 L2
+b(R+; Hµ(Y ; E)) ∩ L2(R+ × Y ; E),
+Dmin(A) = x
+1
+2 Hµ
+b (M; E ),
+and
+Dmax(A∧) = Dmin(A∧) ⊕
+�
+σ0∈specb(p)∩R
+Eσ0(p),
+Dmax(A) = Dmin(A) ⊕
+�
+σ0∈specb(p)∩R
+Eσ0(p),
+where Eσ0(p) is defined as in (3.3) based on a cut-off function ω ∈ C∞
+c ([0, ε
+4)) with
+ω ≡ 1 near x = 0 so that elements in Eσ0(p) can interchangeably be regarded both
+as sections of E on R+ × Y , as well as sections of E on M supported near the
+boundary. In particular, this implies that
+�
+Dmax(A)/Dmin(A), [·, ·]A
+� ∼=
+�
+Dmax(A∧)/Dmin(A∧), [·, ·]A∧
+�
+
+16
+THOMAS KRAINER
+because [u, v]A∧ = [u, v]A for u, v ∈
+�
+σ0∈specb(p)∩R
+Eσ0(p) by construction. Finally, it
+remains to note that Dmax ֒→ x− 1
+4 Hµ
+b (M; E ), and the embedding x− 1
+4 Hµ
+b (M; E ) ֒→
+x− 1
+2 L2
+b(M; E ) = L2(M; E ) is compact.
+□
+Theorem A.4 and Lemma 3.9 imply:
+Corollary A.7 (Cobordism Invariance of the Index). Suppose that E = E− ⊕ E+
+is an orthogonal direct sum, and that the family (A.1) is of the form
+D(σ) =
+�σ
+D∗
+D
+−σ
+�
+: C∞
+�
+Y ;
+E−
+⊕
+E+
+�
+→ C∞
+�
+Y ;
+E−
+⊕
+E+
+�
+, σ ∈ R,
+where D : C∞(Y ; E−) → C∞(Y ; E+) is an elliptic differential operator of first
+order, and D∗ : C∞(Y ; E+) → C∞(Y ; E−) is its (formal) adjoint. Then
+SF[D(σ)] = ind D = dim ker(D) − dim ker(D∗).
+In particular, if the assumptions of Theorem A.4 hold, then ind(D) = 0.
+References
+[1] P. Albin, E. Leichtnam, R. Mazzeo, and P. Piazza, The signature package on Witt spaces,
+Ann. Sci. ´Ec. Norm. Sup´er. (4) 45 (2012), no. 2, 241–310.
+[2]
+, Hodge theory on Cheeger spaces, J. Reine Angew. Math. 744 (2018), 29–102.
+[3] M. Braverman, New proof of the cobordism invariance of the index, Proc. Amer. Math. Soc.
+130 (2002), no. 4, 1095–1101.
+[4] M. Braverman and P. Shi, Cobordism invariance of the index of Callias-type operators,
+Comm. Partial Differential Equations 41 (2016), no. 8, 1183–1203.
+[5] C. Carvalho, A K-theory proof of the cobordism invariance of the index, K-Theory 36 (2005),
+no. 1-2, 1–31.
+[6] L. Hartmann, M. Lesch, and B. Vertman, On the domain of Dirac and Laplace type operators
+on stratified spaces, J. Spectr. Theory 8 (2018), no. 4, 1295–1348.
+[7]
+, Resolvent trace asymptotics on stratified spaces, Pure Appl. Anal. 3 (2021), no. 1,
+75–108.
+[8] M. Hilsum, Bordism invariance in KK-theory, Math. Scand. 107 (2010), no. 1, 73–89.
+[9] N. Higson, A note on the cobordism invariance of the index, Topology 30 (1991), no. 3,
+439–443.
+[10] T. Krainer, Extensions of symmetric operators that are invariant under scaling and applica-
+tions to indicial operators, New York J. Math. 28 (2022), 705–772.
+[11] M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singu-
+larities, Topology 32 (1993), no. 3, 611–623.
+[12]
+, Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-
+Texte zur Math. vol 136, B.G. Teubner, Stuttgart, Leipzig, 1997.
+[13] R. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics,
+A K Peters, Ltd., Wellesley, MA, 1993.
+[14] S. Moroianu, Cusp geometry and the cobordism invariance of the index, Adv. Math. 194
+(2005), no. 2, 504–519.
+[15] V.E. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, and B.Yu. Sternin, Elliptic theory on singular
+manifolds, Differential and Integral Equations and Their Applications, vol. 7, Chapman &
+Hall/CRC, Boca Raton, FL, 2006.
+[16] L. Nicolaescu, On the cobordism invariance of the index of Dirac operators, Proc. Amer.
+Math. Soc. 125 (1997), no. 9, 2797–2801.
+[17] R. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies,
+No. 57, Princeton University Press, Princeton, NJ, 1965.
+[18] B.-W. Schulze,
+Pseudo-differential calculus on manifolds with geometric singularities,
+Pseudo-differential operators:
+Partial differential equations and time-frequency analysis,
+pp. 37–83, Fields Inst. Commun., vol. 52, Amer. Math. Soc., Providence, RI, 2007.
+
+COBORDISM INVARIANCE OF THE INDEX REVISITED
+17
+[19] C. Wulff, Bordism invariance of the coarse index, Proc. Amer. Math. Soc. 140 (2012), no. 8,
+2693–2697.
+Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601-3760
+Email address: krainer@psu.edu
+

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+ON GRANULARITY OF PROSODIC REPRESENTATIONS IN EXPRESSIVE
+TEXT-TO-SPEECH
+Mikolaj Babianski, Kamil Pokora, Raahil Shah, Rafal Sienkiewicz, Daniel Korzekwa, Viacheslav Klimkov
+Amazon Text-to-Speech Research
+{babiansk, kamipoko, vklimkov}@amazon.com
+ABSTRACT
+In expressive speech synthesis it is widely adopted to use
+latent prosody representations to deal with variability of the
+data during training.
+Same text may correspond to vari-
+ous acoustic realizations, which is known as a one-to-many
+mapping problem in text-to-speech.
+Utterance, word, or
+phoneme-level representations are extracted from target sig-
+nal in an auto-encoding setup, to complement phonetic input
+and simplify that mapping. This paper compares prosodic
+embeddings at different levels of granularity and examines
+their prediction from text. We show that utterance-level em-
+beddings have insufficient capacity and phoneme-level tend
+to introduce instabilities when predicted from text. Word-
+level representations impose balance between capacity and
+predictability. As a result, we close the gap in naturalness by
+90% between synthetic speech and recordings on LibriTTS
+dataset, without sacrificing intelligibility.
+Index Terms— speech synthesis, TTS, prosody, Text-to-
+Speech, representation learning
+1. INTRODUCTION
+Neural Text-to-Speech (NTTS) [1] is characterized by syn-
+thesizing speech waveform solely with deep neural networks.
+This paradigm greatly enhanced naturalness and flexibility of
+speech synthesis. It enables new applications such as expres-
+sive [2, 3] and low-resource [4] speech generation, speaker
+identity [5] and prosody transplantation [6, 7]. This paper
+focuses on expressive speech synthesis, i.e.
+generation of
+speech that originally contains great degree of variation in
+terms of intonation and inflections.
+This variation is not described by phoneme sequence, typ-
+ically used as input to NTTS. Thus, the statistical model has
+to perform a one-to-many mapping, where the same input
+text can correspond to different acoustic realizations. Vanilla
+modelling approaches suffer from averaging and fail to repro-
+duce the original variability of the training data.
+To avoid averaging, it is common to use additional input
+that describes variability in the data. Initially, it was proposed
+to extract a single latent representation of the target speech in
+an auto-encoder manner for the whole utterance [8]. Target
+speech is not available during inference, so either the cen-
+troid representation is used [9] or it is separately predicted
+from text [10, 11].
+A single representation for the whole
+utterance can’t store temporal information effectively, thus,
+it was proposed to use more fine-grained representations at
+the phoneme-level for the task of prosody transplantation [6,
+12]. This idea was further expanded to text-to-speech, where
+word-level [13, 14] and phoneme-level [15, 16, 17] represen-
+tations were utilized. At the fine-grained level, prosody can
+be represented with pre-extracted features such as pitch, en-
+ergy, spectral tilt, but learnt representations can convey more
+information and represent more abstract aspects of prosody
+such as emotions. Therefore, in the rest of the paper we focus
+on learnt representations.
+This paper provides a systematic comparison of prosodic
+representations at different levels of granularity. We compare
+performance of utterance, word, and phoneme-level prosody
+embeddings in terms of a) capacity: what if we have a perfect
+prosody predictor; b) predictability: how sensitive is the ap-
+proach to inaccurate prosody predictions. Main contributions
+of this study are:
+• We systematically compare prosody embeddings at dif-
+ferent levels of granularity.
+• A solution to intelligibility issues in the case of phoneme-
+level prosody reference is proposed.
+• We show the trade-off between capacity and pre-
+dictability of prosody embeddings, advocating the use
+of word-level representations.
+• We examine data quantity and input features needed for
+robust prosody prediction from text.
+The rest of the paper is organized as follows: Section 2
+describes the text-to-speech framework used; Section 3 elab-
+orates on prosody embedding prediction from text; Section 4
+compares prosody embeddings at different levels of granular-
+ity in objective and subjective evaluations; Section 5 presents
+ablation studies on prosody embedding prediction; Section 6
+concludes the paper.
+978-1-6654-7189-3/22/$31.00 ©2023 IEEE
+arXiv:2301.11446v1  [eess.AS]  26 Jan 2023
+
+(a)
+(b)
+Fig. 1. Schematic diagram of the TTS model during a) training and b) inference. The dashed arrow denotes sampling from
+parametric distribution. Components in red are of prosody embeddings granularity (utterance/word/phoneme). Green, dashed
+lines denote loss functions.
+2. ACOUSTIC MODEL
+The backbone of our acoustic model architecture (Figure 1) is
+similar to the explicit duration TTS model presented in Shah
+et al. [4]. It follows the encoder-decoder paradigm, where the
+input phoneme sequence x is encoded by a phoneme encoder
+presented in the Tacotron2 [1] paper. We concatenate the en-
+coded phoneme sequence with both speaker s and prosody z
+embeddings upsampled by repetition [15, 16] to the phoneme-
+level. Speaker embeddings are represented as corresponding
+entries in the embedding look-up table. Prosody embeddings
+are obtained via compression of the mel-spectrogram y with
+the use of variational prosody reference encoder described in
+Section 2.1. During inference, the encoded sequence is up-
+sampled accordingly to alignments produced by the duration
+model, described in Section 2.2. The upsampled sequence is
+then passed to the decoder to map the disentangled linguistic
+features, speaker and prosodic contents into acoustic param-
+eters represented as mel-spectrograms. In this work, we use
+the non-autoregressive decoder presented in Shah et al. [4].
+2.1. Variational Prosody Reference Encoder
+To alleviate the one-to-many problem of TTS we use the vari-
+ational prosody reference encoder [16]. We aim to learn the
+latent representation of the information, which cannot be de-
+rived from the other input streams - phoneme sequence and
+speaker embedding. For clarity of the architecture presenta-
+tion, here we describe only one level of granularity - word-
+level. Modification of the model architecture to adjust for dif-
+ferent prosody embedding granularities is described in Sec-
+tion 2.3. The variational reference encoder (Figure 1a) takes
+target mel-spectrogram frames as input and converts them
+into a sequence of n latent vectors z, which corresponds to
+the number of words in the utterance. We refer to this repre-
+sentation as word-level prosody embeddings.
+The encoder comprises a stack of six residual gated con-
+volution blocks [18]. Each residual gated convolution block
+is composed of a 1D-convolution with a kernel size of 15 and
+a hidden dimension of 512, followed by a tanh filter and a
+sigmoid activation gate which are element-wise multiplied
+and then added to a residual connection. The convolution
+stack is followed by a BiLSTM layer with a hidden dimension
+of 128. We use a dropout of 0.1 in convolutional and BiLSTM
+layers. The BiLSTM layer output is firstly aggregated to the
+word-level by taking a middle frame of each word. Then,
+after a dense projection we obtain a sequence of Gaussian
+distribution parameters µ and σ, which we use to sample a
+sequence of prosody embeddings corresponding to words z
+of dimension 8. Finally, we upsample the word-level prosody
+embeddings by repetition to the phoneme-level and concate-
+nate them with the phoneme encoder output (Figure 1a).
+As we do not have access to target mel-spectrograms at
+inference time (Figure 1b), a separate model is introduced to
+predict prosody embeddings z from text. The architecture for
+this model is described in Section 3.
+2.2. Duration Model
+Neural TTS requires learning the alignment between two dif-
+ferent length sequences, which are the text, represented by
+phonemes, and speech, represented by acoustic parameters
+i.e. mel-spectrogram frames. There are two major approaches
+to obtain the alignment: attention-based [1] and explicit-
+duration-based [4, 19, 16, 20]. Attention-based components
+typically used for this task have known instabilities, which
+exhibit in synthesised speech as mumbling, early cut-offs,
+word repetition and word skipping [21, 22, 23]. Following
+recent research in the field inspired by traditional parametric
+speech synthesis techniques [24, 25], these issues are mit-
+igated by explicitly modelling the durations of phonemes
+[4, 19, 16, 20] which we decide to use in this work.
+
+mel-spectrogram
+y
+6 x Gated Conv
+Duration Model
+BiLSTM
+Aggregation
+predicted duration
+Dense Proj.
+d
+N(0,1)
+: L2
+μ,a
+DKL
+predicted
+oracle duration
+mel-spectrogram
+d
+y
+prosody
+embeddings (z)
+Concat
+Upsampling
+Encoder
+Decoder
+speaker embedding
+phoneme sequence
+x
+sBERT embeddings
+phoneme sequence
+b
+x
+Prosody
+Embeddings
+Duration Model
+Predictor
+predicted
+predicted duration
+mel-spectrogram
+p
+y
+prosody embeddings
+Encoder
+Concat
+Upsampling
+Decoder
+phoneme sequence
+speaker embedding
+x
+sWe use forced alignment from a Gaussian Mixture Model
+(GMM) based external aligner in the Kaldi Speech Recog-
+nition Toolkit [26] to produce ground truth duration for
+each phoneme, represented as the integer number of mel-
+spectrogram frames it corresponds to. To predict these du-
+rations, we train a duration model component following the
+architecture detailed in Shah et al. [4], with the addition of
+speaker and prosody embeddings conditioning. The duration
+model is trained jointly with the acoustic model by minimiz-
+ing L2 loss function in the logarithmic domain between pre-
+dicted and ground truth phoneme durations. During training,
+teacher forcing is used, i.e. the acoustic model uses ground
+truth duration values to upsample a phoneme’s encoding to
+the respective number of mel-spectrogram frames.
+2.3. Prosody Embeddings Granularity
+The model described above uses word-level prosody embed-
+dings. Which means that there is one embedding correspond-
+ing to each word in the input text. In this work we also explore
+two other levels of prosody modelling granularity: phoneme-
+level (one embedding per each phoneme) and utterance-level
+(single embedding for the whole utterance). For the phoneme-
+level prosody modelling we change the reference encoder to
+output one embedding per each phoneme and reduce prosody
+embedding dimension from 8 to 3, which we found optimal
+in terms of stability, for this level of granularity. In the case
+of utterance-level prosody modelling we use a stride of 2 in
+the residual gated convolution blocks of the reference encoder
+in order to gradually downsample the time resolution [27].
+Then, we project the first and last state of the BiLSTM layer
+into two vectors of dimension 64, which represent mean µ
+and standard deviation σ of the posterior distribution. Finally
+we sample a single 64-dimensional prosody embedding z cor-
+responding to the whole utterance.
+2.4. Training Procedure
+To train the acoustic and duration models we use Adam opti-
+miser with β1 = 0.9 and β2 = 0.98. We use a linear warm-up
+of the learning rate from 0.1 to 1 for the first 10k steps, fol-
+lowed by an exponential decay from 10k steps to 100k steps
+with a minimum value of 10−5. Acoustic and duration models
+are trained jointly for 500K steps with a batch-size equal 32
+and are optimized with respect to the following loss function:
+Ltotal = L1melspectrogram + L2logduration + γ ∗ DKL (1)
+where L1melspectrogram is the L1-distance between pre-
+dicted and oracle mel-spectrograms and L2logduration is the
+L2-distance between predicted and ground truth durations
+calculated in the logarithmic domain.
+DKL is the Kull-
+back–Leibler divergence between outputs of the variational
+prosody reference encoder and N(0, 1). We find the optimal
+value of γ to be 10−3 for the phoneme-level and 10−5 for
+the utterance and word-level prosody modelling. We present
+ablation of the γ parameter in Section 4.4.
+3. PROSODY EMBEDDINGS PREDICTOR
+At inference time we do not have access to target mel-
+spectrograms, therefore, we use a separate model to pre-
+dict prosodic representations z from text (Figure 1b). The
+prosody embeddings predictor model (Figure 2) has three in-
+put streams: phoneme sequence, contextual word embeddings
+extracted with a pre-trained BERT model [28] and speaker
+embedding. Phoneme sequence and contextual word embed-
+dings are encoded by separate Tacotron2-based encoders. We
+upsample the encoded BERT embeddings from the word-
+level to the phoneme-level and the speaker embedding to the
+phoneme-level, before concatenating them with the encoded
+phoneme representations.
+Next, we pass the concatenated
+sequence to another Tactotron2-based encoder block. After
+that, in the case of word-level embeddings prediction, the
+encoded phoneme-level representation is aggregated to the
+word-level by taking the middle frame of each word.
+Fi-
+nally, we use an autoregressive decoder to predict prosody
+embeddings. The autoregressive decoder is inspired by the
+architecture of the Tacotron2 mel-spectrogram decoder. In
+order to adapt the decoder to the prosody prediction task, we
+reduce the hidden dimension of the LSTM-layers to 128 and
+pre-net hidden dimension to 6. To predict the utterance-level
+prosody representation, instead of the autoregressive decoder,
+we use a simple linear projection layer.
+We train this model using prosody embeddings extracted
+with previously trained acoustic model as target labels.
+Specifically, the model is trained to predict posterior mean µ
+for each target prosody embedding using L2 loss and teacher-
+forcing framework.
+Fig. 2. Schematic diagram of the prosody embeddings predic-
+tor model. Components with dashed border are used only for
+fine-grained (word or phoneme-level) prosody embeddings
+prediction.
+
+BERT embeddings
+phoneme sequence
+b
+x
+Encoder
+Encoder
+speaker embedding
+s
+predicted
+prosody embeddings
+Concat
+Autoregressive
+Encoder
+Decoder
+phoneme-level
+word-level
+Aggregation
+representation
+representation4. EXPERIMENTS - PROSODY EMBEDDINGS
+GRANULARITY
+In this section we conduct a systematic study of prosodic rep-
+resentations at different levels of granularity applied to the ex-
+pressive TTS task. The performance of utterance, word, and
+phoneme-level prosody embeddings is compared in terms of
+capacity and predictability. We evaluate naturalness as well
+as stability and intelligibility of synthesized speech.
+4.1. Data
+Evaluations are conducted on a publicly available corpus of
+audiobook recordings - LibriTTS [29]. From which we use
+only recordings marked as clean. The training set consists
+of approximately 250 hours of speech (split into 140,000 ut-
+terances) narrated in an expressive manner by 1229 speakers.
+For validation we use a held-out set of 1000 randomly se-
+lected utterances from the 100 most frequent speakers. We
+extract 80-band mel-spectrograms with a 12.5 ms frame-shift
+as acoustic features.
+4.2. Systems
+We use our acoustic model (Section 2) along with the prosody
+embeddings predictor (Section 3) to test 3 levels of prosody
+embeddings granularity:
+1) G-VAE - utterance-level.
+2)
+W-VAE - word-level. 3) P-VAE - phoneme-level. All mel-
+spectrogram prediction systems are used in combination with
+the Universal Neural Parallel WaveNet Vocoder [30] in order
+to obtain a 24kHz audio signal.
+4.3. Subjective Evaluation Protocol
+For the subjective evaluation we conduct MUSHRA tests [31]
+with the Amazon Mechanical Turk platform. 60 native En-
+glish speakers are presented with the samples in a random
+order side-by-side, and are asked to “Evaluate naturalness of
+the samples on the scale from 0 to 100.” A total of 1000 ut-
+terances are used for testing and the test is balanced in such a
+way that each test case is scored by 3 listeners independently.
+Ground truth mel-spectrograms vocoded with the Universal
+Parallel WaveNet Vocoder (Ref system) are used as an upper
+anchor. The significance of the MUSHRA results is analyzed
+using a Wilcoxon signed-rank test with Bonferroni-Holm cor-
+rection applied [32].
+4.4. Stability
+To quantify the stability of tested systems and the intelligibil-
+ity of synthesized speech we conduct Word Error Rate (WER)
+analysis. The whole test set of 1000 utterances described in
+Section 4.1 is used for the evaluation. We transcribe speech
+generated in the TTS mode (prosody embeddings predicted
+from text) with the ASpIRE Chain ASR model from Kaldi.
+Then the WER is computed between the sentence text and the
+corresponding transcription.
+The G-VAE and W-VAE models have WER scores (Ta-
+ble 1) comparable to recordings when trained with DKL loss
+weight (γ) equal to 10−5. Whereas, training the P-VAE model
+in an analogical setup results in significant stability issues.
+We believe that this is caused by the phoneme-level prosody
+embeddings distribution being very hard to predict from text.
+Only after applying more regularization during training by in-
+creasing DKL loss weight we are able to obtain a phoneme-
+level model matching other systems in terms of WER score.
+Intelligibility is a crucial property of TTS system. Therefore,
+for all the other experiments we use the G-VAE and W-VAE
+models trained with γ = 10−5 and the P-VAE model trained
+with γ = 10−3, as they are matching our stability require-
+ments. We found that further increasing γ parameter does not
+bring any significant improvements and may lead to degrada-
+tion in segmental quality of synthesized audio.
+System
+DKLγ
+WER ↓
+G-VAE
+1e − 5
+2.18% ± 0.30
+W-VAE
+1e − 5
+2.13% ± 0.30
+P-VAE
+1e − 5
+3.59% ± 0.36
+1e − 4
+2.47% ± 0.31
+1e − 3
+2.17% ± 0.29
+1e − 2
+2.17% ± 0.29
+Ref
+−
+2.29% ± 0.31
+Table 1. Word Error Rate with 95% confidence intervals [33]
+computed across the 1000 test utterances, along with DKL
+loss weight (γ).
+4.5. Capacity
+We analyse the best-case performance of acoustic models by
+simulating perfectly predicted prosody embeddings in the Or-
+acle Resynthesis setup. That is, at inference time we pro-
+vide ground truth latent representations from the variational
+reference encoder for all tested systems. We evaluate natu-
+ralness in a subjective test as described in Section 4.3 and
+summarize results in Figure 3a. In this setup, the G-VAE
+model scores significantly lower than the other systems (p-
+value < 0.01), suggesting that fine-grained embeddings are
+required for natural prosody modelling. There is no statisti-
+cally significant difference between W-VAE and P-VAE sys-
+tems (p-value > 0.01). It is worth mentioning, that we have
+also conducted an analogical experiment with a P-VAE model
+trained with lower DKL loss weight (γ = 10−5). With such
+model, when ground truth prosody embeddings provided dur-
+ing inference, we are able to reconstruct speech almost per-
+fectly. However, it comes at a cost of stability issues, when
+prosody embeddings predicted from text are used as described
+in Section 4.4.
+
+(a) Ground truth prosody embeddings
+(b) Prosody embeddings predicted from text
+Fig. 3. Subjective listeners ratings from the naturalness MUSHRA tests with a) ground truth prosody embeddings and b) prosody
+embeddings predicted from text (TTS). Mean scores are reported below the system names.
+To gain a deeper understanding of prosody representa-
+tions, we evaluate our acoustic models in the Oracle Resyn-
+thesis setup with changed speaker embedding. That is, we
+extract prosody from a source recording and resynthesize the
+same text with changed voice - a so-called Voice Conversion
+(VC). We convert all 1000 test utterances into 4 selected (two
+male and two female) target voices.
+To effectively measure how close the prosody patterns
+of converted speech are to the source recordings, we first
+extract fundamental frequency (F0) at the frame-level from
+source and converted audio pairs with the RAPT algorithm
+[34]. Then we calculate two metrics commonly used to mea-
+sure the linear dependence of prosody contours: Pearson
+Correlation Coefficient (PCC) and Root Mean Squared Error
+(RMSE) [35]. We can see that the results (Table 2) are very
+similar for the W-VAE and P-VAE models, while the G-VAE
+performs much worse in both metrics. This reinforces the
+subjective evaluation findings that utterance-level embed-
+dings do not provide sufficient capacity to capture expressive
+prosody and fine-grained modelling is required for expressive
+speech generation.
+System
+F0 RMSE ↓
+F0 PCC ↑
+G-VAE
+1.693 ± 0.012
+0.760 ± 0.003
+W-VAE
+1.535 ± 0.011
+0.801 ± 0.002
+P-VAE
+1.539 ± 0.012
+0.802 ± 0.003
+Table 2. Objective Voice Conversion evaluation metrics with
+95% confidence intervals computed between source and con-
+verted speech: F0 Root Mean Square Error (RMSE), F0 Pear-
+son Correlation Coefficient (PCC) [35].
+4.6. Predictability
+Finally, we evaluate our systems in the TTS scenario. That
+is, we generate speech from textual input only and provide
+prosody embeddings predicted from text during inference.
+The naturalness of synthesized speech is evaluated subjec-
+tively as described in Section 4.3. The results are summarised
+in Figure 3b (all comparisons are statistically significant with
+p-value < 0.01). The G-VAE model scores much worse
+than other systems. It fails to reproduce expressive speech
+variability and tends to output flat prosody contours due to
+averaging. We also observe issues with accurate phoneme
+duration prediction for the G-VAE model, e.g.
+unnatural,
+fast-paced speech. We hypothesise, that a single representa-
+tion for the whole utterance can’t store temporal information
+effectively. Such issues are not visible in the models using
+fine-grained prosody representations.
+While both of them
+score much higher than the G-VAE, the word-level model
+performs better in terms of fidelity and prosody naturalness
+and closes the gap between the P-VAE model and the ref-
+erence system by over 90%. We conclude that word-level
+representations impose a good compromise for expressive
+prosody modelling granularity. They have enough capacity to
+produce varied and natural speech and still can be accurately
+predicted from text.
+5. EXPERIMENTS - PROSODY PREDICTION
+In this section we focus on semantically concerted prosody
+prediction. First, we investigate the impact of data quantity
+used in the training procedure. Second, we conduct an abla-
+tion study of the prosody embeddings predictor input streams.
+5.1. Data Quantity
+We investigate the impact of training data quantity on our
+system, by looking into a single-speaker scenario with lim-
+ited amount of data.
+We build a dataset by taking 15000
+utterances from the HiFi corpus [36] coming from one male
+speaker (id 6097). Again, we keep a held-out set of 1000
+randomly selected utterances for validation.
+We train the
+
+100
+80
+ Score
+60
+MUSHRA
+40
+20
+0
+G-VAE
+W-VAE
+P-VAE
+Ref
+67.77
+72.72
+72.03
+74.00100
+80
+Score
+60
+MUSHRA
+40
+20
+0
+G-VAE
+W-VAE
+P-VAE
+Ref
+62.99
+71.65
+68.62
+71.80word-level acoustic model and the prosody embeddings pre-
+dictor in two setups: 1) HiFi - using only 15000 utterances
+from a single HiFi speaker. 2) HiFi+LibriTTS - addition-
+ally adding LibriTTS corpus, which results in approximately
+155000 utterances in the training set.
+We evaluate both
+scenarios using a MUSHRA subjective naturalness test anal-
+ogously to Section 4.6 and summarize results in Table 3.
+Using an additional, large-scale dataset during training sig-
+nificantly improves naturalness of synthesized speech (p-
+value < 0.01). Per-case analysis of listeners judgements
+reveal that both systems produce audio of similar segmental
+quality, but the HiFi+LibriTTS system provides more stable
+prosody, especially for longer utterances. We conclude that
+semantically concerted prosody prediction is a data hungry
+problem and limited amount of training data can result in less
+stable prosody of generated speech. However, using auxiliary
+data in the training procedure allows to obtain a more robust
+prosody predictor and mitigate this issue.
+HiFi
+HiFi+LibriTTS
+Ref
+64.26
+67.01
+67.17
+Table 3. Mean MUSHRA scores for the word-level models
+trained on a single speaker form the HiFi corpus with and
+without auxiliary LibriTTS data. All comparisons from this
+Table are statistically significant (p-value < 0.01).
+5.2. Prosody Predictor Input Streams Ablation Study
+In previous works, word-level prosody embeddings are typ-
+ically predicted at inference time from one of the following
+representations derived from text: word-level contextual em-
+beddings [13, 37] or phoneme sequence [14, 38]. We use both
+of them in our prosody embeddings predictor model. To de-
+termine the contribution of each input stream to the model
+performance we conduct an ablation study. We train the word-
+level prosody embeddings predictor model in three configura-
+tions: BERT & Phoneme - trained exactly as described in sec-
+tion 3; BERT - with only BERT embeddings input; Phoneme -
+with only phoneme sequence input. All three predictor mod-
+els are used in combination with the same W-VAE acoustic
+model to synthesize 1000 validation utterances from section
+4.1. As a subjective naturalness evaluation, a preference test
+is carried out using Amazon Mechanical Turk platform. Na-
+tive English speakers are asked to ”Select which audio sounds
+more natural” for pairs of audio samples generated with dif-
+ferent systems. We find that removing fine-grained phoneme
+sequence input stream (Figure 4a) causes less stable prosody
+prediction and results in significant degradation in naturalness
+(p-value < 0.01). Whereas, the difference between BERT &
+Phoneme and Phoneme systems (Figure 4b) is not statistically
+significant. Listening to samples revealed that the improve-
+ment of using contextual word embeddings comes mainly in
+better phrasing and pause prediction. However, differences
+are quite subtle and therefore are not reflected in the crowd-
+sourced naturalness evaluation results.
+This result contra-
+dicts the findings from [13], where significant quality degra-
+dation when removing BERT embeddings input was reported.
+However, in [13] authors proposed to use word-level syntax
+features e.g. part-of-speech labels, compound-noun flag and
+punctuation flag as a second input stream to the model pre-
+dicting prosody embeddings.
+We believe that fine-grained
+phoneme sequence input is more correlated with prosody than
+those syntax features, therefore, removing BERT input stream
+is less harmful in our work.
+(a)
+(b)
+Fig. 4.
+Results of naturalness preference tests.
+On the
+right the word-level prosody embeddings predictor model de-
+scribed in Section 3. On the left the same model but with
+only one input stream: a) BERT embeddings, b) phoneme se-
+quence.
+6. CONCLUSIONS
+In this paper we explored the design of prosodic representa-
+tions learned in an auto-encoder manner. First, we introduced
+a TTS framework allowing for a fair comparison of prosody
+embeddings of different granularity. Then a systematic study
+of utterance, word and phoneme-level representations was
+conducted on a large-scale, publicly available dataset - Lib-
+riTTS. Through our experiments, we demonstrated the trade-
+off between capacity and predictability of prosody represen-
+tations.
+We showed that utterance-level embeddings have
+insufficient capacity to model expressive speech variability.
+Whereas phoneme-level representations require strong reg-
+ularization for stable prediction from text at inference time.
+We found that word-level embeddings impose a good balance
+between capacity and predictability. As a result, we closed
+the gap in naturalness by 90% between synthetic speech
+and recordings without sacrificing intelligibility.
+Finally,
+we looked into applying the proposed approach in a single-
+speaker scenario.
+We showed that semantically concerted
+prosody prediction is a data hungry problem and limited
+amount of training data can result in less stable prosody of
+generated speech. However, this issue can be mitigated by
+using auxiliary data in the training procedure.
+
+BERT
+No-Pref
+BERT & Phoneme
+(32.87%)
+(31.78%)
+(35.35%)
+0
+25
+50
+75
+100Phoneme
+No-Pref
+BERT & Phoneme
+(34.26%)
+(31.27%)
+(34.47%)
+0
+25
+50
+75
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+

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+Conversational Turn-taking as a Stochastic Process
+on Networks
+Lisa O’Bryan1, Santiago Segarra2, Jensine Paoletti1, Stephanie Zajac1, Margaret E. Beier1,
+Ashutosh Sabharwal2, Matthew Wettergreen3, and Eduardo Salas1
+Abstract—Understanding why certain individuals work well
+(or poorly) together as a team is a key research focus in
+the psychological and behavioral sciences and a fundamental
+problem for team-based organizations. Nevertheless, we have a
+limited ability to predict the social and work-related dynamics
+that will emerge from a given combination of team members.
+In this work, we model vocal turn-taking behavior within
+conversations as a parametric stochastic process on a network
+composed of the team members. More precisely, we model
+the dynamic of exchanging the ‘speaker token’ among team
+members as a random walk in a graph that is driven by both
+individual level features and the conversation history. We fit our
+model to conversational turn-taking data extracted from audio
+recordings of multinational student teams during undergraduate
+engineering design internships. Through this real-world data we
+validate the explanatory power of our model and we unveil
+statistically significant differences in speaking behaviors between
+team members of different nationalities.
+I. INTRODUCTION
+Data consisting of entities in interconnected systems are
+ubiquitous in multiple fields. Thus, network structures are
+commonly used across many disciplines for representation and
+analysis of complex information [1], from neuroscience [2]
+to wireless communications [3]. In this work, we represent
+interactions within teams as small in-person social networks.
+Computational models can be an effective way to study
+the social dynamics that emerge from individuals interact-
+ing within groups [4]. In particular, a variety of modeling
+approaches have been used to try to replicate natural turn-
+taking behaviors observed in conversation [5]–[8]. Many of
+these models have been successful in replicating realistic
+patterns of conversational turn-taking. However, the ability to
+understand the driving mechanisms underlying these patterns
+and generalize to novel team compositions is lacking. As
+a step towards this goal, we develop a stochastic model of
+conversations that can be used to explore how individual dif-
+ferences impact the emergence of turn-taking patterns within
+teams. More precisely, we propose a parametric model that
+captures the individuals’ innate tendency to speak as well
+as the effect that having spoken recently has on speaking
+again. At every point in time, the next speaker is drawn
+1Department of Psychological Sciences, Rice University, Houston, TX,
+USA. 2Department of Electrical and Computer Engineering, Rice University,
+Houston, TX, USA. 3Department of Bioengineering, Rice University, Hous-
+ton, TX, USA. Subsets of the team data used in this manuscript have been
+previously published as part of dissertations by Stephanie Zajac, Department
+of Psychological Sciences, Rice University and Jian Cao, Department of
+Electrical and Computer Engineering, Rice University. Funding for this project
+was provided by a Microsoft Productivity Research Grant, the National
+Science Foundation (Award Number: 1910117), and the Army Research
+Institute (Grant Number: W911NF-22-1-0226).
+from a probability distribution determined by the history
+of speakers and the aforementioned parameters. The model
+replicates the majority of conversational turn-taking patterns
+observed in our real-world data, and our results highlight the
+important role the memory function plays in replicating these
+patterns. Furthermore, our results indicate that differences in
+team member nationality can play a strong role in shaping
+communication patterns within multinational teams.
+Contributions. The contributions of our work are twofold:
+i) We propose a simple parametric stochastic process that can
+capture complex behaviors observed in real data.
+ii) We present a novel dataset of conversational turn-taking
+in undergraduate teams and we apply our model to reveal
+significant differences in speaking behavior between student
+nationalities.
+II. CONVERSATION MODEL
+Inspired by a model by Stasser and Taylor [7], our model
+incorporates two key notions: i) the relative likelihood πi that
+team member i speaks on a given turn independent of their
+speaking history and ii) the effect mi that an individual’s
+current speaking turn has on their likelihood of speaking
+on subsequent turns. We consider the inherent likelihood of
+speaking πi of each member i as independent of the history
+of exchanges and, thus, a constant throughout the conversation.
+In contrast, we encode dependencies within a conversation
+through the (turn-dependent) memory function mi(t). More
+precisely, for a given member i and a turn t, the memory
+value is given by
+mi(t) = di e−0.5 (t−tlast
+i
+),
+(1)
+where di is a learnable parameter that controls the scale of the
+memory effect for each individual and tlast
+i
+denotes the last
+turn on which member i spoke. The negative exponential form
+in (1) reveals that the memory value asymptotically decreases
+to 0 as (t − tlast
+i
+) increases, i.e., as more turns occur since
+the last time that member i spoke. This encodes the natural
+assumption that whether or not an individual spoke many turns
+ago is inconsequential to their likelihood of speaking next. The
+memory function in (1) is combined with the innate speaking
+tendencies πi to compute the likelihood ℓi(t) that member i
+speaks at turn t as follows
+ℓi(t) =
+�
+0,
+if tlast
+i
+= t − 1,
+πi + mi(t),
+otherwise.
+(2)
+Speakers are not allowed to speak on two consecutive turns
+since these would simply be considered part of the same turn.
+arXiv:2301.04030v1  [cs.SI]  10 Jan 2023
+
+This is enforced in (2) by setting the likelihood to zero for the
+member that has just spoken. Lastly, denoting by N the total
+number of team members, the likelihoods ℓi(t) are normalized
+to sum up to 1 so that they define bona fide probabilities pi(t)
+as follows
+pi(t) =
+ℓi(t)
+�N
+j=1 ℓj(t)
+.
+(3)
+The speaker at turn t is then drawn from this probability
+distribution across team members.
+In summary, the conversational behavior of each individual
+i within our model is given by two parameters (πi, di). Given
+these parameters for every team member, the model provides
+a well-defined stochastic process to generate conversations
+by the team. More precisely, to determine the speaker at
+turn t, we first compute the memory values of each member
+following (1), we then compute likelihoods and transform
+those into probabilities following (2) and (3), respectively,
+and we finally draw the next speaker from that probability
+distribution.
+Our main departure from Stasser and Taylor [7] is that
+our model is based on individual-level parameters whereas
+theirs is based on team-level parameters. Specifically, Stasser
+and Taylor’s [7] model depends on a single parameter r
+that determines the inherent speaking probability of every
+individual (what we denote by πi) as well as a single parameter
+d that determines the scale of the memory function for every
+team member. This fundamental difference is a key enabler for
+our study of how individual traits relate to each team member’s
+conversational behavior since, given observed conversations of
+a team, our model enables the estimation of the parameters
+(πi, di) for every team member.
+Given observed turn-taking data, we can fit our model by
+selecting the parameters (πi, di) for every team member that
+maximize the probability of generating the observed data.
+More precisely, if we denote by Ht−1 the history of turn-
+taking up to turn t − 1 in the observed data for a given
+team, and by ht the speaker at turn t, we can compute the
+probability that our model selects that true speaker ht [cf. (3)].
+Following the notation in (3), we denote this probability by
+pht(t | Ht−1, {(πi, di)}n
+i=1), i.e., the probability of selecting
+the true speaker ht at turn t but where we have now made
+explicit that this value depends on the past history Ht−1 and
+the parameters (πi, di) for each of the n members in the team.
+With this notation in place, the log-likelihood of observing the
+true history of T turns is given by
+L(HT | {(πi, di)}n
+i=1) =
+T
+�
+t=1
+log pht(t | Ht−1, {(πi, di)}n
+i=1).
+(4)
+We fit our model by finding the parameters {(πi, di)}n
+i=1
+that maximize (4). We also fit a reduced model that does
+not contain the memory parameters {di}n
+i=1 or, equivalently,
+where di = 0 for all i. We did this to determine the minimal
+viable model that can explain our observed data.
+III. DATASET
+In 2016 and 2017, we collected data on team interactions
+in student engineering design teams during 7-week internships
+at a private university in the southern United States. The first
+week of the internship consisted of a condensed course on
+the engineering design process, which helped to ensure all
+participants had a similar baseline level of knowledge. During
+the remaining six weeks, team members worked together to
+plan and execute their project which sought to meet a real-
+world need. We collected data from 7 multi-national teams
+with team members from the United States (n = 13), Malawi
+(n = 7), and Brazil (n = 4), with equal numbers of female and
+male participants. After consenting to the study, participants
+completed a self-report survey of their personality traits,
+attitudes, and demographic information. From these data we
+extracted five features for each individual that we hypothesized
+could relate to individual differences in speaking patterns,
+namely, extraversion, agreeableness, conscientiousness, sex
+(male, female), and nationality (American, Non-American).
+Our dataset includes multiple meetings from throughout the
+internships for all teams. Audio streams were processed by
+annotating the start and end times of speaking turns by each
+team member during the meetings. Overall, we extracted a
+mean (SD) of 1941 (1416.5) speaking turns per team.
+IV. NUMERICAL EXPERIMENTS
+To assess the predictive power of the full and reduced
+(i.e., without the memory component) models, we perform the
+following three classes of experiments.
+Predicting the next speaker. For each team, we split their
+turn-taking history into a training and a testing set. The
+training data contains the first 80 percent of the total turns
+whereas the testing data contains the remaining turns. As
+previously explained, we compute the maximum likelihood
+estimates of the model parameters but this time based only
+on maximizing the probability of observing the history of
+the training dataset. We then compute the log-likelihood of
+observing the history of the testing dataset, as in (4), for
+both the full and reduced fitted models. The larger (less
+negative) the attained value, the better predictive power of the
+corresponding model.
+Overall, the full simulation model (i.e., with memory pa-
+rameter) predicts the observed data better than the reduced
+simulation model. Table I shows the log-likelihoods attained
+for the testing dataset (last 20 percent of speaking turns)
+for each team and simulation model. The full simulation
+model consistently yields larger (less negative) log-likelihoods,
+indicating a better predictive performance.
+Reproducing relevant conversation patterns.
+We test the fit of both the full and reduced simulation
+models by comparing three measures between our observed
+and simulated datasets: 1) the proportion of time in which each
+team member spoke, 2) the proportion of a given speaker’s
+speaking turns following an ABA format in which there was
+one turn by a different speaker between a given speaker’s
+sequential turns (reflecting the proportion of turns that were
+part of dyadic exchanges), and 3) the proportion of turns that
+
+TABLE I
+LOG-LIKELIHOOD ATTAINED BY BOTH MODELS
+Team
+No Memory
+Memory
+Team 1
+-159.1586
+-154.4533
+Team 2
+-45.3267
+-43.2123
+Team 3
+-196.5418
+-188.6291
+Team 4
+-278.8524
+-255.2545
+Team 5
+-909.8953
+-617.3478
+Team 6
+-623.9848
+-589.0674
+Team 7
+-105.8199
+-104.0864
+were part of long dyadic exchanges (4 or more consecutive
+speaking turns (e.g. ABAB) between two team members).
+We calculate these measures for each of 10,000 replications
+of our simulation models and compare them to the values
+found in our observed data from each team. For each of our
+three measures, we find the proportion of model replications
+in which the observed value in the real data fell within the
+95 percent confidence interval for values produced by each
+simulation model. For each of the three speaking patterns of
+interest, we use chi-squared tests to compare the number of
+individuals or dyads across teams whose behaviors are not
+significantly different from those displayed by the full and
+reduced simulation models.
+The full simulation model matches the patterns displayed
+by significantly more individuals and dyads across teams
+than the reduced simulation model. The full simulation model
+correctly estimates the proportion of speaking turns spoken
+by each team member for 100% (24/24) of team members
+whereas the reduced simulation model correctly estimates the
+proportion for 70.8% (17/24) of team members (χ2 = 6.0, p
+= 0.014; Figure 1(a)). Moreover, the full simulation model
+correctly estimates the proportion of each team member’s
+speaking turns with an ABA format for 87.5% (21/24) of team
+members, but the reduced simulation model correctly estimates
+the proportion for 29.2% (7/24) of team members, with the
+tendency to underestimate the proportion of turns (χ2 = 14.5,
+p = 0.00014; Figure 1(b)). Finally, the full simulation model
+correctly estimates the proportion of speaking turns that were
+part of dyadic exchanges of length 4 turns or greater for 86.7%
+(26/30) of team member dyads, and the reduced simulation
+model correctly estimates the proportion for 36.7% (11/30) of
+team member dyads, with the tendency to underestimate the
+proportion of turns (χ2 = 13.8, p = 0.00020; Figure 1(c)).
+Relating individual traits and speaking behavior. To gain
+insight into the relative importance of different individual traits
+in understanding speaking behaviors, we use an information-
+theoretic approach [9] to determine which trait(s) best explain
+between-individual variation in model parameters πi (baseline
+likelihood of speaking) and di (likelihood of speaking again
+after speaking recently). Using the MuMIn function [10] in
+R, we examine which linear model (i.e., a null model and 5
+uni-variate models consisting of each of our individual-level
+predictor variables; see Section III) best explains variation in
+each parameter value across team members. We group-mean
+center our three continuous variables (extraversion, agreeable-
+ness, conscientiousness) to reflect the relative values of these
+personality traits among team members. We limit the number
+of variables per linear model to one to avoid overfitting. We
+rank our linear models according to the Akaike information
+criterion adjusted for small sample sizes (AICc) [9]. We con-
+sider top-performing linear models to be the best performing
+model (i.e. the model with the lowest AICc value) in our model
+set and any model less than 2 AICc points greater than the
+best performing linear model [9]. We examine the correlation
+between individual traits and simulation model parameters for
+all top-performing linear models to determine how these traits
+shape speaking behaviors.
+Tables II and III display the results of our model selection
+analysis that compares the relative ability of each of our five
+univariate models to explain between-individual variation in
+πi and di. The tables display the AICc values for each model
+as well as the ∆AICc values (relative to the best model)
+and corresponding model weights. Model weights reflect the
+relative support for a given linear model compared to the other
+candidate models, with 1 indicating full support.
+TABLE II
+MODEL SELECTION FOR BASELINE LIKELIHOOD OF SPEAKING πi
+model
+df
+AICc
+∆
+wt
+Nationality
+3
+-19.6
+0.0
+0.94
+Null
+2
+-12.4
+7.2
+0.025
+Agreeableness
+3
+-11.4
+8.2
+0.015
+Sex
+3
+-10.3
+9.3
+0.010
+Extraversion
+3
+-10.0
+9.6
+0.010
+Conscientiousness
+3
+-9.9
+9.7
+0.010
+TABLE III
+MODEL SELECTION FOR SHAPE OF MEMORY FUNCTION di
+model
+df
+AICc
+∆
+wt
+Null
+2
+105.2
+0.0
+0.42
+Extraversion
+3
+107.5
+2.4
+0.13
+Nationality
+3
+107.7
+2.5
+0.12
+Conscientiousness
+3
+107.8
+2.6
+0.11
+Sex
+3
+107.8
+2.6
+0.11
+Agreeableness
+3
+107.8
+2.6
+0.11
+The linear model that best explains between-individual dif-
+ferences in πi, the parameter reflecting the baseline likelihood
+of initiating a speaking turn, has nationality as the predictor
+variable. This linear model has a cumulative model weight
+of 93.5%. The second best linear model is the null model,
+which has a ∆AICc value 7.2 higher than the best model
+(Table II). Since this ∆AICc value is greater than our criterion
+of ∆AICc = 2 [9], we only consider the linear model with
+nationality as a predictor variable as a top-performing model
+within our model set. Overall, this model is supported 37.6
+times more strongly (evidence ratio = wi/wj = 0.94/0.025 =
+37.6) than the null model. When we analyze our top model,
+we find that Americans have significantly higher likelihoods
+of initiating speaking turns than non-Americans (β = 0.20, p
+< 0.01, Figure 2).
+The linear model that best explains between-individual
+differences in di, the change in likelihood of speaking after
+having just spoken, is the null model. This linear model has
+a cumulative model weight of 41.6%. The second best linear
+
+Fig. 1. Black points represent a) observed proportion of speaking turns by each team member, b) observed proportion of speaking turns
+with one turn in between (e.g. ABA) for each team member, c) observed proportion of speaking turns that were part of consecutive dyadic
+exchanges of length 4 turns or greater. Error bars represent the 95 percent confidence intervals for the proportions estimated by the reduced
+simulation model (i.e., without memory parameter) (red) and full simulation model (blue). Y-axis scale varies by team to improve visibility
+Fig. 2. Boxplot of baseline likelihood of speaking (parameter πi) by
+nationality across all teams.
+model has extraversion as a predictor variable and a ∆AICc
+value of 2.4 (Table III). Thus, only the null model is con-
+sidered a top-performing linear model within our model set,
+indicating that none of the predictor variables we considered
+explain between-individual variation in di. Overall, the null
+model is supported approximately 3.2 times more strongly
+(evidence ratio = wi/wj = 0.42/0.13 = 3.2) than the model
+with extraversion as a predictor.
+V. DISCUSSION
+The presence of the memory parameter is important in
+simulating the patterns of vocal turn-taking we observed in
+our study. Compared to the reduced simulation model with no
+memory parameter, the full simulation model more accurately
+predicts future speaking turns and better captures individual
+and dyadic speaking patterns. This result supports the findings
+by Stasser and Taylor [7] and Parker [5] that an individual’s
+current likelihood of speaking is impacted by their recent
+speaking behaviors. Nevertheless, our results differ from those
+of Stasser and Taylor in that we find different parameter values
+controlling memory function shape for different individuals.
+Our study finds evidence for consistent between-individual
+differences in speaking behaviors supporting previous find-
+ings that individual traits can correlate with communication
+behaviors [11]–[13]. Our finding that non-Americans initiate
+speaking turns less frequently than Americans is consistent
+with a recent study by Li et al. [11] which found that Chinese
+team members, who tended to be less proficient in English, ini-
+tiated fewer speaking turns than the American team members.
+Although we did not measure English language proficiency in
+our study, our finding could be related to language proficiency
+since the non-American students in our study were non-native
+English speakers. Another reason why non-Americans may not
+have initiated speaking turns as frequently could be that they
+had a perceived lower status than American team members.
+Social status may be awarded to the ethnic subgroup with
+the greatest numerical majority [14]. Since both the Brazilian
+and Malawian students were completing the internship at
+
+a)
+Team 1
+Team 2
+Team 3
+Team 4
+Team 5
+Team 6
+Team 7
+王
+王
+工工
+0.40-
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+0.32
+0.4-
+0.4 -
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+0.32
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+0.2
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+0.2
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+0.20-
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+2
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+0.8-
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+0.75
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+0.6
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+0.6
+0.6
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+0.6
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+(2
+0.3
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+0.6 -
+FI
+T
+0.4
+0.5
+Exchanges
+0.5-
+工
+Long
+0.20
+0.6
+0.4 
+0.4-
+0.3-
+0.2
+0.4 
+0.15
+0.3
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+0.10
+0.2
+0.2
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+.
+0.1-
+0.05
+0.1
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+0.1-
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+1
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+工
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+0.0
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+0.0 -
+0.00
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+2.3.4.3.4
+2.3.
+4
+2. 3.
+3
+3
+2'2'
+3
+Dyad0.6
+Baseline Likelihood
+0.4
+0.2
+American
+Nonamerican
+Nationalityan American university and were outnumbered by American
+students, they may have demonstrated lower status behaviors
+like speaking up less frequently [12].
+Although nationality best explained differences in baseline
+frequency of speaking turn initiation, none of our predictor
+variables explained variation in memory function shape. Future
+studies are needed to determine whether other traits may
+explain the observed variation in this speaking tendency.
+Nevertheless, since Americans were more likely to initiate
+speaking turns, the broad tendency to speak again after hav-
+ing recently spoken further enhanced individual differences
+in speaking frequency across team members. Overall, these
+results help expand knowledge of the impact cultural diversity
+can have on team processes [15].
+A limitation of our study was that we only had data on a
+relatively small number of teams and team members. This
+lack of power prevented us from exploring more complex
+relationships between individual traits and their impacts on
+speaking behaviors. For example, Neubert and Tagger [16]
+found that gender moderated the relationship between indi-
+vidual traits and leadership, with certain traits being more
+important for leadership in males than females and vice
+versa. Since we tested each of our predictor variables on its
+own, the strong effect of nationality may have overpowered
+more subtle or complicated effects of other variables, such as
+personality and gender. This could be a reason why individual
+traits like extraversion, which can be strongly correlated with
+communication tendencies [17]–[19], did not correspond to
+individual differences in speaking behaviors in our study.
+Extending our study to more teams would enable a greater
+understanding of how multiple traits may interact to impact
+speaking behaviors.
+VI. CONCLUSION AND FUTURE WORK
+Our study develops a model of conversational turn-taking
+that can provide a mechanistic understanding of how patterns
+of communication emerge within teams and can be used to
+investigate the relationship between team member traits and
+specific speaking behaviors. Future extensions of our model
+could integrate more fine-grained speaking behaviors such as
+the timing between turns and turn overlap, which may enable
+the study of more complex or subtle turn-taking dynamics. For
+example, individuals higher in dominance have been found
+to interrupt more often, which can have a suppressive effect
+on the speaking behaviors of others [20]. Ultimately, through
+extensions of our modeling approach, it could be possible to
+predict the conversational interactions among team members
+based on their trait composition alone. This ability could
+enable the anticipation of undesirable team outcomes (e.g., de-
+velopment of subgroups) so that interventions could be applied
+ahead of time. Similarly, for established teams, it could also be
+possible to predict the effects team composition changes may
+have on communication patterns, thus providing guidelines for
+restaffing or retraining team members. Such predictive models
+would represent a significant advancement in teams research,
+enabling a more mechanistic understanding of the connection
+between team composition and team processes [21], [22].
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+

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+WuYun: Exploring hierarchical skeleton-guided
+melody generation using knowledge-enhanced deep
+learning
+Kejun Zhang1,2,3,∗,
+Xinda Wu1,∗,
+Tieyao Zhang1,
+Zhijie Huang1,
+Xu Tan4,
+Qihao Liang1,
+Songruoyao Wu1,
+Lingyun Sun1,2,†
+1College of Computer Science and Technology, Zhejiang University, China.
+2Alibaba-Zhejiang University Joint Institute of Frontier Technologies, China.
+3Innovation Center of Yangtze River Delta, China.
+4Microsoft Research Asia
+{zhangkejun, wuxinda, kreutzer0421, zj_huang,
+qhliang, 12221193, sunly}@zju.edu.cn
+xuta@microsoft.com
+Abstract
+Although deep learning has revolutionized music generation, existing methods
+for structured melody generation follow an end-to-end left-to-right note-by-note
+generative paradigm and treat each note equally. Here, we present WuYun, a
+knowledge-enhanced deep learning architecture for improving the structure of
+generated melodies, which first generates the most structurally important notes to
+construct a melodic skeleton and subsequently infills it with dynamically decorative
+notes into a full-fledged melody. Specifically, we use music domain knowledge
+to extract melodic skeletons and employ sequence learning to reconstruct them,
+which serve as additional knowledge to provide auxiliary guidance for the melody
+generation process. We demonstrate that WuYun can generate melodies with better
+long-term structure and musicality and outperforms other state-of-the-art methods
+by 0.51 on average on all subjective evaluation metrics. Our study provides a
+multidisciplinary lens to design melodic hierarchical structures and bridge the
+gap between data-driven and knowledge-based approaches for numerous music
+generation tasks.
+1
+Introduction
+Automatic music generation is one of the popular multidisciplinary research topics in generative
+art and computational creativity (1), which has achieved revolutionary advances in various artificial
+intelligence-generated content applications by utilizing deep learning techniques (2, 3), including
+interactive music production collaboration tools (4, 5), video background music generation (6), music
+education (7), and music therapy (8). As one of the crucial components of music generation, melody
+generation has drawn much attention from both the academic and industrial fields. Although melodies
+appear to be a simple linear succession of notes unfolding over time, the organizational structure of
+the melodic notes is hierarchical, like a tree resulting in intricate long-distance dependencies (9, 10).
+Hence, the complex long-distance dependencies make it difficult for neural networks to discover and
+learn the hierarchical structure relationships among these musical elements and generate long-term
+structured melodies. In recent years, language models in natural language processing (NLP) have
+∗Equal contribution.
+†Corresponding author.
+Preprint. Under review.
+arXiv:2301.04488v1  [cs.SD]  11 Jan 2023
+
+been employed to capture long-distance dependencies for structured melody generation with the
+advantages of an easy-to-use end-to-end deep learning framework, effective representation learning,
+and arbitrary sequence length generation. Their powerful ability to automatically learn the latent
+knowledge from big data, without explicitly codifying the domain-specific rules, has been proved and
+applied in multiple disciplines (11–14).
+Numerous specialized architectures of the language model for music generation have demonstrated
+promising performance in generating long-range coherent melodies, including effective attention
+mechanisms (15, 16), enhanced memory networks (17–19), large-scale deep neural networks (20),
+and explicit musicality regularization (21). Furthermore, various MIDI-derived symbolic music
+representation methods designed auxiliary musical spatiotemporal symbols (e.g., BAR, POSITION,
+and CHORD) for the input symbolic music data to help music generation models learn the long-
+distance dependencies better, longer, and faster (17–19, 22, 23). However, the scarcity of publicly
+available melody data limits the usage of the power of language-based music generation models.
+Moreover, the process of melody generation still lacks controllability. These models are trained in the
+dominant end-to-end and data-driven learning paradigms, which optimize the network’s large-scale
+parameters via learning to map the input data to output data, thus occasionally resulting in excessive
+repetition or boring sounds in the generated music (21).
+Recent studies used a deep learning-based hierarchical generation strategy to first hallucinate or
+predict the object’s structure and then use it to constrain downstream generation tasks (e.g., protein,
+font, or music) (24–33), which enables the neural networks to learn from the limited data far
+more efficiently and improves the controllability of the generation process. For structured melody
+generation, some scholars first generate a melody’s hierarchical music structure representation (31)
+or bar-level musical structure relationship graph (32, 33) and then generate melodies conditioned
+on the generated parallel structure information as additional knowledge. Such a strategy requires
+recognizing the group structure of the musical syntax in a melodic surface (e.g., phrases and sections)
+to extract music features for building a structure generation model. Nonetheless, inadequate music
+structure boundary detection algorithms hinder the extraction of accurate melodic group structure.
+Conversely, little attention has been paid to the organizational logic of the deep structure beneath
+the melodic surface, organized by different levels of structural importance among various musical
+events (34–36) with the potential to enhance structured melody generation. Typically, the majority
+of existing melody generation methods for pursuing long-term structure follow an end-to-end left-
+to-right note-by-note generative paradigm and treat each note equally. So far, however, there is still
+an insufficient investigation into an alternative order of melody generation and the difference in the
+relative structural importance among musical notes.
+In this study, we propose WuYun, a hierarchical skeleton-guided melody generation architecture
+based on knowledge-enhanced deep learning that incorporates the melodic skeleton as deep structural
+support to provide explicit guidance on the development direction of melody generation (Fig. 1A).
+WuYun follows the hierarchical organization principle of structure and prolongation (35, 37), thus
+dividing traditional single-stage end-to-end melody generation into two stages: melodic skeleton
+construction and melody inpainting (Fig. 1B). At the stage of melodic skeleton construction, we first
+extract the most structurally important notes in a musical piece from rhythm and pitch dimensions
+as melodic skeletons on the basis of the music domain knowledge. We then train an autoregressive
+decoder-only Transformer-based network (38) on the collected melodic skeleton data to construct
+new melodic skeletons (Fig. 1C, a). We treat the melodic skeleton as the underlying framework of the
+final generated melody. At the stage of melody inpainting, we adopt a Transformer encoder–decoder
+architecture (39) to elaborate the melodic skeleton into a full-fledged melody by encoding the melodic
+skeleton as additional knowledge into the decoder to guide the melody generation process (Fig. 1C,
+b). To prove the effectiveness of the architecture, we evaluate WuYun on a publicly available melody
+dataset. Experimental results show that the generated melodic skeleton has comparable quality with
+the real one extracted by our proposed melodic skeleton extraction framework. The hierarchical
+skeleton-guided melody generation architecture effectively improves generated melodies’ long-term
+structure and musicality and outperforms other state-of-the-art methods by 0.51 on average on all
+subjective evaluation metrics.
+2
+
+Melodic skeleton
+Melody
+Melody
+Database
+Melodic Skeleton Extraction
+Framework
+Pairing
+B
+Stage 1: melodic skeleton construction
+Stage 2: melody inpainting
+Melodic
+Skeleton
+Database
+Hierarchical structure analysis in music
+(Music domain knowledge)
+Input Module
+(Embedding)
+Positional 
+Encoding
+Transformer-XL
+（4 blocks）
+Output Module
+(Classifier)
+C
+a) Melodic Skeleton Generation Module
+Input
+(Melodic skeleton)
+Representation
+Input Module
+(Embedding)
+Encoder
+（4 blocks）
+Decoder
+（4 blocks）
+Recurrent Transformer
+b) Melodic Prolongation Generation Module
+Representation
+Input Module
+(Embedding)
+Input
+(Melody)
+Representation
+Input
+(Melodic skeleton)
+Output Module
+(Classifier)
+Train
+A
+Melodic Skeleton
+Melody (example)
+Melodic skeleton
+Sequence learning model
+Y1
+Y2
+Yn
+···
+···
+X1
+X2
+Xn
+···
+···
+Partial sequence
+Melodic skeleton
+···
+X1
+X2
+Xn
+···
+Encoder
+Y2
+Y3
+Yn
+···
+Y1
+Y2
+Yn-1
+···
+···
+Decoder
+…
+Predicted tokens
+Predicted tokens
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+···
+Sequence-to-sequence
+learning model
+Melody
+Figure 1: Architecture of WuYun. (A) The first eight bars of the melody of “Hey Jude” from The Beatles
+(excluding anacrusis). The upper part of the figure shows the basic shape of the melodic motion, and the low
+part of the figure shows the melodic skeleton in the rhythm dimension. Every melody has an underlying melodic
+skeleton that provides structural support and connections among musical elements to guide the melodic motion.
+(B) Hierarchical melody generation process. WuYun divides the melody generation process into melodic skeleton
+construction and melody inpainting stages following the hierarchical organization principle of structure and
+prolongation. At the melodic skeleton construction stage, the melodic skeleton extraction framework is proposed
+to extract the melodic skeleton in the rhythm and pitch dimensions by the hierarchical structure theory from
+music domain knowledge. A neural network for sequence learning trained on melodic skeletons can generate
+novel ones. At the melody inpainting stage, another neural network for sequence-to-sequence learning would
+fill the generated melodic skeleton into a full-fledged melody. (C) Architecture details of WuYun. WuYun is
+composed of a melodic skeleton generation module and a melodic prolongation generation module; the former
+is used for the melodic skeleton construction stage, and the latter is used for the melody inpainting stage with the
+guidance of the melodic skeleton.
+3
+
+OO4车
+42
+Result
+2.1
+Hierarchical organization principle of structure and prolongation
+Most AI artistic generative models differ significantly from humans in their artistic creation process,
+especially in music generation. For example, a typical music generation model generates music
+content sequentially from left to right at once (40). However, human artworks tend to develop
+iteratively from a basic underlying idea or structure through elaboration, expansion, and individual
+shaping. Human artistic creation follows an age-old fundamental principle of creative thinking,
+namely, structure and prolongation, which has significantly contributed to human thinking and
+creativity. In the art of music, this principle governs the underlying logic in musical composition
+and makes musical reasoning and explanation comprehensible and acceptable (37). It conforms
+to the brain’s cognitive processing mechanism of structurally organizing sequential information
+(41–44), which makes the brain encode and process information more efficiently and improves
+musical memories (45, 46). For example, musicians use this principle, consciously or unconsciously,
+to study, organize, and perform their musical works.
+The hierarchical structure is a key feature of the tonal musical syntax system, where musical elements
+are almost always hierarchically organized by strict rules at a fundamental level rather than unlimited
+creative expression (36). Some researchers have investigated the patterns of structural organization
+and generalized them into music theories regarding the hierarchical structure in music from the
+perspective of the structure and prolongation principle. Schenker (47) was the first to introduce this
+principle to describe the musical structure in a hierarchically organized way. The central idea of
+Schenkerian theory about the hierarchical structure is that some musical events are elaborated by
+other musical events in a recursive and embedded fashion (9, 34). That is, not all musical events
+are equally important. Some musical events have structural importance as stable factors in music,
+whereas others are more decorative as dynamic factors. Therefore, Schenker proposed different
+levels of structure hierarchy to organize tonal music and analyze its motion. Based on Schenker’s
+ideas, the generative theory of tonal music (GTTM), proposed by Lerdahl and Jackendoff (34), is
+one of the most influential theories in current music theory and music psychology. GTTM provides
+a systematic analysis and description of the hierarchical structure in music. Based on the listeners’
+perception of tonal music, GTTM lists four hierarchical structure relationships from rhythm and pitch
+dimensions: grouping structure, metrical structure, time span reduction, and prolongational reduction.
+The term “reduction” refers to the stepwise reduction of less important musical events from the
+musical surface, revealing the underlying framework or skeleton that plays an essential role in the
+music’s qualities and developmental direction. In summary, under the surface of the music, musical
+events are hierarchically organized based on the structural stability in rhythm and pitch dimensions
+(48, 49).
+Inspired by the iterative mode of human composition guided by the principles of structure and
+prolongation, the whole process of melody creation can be seen as progressively filling individual
+decorative notes among the melodic skeleton; it is an effective modern composition technique that
+perfectly combines rules and composers’ personality (35). This composition technique has been
+developed and applied in music teaching for a long history. In the following, we elaborate on the
+melodic skeleton extraction framework from rhythm and pitch dimensions and introduce the design of
+WuYun melody generation architecture that first constructs the melodic skeleton and then completes
+the melody instead of sequentially generating a melody note-by-note at once. The manner of WuYun’s
+melody generation process is more musically meaningful than the dominant end-to-end left-to-right
+note-by-note melody generation paradigm.
+2.2
+Melodic skeleton extraction framework
+Music theories present that there is an underlying identifiable framework beneath the melody surface
+called the melodic skeleton (35, 50). The melodic skeleton is composed of certain notes, which
+sound more structurally important from rhythm and pitch dimensions (34, 48) and are called the
+skeleton notes. The skeleton note attracts the audience’s attention and makes a deeper impression
+on them. By contrast, the remaining part of the notes plays a decorative role, giving the melody
+personalities or styles, and are called the decorative notes or prolongation notes. The melodic skeleton
+serves as the crucial structural support of rhythm and harmony, indicating the direction of melody
+development. Knowledge of melodic skeleton information can help humans and machines better
+4
+
+Figure 2: Melodic skeleton extraction framework. (A) Rhythm pattern of strong and weak beat distribution
+in the 4/4 time signature with different note resolutions. (B) Rhythmic skeleton extraction. The rhythmic
+skeleton consists of the metrical accents, agogic accents on metrical accents, and agogic accents on syncopations
+in each measure. (C) Illustration of the tension measure in the pitch class helix of the spiral array with a C major
+chord. The right part of the figure presents the tension value of the notes in the first eight bars of the melody of
+“Hey Jude.” (D) Tonal skeleton extraction. The tonal skeleton consists of the notes with the minimum tension
+value in the rhythm cell.
+5
+
+A
+ Strong
+4th
+Weak
+Sub.
+Weak
+ Strong
+Strong
+8th
+Weak
+Weak
+Weak
+Strong
+Strong
+Weak
+Weak
+ Strong
+Weak
+ Strong
+Weak
+Weak
+Weak
+Weak
+ Strong
+ Strong
+Weak
+16th
+Weak
+Sub.
+Sub.
+Sub.
+Sub.
+Grid
+1
+2 
+3
+5
+6
+7
+8
+9
+10
+11
+12 
+13
+14
+15
+16
+4
+B
+Original
+Metrical Accent
+Agogic
+Synopation
+Rhythmic skeleton
+c
+1.9
+A#
+D#
+○ Tonal Skeleton Note
+1.8
+ Prolongation Note
+1.7
+G#
+F#
+B
+1.6
+C#
+1.5
+AD
+V
+D
+1.4
+G
+E
+E
+A
+E-
+B-
+C
+1.1
+IV
+F
+G-
+0.9
+G
+A-
+0.8
+D.
+0.7
+12
+14
+24
+26
+28
+30
+32
+40
+0
+16
+20
+22
+38
+Note Sequence
+D
+Rhythm Cell
+Tonal Skeletonanalyze, understand, and learn the logic of melodic hierarchy organization from surface to deep
+layers.
+Here, we introduce a framework for melodic skeleton extraction based on the knowledge of music
+theory (34, 35, 50–53) and music psychology (48, 49) to identify the dominant and subordinate
+relationship of structural importance from rhythm and pitch dimensions between melody notes. The
+theoretical basis and implementation details are described briefly below.
+2.2.1
+Rhythmic skeleton extraction
+Before describing the theoretical basis of this work, we would have to cover some basic musical
+terms and concepts regarding meter and rhythm in the time aspect of music. In music theory, the
+pulse splits time into a series of uniformly spaced chunks called beats. Not all beats are created equal,
+and certain beats are felt stronger than others. The first beat of each measure is a downbeat, and the
+one that follows is an upbeat. The meter measures the number of beats in the regular and repeated
+pattern of the downbeat. For example, the most common meter in music is 4/4; each measure has
+four quarter note beats. The distribution pattern of strong and weak beats is “strong weak sub-strong
+weak,” as shown in Fig. 2A. Rhythm can be defined as the organization pattern of one downbeat with
+one or more upbeat (54). Therefore, the meter provides the temporal framework for organizing music
+rhythm.
+To draw listeners’ attention, musicians often use accents in musical compositions or performances
+to emphasize a particular note. Accents can be expressed in various ways to increase musical
+expressiveness and add character to the movement of music. Metrical and rhythmic accents are the
+two main accents in the symbolic melody data. A metrical accent is an accent that falls on the strong
+beat position within a measure. The metrical accent is periodic and cyclic, whose distribution pattern
+depends on the type of meter. A rhythmic accent is an accent in a strong position within the rhythm,
+which emphasizes a point that is not constrained by the meter’s structure. Consequently, it is flexible
+and changeable rather than static and fixed. Rhythmic accents can be created by increasing the notes’
+dynamic (i.e., dynamic accent), extending the notes’ duration (agogic accent), using syncopation,
+and so forth. The agogic accent can be easily distinguished by comparing the surrounding notes on
+duration. Additionally, the musician can use syncopation to change the normal rhythm pattern by
+extending the duration of notes from the weak beat or weak beat part to the subsequent strong beat
+or strong beat part. Note that a note cannot be both a metrical accent and a syncopation, which are
+conflicted with each other. The dynamic accent is not used since it has no obvious changes in most
+symbolic melody data.
+In this study, we extract the metrical accents, the agogic accents falling on the metrical accent, and
+the agogic accents falling on the syncopation as the rhythmic skeleton notes, as illustrated in Fig.
+2B. Metrical accents are the foundation of other types of accent (34). When two or more different
+types of accents work together, the listener will experience a particularly more prominent accent.
+Therefore, when a rhythmic accent and a metrical accent overlap, the rhythmic accent is generally
+more perceptible and is the one that is preferred. If there were continuous rhythmic skeleton notes,
+we chose the most structurally important note as the rhythmic skeleton note according to the intensity
+of the accent.
+2.2.2
+Tonal skeleton extraction
+In tonal music, the pitch of the melody moves around a centrally stable note (i.e., the tonal center or
+tonic), repeatedly moving away from it and back to it. Pitches are essentially organized into a distinct
+hierarchy scale based on tonal stability. There is a mutual attraction between pitches with different
+stable levels, which can stimulate different emotional experiences (47). Specifically, an unstable
+pitch tends to be a stable pitch, which would make the listeners feel relaxed or dismissed. Moving
+from a stable to an unstable pitch would increase the listener’s sense of tension. The prolongational
+reduction theory of GTTM suggests that the more important music event has less tension and vice
+versa (53). Note that the same tone may have different feelings in different contexts, which may be
+pleasant or anxious.
+For the tonal skeleton extraction method, we use the tension level as a metric to quantify the relative
+importance of the pitch. The specific recognition procedure is as follows.
+6
+
+• First, we used the position of the rhythmic skeleton notes as the boundary of the individual
+context because the metrical structure is the important basis of all hierarchical structure
+types (34).
+• Second, we combined two or three successive notes as the minimum rhythmic cell in each
+segment, according to the repetition frequency in the melody (55) and the number of notes
+of this rhythmic cell. The term “rhythmic cell” defines as a “small rhythmic and melodic
+design that can be isolated or can make up one part of a thematic context” (56). Therefore,
+each rhythmic cell can be seen as an isolated thematic context for calculating the tension
+profile.
+• Finally, we adopted a mathematical tonal tension model to quantify each note’s tension
+value by calculating the distance between every single tone and global key in the spiral array
+(57–59), as shown in Fig. 2C. We selected the note with the minimum tension value in each
+rhythmic cell as the tonal skeleton note. For example, Fig. 2D shows the tonal skeleton of
+the first eight bars from the song “Hey Jude.”
+2.3
+Design of WuYun
+Figure 1C shows the diagram of the proposed hierarchical melody generation architecture called
+WuYun. First, we convert the melody MIDI files and their melodic skeletons into musical event
+sequences as the input data for model training using the MeMIDI symbolic music representation
+method. Then, we design a hierarchical melody generation architecture with two generative modules
+responsible for melodic skeleton construction and melody inpainting, respectively. In this subsection,
+we will introduce the hierarchical melody generation architecture about how we generate the melodic
+skeleton and incorporate it to guide the melody generation process. The details about the MeMIDI
+symbolic music representation method and the word embedding technique used in this architecture’s
+input module are described in the Materials and Methods section.
+WuYun is designed to generate melodies in two stages hierarchically: melodic skeleton construc-
+tion and melody inpainting, instead of the dominant end-to-end left-to-right note-by-note melody
+generation paradigm. At the stage of melodic skeleton construction, we use the Transformer-XL
+model with only the decoder as the melodic skeleton generation module (19), which has the ad-
+vantage of remarkable performance in capturing long-term dependence. To develop the capacity
+of melodic skeleton construction, we trained the Transformer-XL model on the extracted melodic
+skeleton database. At the stage of melody inpainting, we employ the recurrent Transformer-based
+encoder–decoder architecture (18) in a sequence-to-sequence setup as the melody inpainting module
+to complete the melody conditioned on the melodic skeleton, i.e., filling the missing information
+between the melodic skeleton notes. In this work, the melody inpainting problem can be defined as
+follows: given a melodic skeleton sequence Cs, generate an inpainted melody sequence Cm. The
+encoder maps the discrete input symbols of the melodic skeleton sequence Cs to a high-dimensional
+continuous vector as conditional input into the decoder, and the decoder then generates an output
+sequence Cm in an autoregressive manner. The melodic skeleton sequence will be saved in the final
+generated melody. This method provides users an entry point to interact with the melody generation
+model by adjusting melodic skeleton notes between two stages to control the melodic motion.
+In this work, we focus on designing a hierarchical skeleton-guided melody generation architecture
+based on knowledge-enhanced deep learning, following the hierarchical organization principle of
+structure and prolongation. Therefore, we used common language models in NLP to make the WuYun
+architecture accessible. The capacity of these two generative modules may be further optimized;
+however, the goal of this study is not to find the most optimal neural network.
+2.4
+Evaluation metrics
+Subjective and objective evaluations are the two essential aspects of evaluating the performance of
+music generation systems. The human listening test is currently an indispensable and viable method
+for subjective evaluation to measure the quality of the generated musical pieces. However, for the
+objective evaluation, many efforts have been made to design quantitative metrics; there is not a set of
+convincing and unified metrics. Although the research field of music generation is multidisciplinary,
+most researchers mainly focus on generative models with different improvement goals rather than
+their contribution to quantifying music complexity. Consequently, almost all the proposed objective
+7
+
+evaluation metrics are difficult to apply for comparing different music creation systems and lack
+sustainability for future development demands. We tried to calculate the averaging overlapped area
+of some musical feature distributions between generated musical pieces and ground-truth musical
+pieces as the objective evaluation metrics like (18, 60) using the public evaluation toolbox. We
+arrived at a similar conclusion as PopMNet (32) that a better result of objective evaluation does not
+mean better structure and musicality of generated music. The same objective evaluation result can be
+calculated and verified with the provided melody MIDI files of this study’s next two experiments.
+Therefore, we conducted two subjective evaluation experiments to evaluate the performance of our
+proposed WuYun, including different melodic skeleton settings in rhythm and pitch dimensions and
+comparisons with public state-of-the-art (SOTA) music generation models.
+We randomly selected ten melodies from the evaluation dataset for the listening materials. Similar to
+previous studies (15, 17, 19), we took the first four bars as prompt and set the maximum number of
+generated bars to 28. We assigned a random order for all musical pieces as the file name, including
+the generated and ground-truth musical pieces. All melody MIDI files were rendered into audio via a
+piano MIDI synthesizer. In the blind listening test, participants were asked to rate each melody on a
+five-point Likert scale (i.e., 1 for bad and 5 for excellent) on five dimensions:
+• Rhythm: Whether the brain can feel the regular accents and rhythm patterns.
+• Richness: Whether the melody sounds rich and interesting in the rhythm and pitch dimen-
+sions.
+• Structure: Whether the brain can feel the boundary of melodic phrases and the balance
+among melodic phrases’ length.
+• Expectation: Whether the direction of melody development meets the audience’s expecta-
+tions for melody development (61).
+• Overall: Overall quality.
+We found that most nonmusicians had heard technical music terms but did not understand what
+they meant. Therefore, before formal experiments, we conducted multiple rounds of discussions,
+testing, and validation with musicians and nonmusicians regarding the above subjective evaluation
+metrics and their descriptions until they could easily understand and grasp them. To ensure that
+the recruited subjects have a common understanding of the metrics and scales in the questionnaire,
+we also conducted evaluation training for them, including the explanation of subjective evaluation
+metrics and preliminary experiments. All audio and MIDI files for evaluation can be found in
+Acknowledgments.
+2.5
+Model performance based on different melodic skeleton settings
+To compare the effectiveness of variants of melodic skeleton extracted from rhythm and pitch
+dimensions, we comprehensively evaluated the performance of WuYun based on different settings of
+the melodic skeleton. Furthermore, we added three control group settings of randomly selected notes
+with different percentages as the melodic skeleton in order to verify the effectiveness of the proposed
+melodic skeleton extraction method based on music domain knowledge. All experimental settings
+and the proportion of melodic skeleton notes in the melody are described below:
+1. Downbeat only uses metrical accents as the melodic skeleton (32.8%).
+2. Long Note only uses agogic accents as the melodic skeleton (27.4%).
+3. Rhythm uses rhythmic skeleton notes as the melodic skeleton (33.8%).
+4. Tonic uses tonal skeleton notes as the melodic skeleton (43.2%).
+5. Interaction uses the intersection of rhythmic skeleton notes and tonal skeleton notes as the
+melodic skeleton (14.2%).
+6. Union uses the union of rhythmic skeleton notes and tonal skeleton notes as the melodic
+skeleton (62.8%).
+7. Random25% randomly selects 25% of melody notes as the melodic skeleton (25%).
+8. Random50% randomly selects 50% of melody notes as the melodic skeleton (50%).
+9. Random75% randomly selects 75% of melody notes as the melodic skeleton (75%).
+8
+
+In this experiment, we obtained 90 musical pieces for rating. We recruited 30 subjects (13 females
+and 17 males, ages 18 and 30 years) from Zhejiang University and Zhejiang Conservatory of Music
+to evaluate the musical pieces with payment. Fifteen subjects among them were professional music
+practitioners with an average of 9 years of music training and 4 years of music performance experience.
+The rest of the subjects have little professional music training or performance experience. Each
+musical piece was assigned to three professionals and three nonprofessional subjects. Each subject
+was required to rate 18 musical pieces, which cost approximately 25 min.
+Figure 3A shows the mean opinion scores of WuYun architecture’s melody generation performances
+with nine different settings on the five subjective evaluation metrics from all subjects in the form of
+histograms. The detailed experimental result is shown in Table S1. Generally, among all melodic
+skeleton settings, the proposed rhythmic and tonal skeleton based on music theory and psychological
+study performs better than other skeletons. The rhythmic skeleton setting (No. 3) achieved the best
+result on all subjective evaluation metrics, followed by the tonal skeleton setting (No. 4). Among
+the three types of melodic skeletons associated with rhythm (Nos. 1, 2, and 3), the melodic skeleton
+composed of a single type of accent (e.g., metrical accents or agogic accents (31)) has a large gap with
+the rhythmic skeleton in richness, expectation, and overall quality and even surpassed by the random
+melodic skeletons (Nos. 7, 8, and 9) on most subjective evaluation metrics. This result indicates that
+a flexible rhythmic skeleton (i.e., including several kinds of musical accents) is essential for melody
+composition to improve musicality. In contrast, a rigid melodic skeleton (i.e., including only one type
+of accent, especially metrical accents) reduces the quality of the generated melody and limits the
+models’ performance. Likewise, as depicted in the right part of Fig. 2C, the pitch classes of the tonal
+skeleton notes are mostly C, D, and E, which also lead to the rigidity of the generated tonal skeletons.
+Additionally, compared to the rhythmic and tonal skeleton settings, the intersection (No.5) and union
+(No. 6) skeleton settings led to a distinct degradation of the melody generation performance. For
+instance, the intersection (No. 5) skeleton setting received the worst scores in most evaluation aspects,
+even worse than the random sampling skeleton settings. This phenomenon can be explained by
+the structure and prolongation proportion tradeoffs in the design of two-stage melody generation
+architecture using an end-to-end learning framework. We can preliminarily see that with the increased
+percentage of melodic skeleton notes, the performance of the two-stage melody generation went up
+first but then down. On the one hand, a low proportion of melodic skeleton notes makes it easier
+to train the melodic skeleton construction model in the first stage. However, the generated skeleton
+notes will be too sparse to guide the second stage of melodic inpainting (such as the intersection
+skeleton setting, only 14.2%). On the other hand, if the proportion of melodic skeleton notes is too
+large, the training difficulty and data dependency of the melodic skeleton construction model will
+increase. Besides, from the perspective of the gestalt theory (62) about the law of the figure–ground
+relationship, during the perception of music, a person’s attention constantly switches between different
+musical elements; sometimes, he/she may be attracted to the rhythm, whereas at other times, to
+the pitch. Therefore, the extraction of the hierarchical dependency structural relationship between
+musical elements is affected by multiple musical dimensions; it will not be like a simple addition or
+subtraction operation but a complex organic combination (63).
+In this study, we chose the setting of the rhythmic skeleton (No. 3) that performed best on all
+subjective evaluation metrics in this experiment as the default skeleton configuration (denoted as
+WuYun-RS) for the next experiment to compare with other melody generation models.
+2.6
+Comparisons with other melody generation methods
+To prove the effectiveness of the proposed hierarchical skeleton-guided melody generation architecture
+based on knowledge-enhanced deep learning, we compared WuYun-RS (i.e., using the rhythmic
+skeleton setting) to five public SOTA Transformer-based melody generation models, namely, Music
+Transformer (15), Pop Music Transformer (17), Compound Word Transformer (19), Melons (33)
+and MeMIDI, that follow an end-to-end left-to-right note-by-note generative paradigm and treat
+each note equally. The MeMIDI setting uses the MeMIDI data representation method like WuYun-
+RS and employs the Transformer-XL model without using the melodic skeleton for the melody
+generation task. Moreover, to prove the effectiveness of the generated melodic skeleton, we added
+the setting of WuYun-RRS, skipped the melodic skeleton construction in the first stage, and directly
+used the real rhythmic skeleton as additional knowledge to guide the melody generation process
+of melody inpainting in the second stage. However, the original music representation of Music
+9
+
+Figure 3: Subjective evaluation results of the WuYun melody generation architecture based on different
+melodic skeleton settings, and the other public melody generation models. (A) Subjective comparison of the
+performance of the WuYun architecture based on different melodic skeleton settings in Experiment 1. Data is the
+mean opinion score. The WuYun architecture with the rhythmic skeleton setting achieves the best performance in
+all melodic skeleton settings on all subjective evaluation metrics. (B) Subjective comparison of the performance
+of different music generation models in Experiment 2. Violin plots show the kernel density estimate of rating
+distribution, the larger the area of the area graph, the greater the probability of the value distribution. The
+black dot within the violin plots indicates the mean; the black line within the violin plot indicates the standard
+deviation. Statistical analyses are done between WuYun-RS and the rest of the models using the one-tailed
+t-test (N = 130). P values of statistical significance are represented as *P < 0.05, **P < 0.01, ***P < 0.001,
+and ****P < 0.0001; ns, insignificant. The subjective comparison results’ data are in Tables S1, 1, and S2,
+respectively.
+Transformer does not include chord progressions. For a fair comparison, we added the CHORD
+events proposed in this work into the MIDI-Like music representation of Music Transformer. Each
+CHORD event was followed by a TIME-SHIFT event and had a higher sorting priority than NOTE-
+related events. Additionally, the MIDI quantization level of the Pop MusicTransformer, Compound
+Word Transformer, and Melons only considered the 16th note time grid. Therefore, in this experiment,
+we applied the 16th note time grid as the MIDI quantization level to the melody dataset for all music
+generation models.
+In this experiment, we obtained 80 musical pieces for rating. Since the second subjective evaluation
+experiment relies on the result of the first subjective evaluation experiment and requires some time to
+collect, process, and analyze, we recruited 13 participants again (i.e., six females and seven males,
+ages 18 and 25 years) to evaluate the musical pieces with payment. Six subjects among them were
+professional music practitioners with an average of 8 years of music training and 4 years of music
+performance experience. Each subject was required to rate all musical pieces. After rating 20 musical
+pieces, subjects were asked to rest for 5 min against hearing fatigue. The average experiment time
+cost each subject about 2 h.
+Figure 3B shows the mean opinion scores and one-tailed t-test results of the different music generation
+systems on the five evaluation metrics in the form of violin plots. The detailed experimental results are
+shown in Tables 1 and S2. Overall, WuYun-RS (No. 6) and WuYun-RRS (No. 7) outperformed the
+other five current SOTA end-to-end left-to-right note-by-note melody generation models on all metrics,
+including MusicTransformer (No. 1), Pop Music Transformer (No. 2), Compound Word Transformer
+(No. 4), Melons (No. 5), and MeMIDI (No. 3). Besides, except for WuYun-RRS, there is a significant
+10
+
+A
+3.25
+ Downbeat  Long Note
+ Rhythm  Tonic Intersection
+ Union  Random25%  Random50%
+■ Random75%
+Mean opinion score (MoS)
+3.00
+2.75
+2.50
+2.25
+2.00
+Rhythm
+Richness
+Structure
+Expectation
+Overall
+Subjective Evaluation Metrics
+B
+Music Transformer
+ Pop Music Transformer
+MeMIDI
+ Compound Word Transformer
+■ Melons
+-WuYun-Rs
+WuYun-RRS
+Mean opinion score(MOS)
+****
+****
+****
+****
+****
+****
+****
+**
+***
+**
+ns
+ns 
+ns
+ns
+5
+3 -
+Richness
+Structure
+Rhythm
+Expectation
+Overall
+Subjective Evaluation MetricsTable 1: Subjective evaluation scores of generated melodies based on different melody generation models in
+Experiment 2 (mean ± standard deviation).
+No.
+Model
+Rhythm
+Richness
+Structure
+Expectation
+Overall
+1
+MT
+2.52 ± 0.93
+2.34 ± 0.83
+2.28 ± 0.86
+2.47 ± 1.01
+2.25 ± 0.88
+2
+PMT
+2.57 ± 0.88
+2.43 ± 0.99
+2.54 ± 1.11
+2.50 ± 1.16
+2.39 ± 1.12
+3
+MeMIDI
+2.61 ± 0.91
+2.55 ± 0.95
+2.53 ± 1.00
+2.51 ± 0.97
+2.42 ± 0.96
+4
+CWT
+2.77 ± 0.83
+2.72 ± 0.85
+2.74 ± 0.81
+2.70 ± 0.83
+2.65 ± 0.90
+5
+Melons
+2.84 ± 0.89
+2.68 ± 0.95
+2.75 ± 0.87
+2.67 ± 0.87
+2.71 ± 0.88
+6
+WuYun-RS
+3.13 ± 0.88
+3.07 ± 0.87
+3.13 ± 0.86
+3.02 ± 0.92
+3.00 ± 0.87
+7
+WuYun-RRS
+3.20 ± 0.81
+3.11 ± 0.85
+3.15 ± 0.88
+3.00 ± 0.96
+3.02 ± 0.88
+8
+Human
+3.54 ± 0.82
+3.65 ± 0.76
+3.68 ± 0.89
+3.55 ± 0.92
+3.57 ± 0.84
+MT, PMT, and CWT stand for Music Transformer, Pop Music Transformer, and Compound Word Trans-
+former, respectively.
+difference (P < 0.01) between WuYun-RS and the other melody generation systems. This result
+demonstrates that WuYun-RS and WuYun-RRS are able to generate melodies with improved long-
+term structure and musicality, which benefit from the rhythmic skeleton as a deep structure to guide
+the melody generation process. Furthermore, WuYun-RS and WuYun-RRS demonstrate highly similar
+performances in terms of the quality of generated melodies on all evaluation metrics. This result
+indicates the effectiveness of the melodic skeletons generated via the melodic skeleton construction
+module. However, there is still an obvious gap between the WuYun melody generation architecture
+and human-composed music, leaving room for improvement. This also shows that designing clever
+decorations for melodic skeletons is another difficult research problem, even for human composers.
+Additionally, when using the same symbolic music representation method, the knowledge-enhanced
+hierarchical skeleton-guided melody generation model of WuYun-RS greatly outperformed the single-
+stage end-to-end left-to-right note-by-note melody generation model of MeMIDI (No. 3). On the one
+hand, this demonstrates that our proposed hierarchical melody generation paradigm can be applied to
+empower the dominant end-to-end left-to-right note-by-note melody generation paradigm. On the
+other hand, although the Compound Word Transformer and Melons (Nos. 4 and 5) were inferior to
+WuYun-RS, their effective compound word representation and the linear Transformer as the backbone
+architecture enable it to process multidimensional music information in one step simultaneously and
+obtain a better result among these five public SOTA melody generation models. Thus, combining the
+proposed knowledge-enhanced hierarchical skeleton-guided music generation architecture with more
+efficient music representation methods and advanced language models can bring a better result for
+melody generation tasks.
+3
+Discussion
+The methodology we have taken in designing WuYun, a hierarchical skeleton-guided melody genera-
+tion architecture based on knowledge-enhanced deep learning, combines music analysis theory and
+musical psychology. Unlike the dominant end-to-end left-to-right note-by-note melody generation
+paradigm, we use the hierarchical organization principle of structure and prolongation to decompose
+the melody generation process into melodic skeleton construction and melody inpainting stages. We
+extract the most structurally important notes based on hearing sensitivity as melodic skeletons and
+incorporate them into the melody generation process as a deep structure to guide the model to learn
+the hierarchical dependency structures among musical event sequences from the limited melody data
+without music boundary detection (31). The human evaluation results demonstrated that our model
+exhibits significant improvement in both long-term structure and musicality across the structured
+melody generation task.
+In practical application scenarios, the ability to obtain real feedback from human users for improving
+the performance and interaction experience of the system is essential for the next generation of
+iterative and interactive music generation systems. In general, WuYun allows human users to edit the
+generated melodic skeleton and adjust its shape to guide and constrain the range of the decorative
+notes at the next stage of melody inpainting. Thus, the proposed generation strategy based on the
+11
+
+hierarchical organization principle of structure and prolongation not only can maintain the long-range
+tonal coherence of generated melodies but also achieve control over the target of melodic motion by
+human users. Additionally, with WuYun and its melodic skeleton analysis framework, human users
+can directly extract the skeleton from existing music compositions for music composition analysis or
+re-creation.
+Our study has some limitations, notably the performance of the melody inpainting model, except
+that the quality of generated melodic skeleton may be poor; even if an original rhythmic skeleton
+is provided, the quality of the completed melody is still far from the real one. Further performance
+improvements could be achieved using pretrained masked language models (23) for music generation,
+especially for the melody inpainting task. Another issue is how to effectively extract an organic
+melodic skeleton from hierarchical musical structures combining two or more musical dimensions
+(e.g., rhythm and pitch) to further improve the structure of generated melodies. According to the
+research in the cognitive psychology in music, while listening to music, only by combining the tonal
+and rhythmic structures can we form a more coherent musical representation and create a complete
+sense of melody (34, 48, 49). However, the brain’s processing mechanism of the hierarchical musical
+structure remains a fundamental research problem in the field of music cognitive psychology (64–66).
+With the help of advanced electroencephalography devices, cognitive musicology can break through
+the human cognition of hierarchical musical structure and apply it to music generation. Additionally,
+we expect to investigate other systematic music analysis theories and gain further psychological
+knowledge to analyze and compare the hierarchical levels of important musical events along different
+musical dimensions for designing a more effective melodic skeleton extraction framework. Another
+direction is to explore explainable AI for music generation to assist end users in making better
+decisions since deep learning methods lack physical transparency of methods. With these potential
+future improvements in mind, we hope that our findings for structured melody generation will optimize
+the dominant melody generation paradigms to improve long-term structure and musicality and provide
+a new lens to develop multidisciplinary research via combining data-driven and knowledge-based
+approaches.
+4
+Materials and Methods
+4.1
+Details of dataset preprocessing
+We evaluate the effectiveness of WuYun architecture on a commonly used and publicly available
+symbolic melody dataset of Wikifonia (32, 33, 67). The Wikifonia dataset contains thousands of
+lead sheets in MusicXML format. It covers various music genres, composed of melody and the
+accompanying chord progression and tonality labels. Here, we describe the procedure below to clean
+up noisy data and artificial errors since the dataset is user-generated.
+• Data Segmentation: To simplify rhythm modeling, we only keep those segments from
+MIDI files with the most commonly used 4/4 time signature (18).
+• MIDI Quantization: For a more beat-accurate timing of sounds, quantization is a useful
+digital music processing of setting MIDI data on beats or exact fractions of beats to eliminate
+some imprecise timing because of expressive musical performance or mistake record. We
+contend that a more precise and adaptable time grid is required to model a more expressive
+metrical context, including the 32nd (18), 64th note, and even triplets. By contrast, most
+prior works only use the 16-note time grid for quantification (17, 19, 31–33); each bar
+is quantized into 16 intervals. In this study, we propose a self-adaptive mixed precision
+quantization method to reduce quantization errors (Fig. 4). This method can automatically
+choose a suitable quantize grid for every single note based on its duration, including straight
+notes and triplets. The difference between straight notes and triplets is dividing the musical
+beat evenly in half or third. First, the notes shorter than a 64th note are discarded, whereas
+notes longer than one bar are saved into the whole note. Second, according to the note
+duration, the rest notes are classified into straight or triplets. However, triplets do not always
+have to have three notes. There are only two notes in triplets is quite common, such as in
+Swing. Additionally, in theory, every note in triplets has an equal rhythm value. However, in
+practice, most notes are slightly different from each other in musical performance. Therefore,
+two or three consecutive notes with approximately the same duration and consistent with
+the duration of triplets are considered as triplets. Based on our experimental statistical
+12
+
+A
+1
+2
+4
+8
+16
+1
+2
+4
+8
+16
+2
+4
+8
+16
+2
+4
+8
+16
+1
+2
+4
+8
+16
+2
+4
+8
+16
+1
+2
+4
+8
+16
+4
+8
+16
+4
+8
+16
+4
+8
+16
+4
+8
+16
+4
+8
+16
+4
+8
+16
+4
+8
+16
+16
+16 16 16
+8
+16 16 16
+8
+16 16 16
+8
+16 16 16
+8
+16
+16
+8
+16 16 16
+8
+16 16 16
+8
+16 16 16
+8
+16
+16 16
+8
+16 16 16
+8
+16 16 16
+8
+16 16 16
+8
+16 16 16
+8
+16
+4
+4
+4
+4
+Basic Note
+Quantization Error
+Shift Backward
+Shift Forward
+Selected Grids
+Standard Grids
+Standard Duration
+Triplet Grids
+Time Signature
+Triplet
+Time 
+Grid
+4
+Whole Note
+16
+16
+16
+16
+8
+16
+16
+16
+8
+8
+8
+16
+8
+16
+16
+16
+16
+32
+64
+Triplet
+Triplet
+16
+16
+16
+8
+16
+16
+16
+64
+Triplet
+Triplet
+Shorter than 64th Note
+16
+16
+16
+16
+32
+8
+8
+16
+4
+16
+8
+8
+Longer than a whole note
+16
+❌
+Figure 4: Illustration of the self-adaptive mixed precision MIDI quantization. The MIDI quantization
+method would automatically choose a suitable quantize grid for every note according to its note length to
+eliminate imprecise timing, including straight notes (minimum 64th note) and triplets (minimum 48th note).
+results, we set the acceptable duration error ratio between the actual triplet and the standard
+triplet to within 20%. Last, the method automatically selects a proper quantize grid for
+every note. In terms of straight notes, the granularity of the time grid depends on the note
+duration. Particularly, the note onset is aligned to its closest 16th note time grid when the
+note duration is greater than or equal to a 16th note, to its closest 32nd note time grid when
+the note duration is between a 16nd and 32th note, and to its closest 64th note time grid
+when the note duration is between a 32nd and 64th note. Moreover, the straight note offset
+is aligned to the 64th note time grid. In terms of triplets, the note onset and offset time is
+aligned to the 48th note time grid.
+• Tonality Unification: For simplicity, the tonalities and chord progressions of those MIDI
+files are transposed to “C major” and “A minor” tonalities (68). We set one chord per beat
+and unify the chord representation of the Wikifonia dataset using the chord dictionary as
+described in the following subsection.
+• Octave Transposition: All melodies are applied octave transposition to shift the pitch into
+the range from C3 to C5 or are removed, which are out of the regular melodic pitch range
+(32).
+After data cleaning, we get 2,921 musical pieces in Wikifonia, including 116,935 bars and 425,223
+notes. Finally, randomly hold out 50 songs for testing and use the remaining for training.
+4.2
+Symbolic melody representation
+In this work, we adopted a modified version of the “MuMIDI” symbolic music representation (18)
+to encode a piece of monophonic melody into discrete musical event sequences. We refer to it as
+MeMIDI. Following is a description of the MeMIDI extensive symbols information, which includes
+bar, position, note, chord, and tempo symbols.
+• Bar and Position
+We use a bar symbol to represent a bar line and a position symbol to represent the onset
+of a note or a chord event. Since the minimum time grids of the straight and triplet note
+are 64th and 48th notes, respectively, and the MIDI files’ time resolution is 480 ticks per
+beat; thus we merge these two kinds of minimum time grids values ({0, 30, 60, ..., 1890} ∪
+{0, 40, 80, ..., 1880}) and use the <Pos_Value> symbol to represent 96 kinds of starting
+positions, such as <Pos_30>. We assign a position symbol for every chord and note music
+event.
+13
+
+Table 2: List of chord events.
+Chord
+Content
+Chord root
+C, Db, D, Eb, E, F, F#, G, Ab, A, Bb, B
+Triad
+M, m, o, +
+Seventh chord
+MM7, Mm7, mM7, mm7, o7, %7, +7, +M7
+Chord quality
+Suspension
+Sus.
+• Note
+A note has three basic attributes: pitch, duration, and velocity. Here, the value of the note
+pitch attribute ranges from 48 (C3) to 83 (C5). The value of the note velocity attribute
+ranges from 0 to 127. Considering both straight notes and triplet notes, the value range of
+the note duration attribute is {30, 60, 90, ..., 1920} ∪ {40, 80, 160, 320, 640} ticks. We use
+a compound word <Pitch_Value, Velocity_Value, Duration_Value> to compress these three
+attributes of one note in one token to shorten the length of the melody events sequence.
+• Chord
+To cover the chord types in the Wikifonia dataset, we use a more comprehensive chord
+event list. As shown in Table 2, we consider 12 chord roots and 13 chord qualities, yielding
+156 possible chord events. We use a chord symbol <Root_Quality> to represent a chord
+musical event. To reduce repetition, we use the same position symbol for Note and Chord,
+which start at the same time. For simplicity, we do not use the Chord symbol in the melodic
+skeleton event sequence.
+• Tempo
+We divide the tempo into three categories: low (below 90), medium (90 to 160), and high
+(above 160).
+4.3
+WuYun architecture
+Here, we briefly elaborate on the configuration details of the two Transformer-based generative
+modules of WuYun architecture, i.e., the melodic skeleton generation module for the melodic skeleton
+construction stage and the melodic prolongation generation module for the melody inpainting stage.
+We refer readers to (17–19) for more details. For reproducibility, we do not tweak the architecture of
+referenced models so that our music generation architecture can be easily assembled with the public
+implementation of Transformers.
+We use an unconditional sequence learning model Transformer-XL for the melodic skeleton genera-
+tion module. We use four self-attention layers, each with eight attention heads. The model hidden
+size and the inner layer of the feed-forward part are set to 512 and 2,048, respectively. All token
+embedding sizes are set to 512, following (19). We use the compound word embedding (Fig. S1) and
+token attribute prediction method for the input and output modules, repectively (18). We employed
+the top-k temperature-controlled stochastic sampling method (k = 10, temperature = 0.9) during
+inference. The length of training input tokens and the memory length are also 512. Here, we used the
+melodic skeleton data extracted from the training part of the Wikifonia dataset to train the melodic
+skeleton generation module.
+We use a conditional sequence-to-sequence model based on Transformer-based recurrent en-
+coder–decoder neural networks for the melodic prolongation generation module (18). We set the
+number of encoder layers, decoder layers, encoder heads, and decoder heads to 4. The size of hidden
+layers and the dimension of token embeddings are set to 256. We keep the same input module, output
+module, sampling method, length of training input tokens, and memory as same as the melodic
+skeleton generation module. For training the melodic prolongation generation module, we use the
+MeMIDI representations of the paired melodic skeleton and melody data as the encoder and decoder
+input data, respectively.
+14
+
+4.4
+Training
+We implemented the WuYun architecture with Pytorch (v1.7.1) (69). The parameters of the WuYun
+architecture were optimized by minimizing the cross-entropy loss on a single NVIDIA GTX 2080-Ti
+GPU with 11 GB memory. Specifically, the training loss was minimized with the Adam optimizer
+(β1 = 0.9, β2 = 0.98), a learning rate of ε = 10−3 , and dropout was applied with a ratio of 0.1.
+The mini-batches of the input data for the melodic skeleton generation module and the melodic
+prolongation generation module were 20 and 44, respectively. It took nearly 2 days to train the two
+modules until training convergence.
+4.5
+Statistical analysis
+All subjective evaluation results were expressed as mean ± standard deviation. The statistical
+significance of the performance difference in WuYun-RS and other melody generation methods
+was analyzed using the one-tailed t-test. Asterisk indicates significant difference at *P < 0.05,
+**P < 0.01, ***P < 0.001, ****P < 0.0001, and ns, not significant.
+Acknowledgements
+Thanks to Huawei Technologies Co., Ltd for the help in dataset collection and comments. We
+thank Jiaxing Yu, Chongjun Zhong, Ruiyuan Tang, and Jiaqi Wang for insightful discussions and
+visualizations.
+Funding:
+This work is supported by the National Natural Science Foundation of China
+(No.62272409), the Key R&D Program of Zhejiang Province (No.2022C03126), the Project of
+Key Laboratory of Intelligent Processing Technology for Digital Music (Zhejiang Conservatory of
+Music), and the Ministry of Culture and Tourism (No.2022DMKLB001).
+Author contributions: Conceptualization: K.Z., L.S. Methodology: X.W., T.Z., Z.H., K.Z. Investi-
+gation: T.Z., Z.H., Q.L. Visualization: S.W., Q.L. Supervision: L.S., K.Z., X.T. Writing—original
+draft: X.W., K.Z., Q.L. Writing—review & editing: K.Z., X.T., X.W., L.S.
+Competing interests: K.Z., X.W., and T.Z. are inventors on a patent application related to this work
+filed by Zhejiang University. The authors declare that they have no other competing interests.
+Data and materials availability: All data needed to evaluate the conclusions in the paper are present
+in the paper and/or the Supplementary Materials. Raw experimental data and the generated symbolic
+melody files are available on Zenodo at DOI 10.5281/zenodo.7480957
+under a Creative Commons Attribution 4.0 International license. The code of the WuYun music
+generation framework is available at https://github.com/NEXTLab-ZJU/wuyun.
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+18
+
+A
+Appendix
+A.1
+Additional Supplementary Figure
+···
+TempM
+Velocity Embeddings
+Duration Embeddings
+Token Embeddings
+Position Embeddings
+Bar Embeddings
+Tempo Embeddings
+Timestep
+A
+➕
+➕
+➕
+➕
+➕
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+TempM
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar1
+Bar2
+Bar2
+Bar1
+Bar1
+Bar1
+Bar2
+Pos0
+Pos0
+Pos720
+Pos720
+Pos960
+Pos960
+Pos960
+Pos1440
+Pos1440
+Pos0
+Pos0
+Pos480
+Pos480
+Pos0
+Bar
+Pos0
+ChC_M
+Pos720
+Pitch62
+Pos960
+ChC_M
+Pitch63
+ChC_M
+Pitch60
+Bar
+Pos0
+Pitch60
+Pos480
+ChD_M
+ChC_M
+Dur240
+Dur480
+Dur480
+Dur720
+Vel127
+Vel127
+Vel127
+Vel127
+Figure S1: MeMIDI encoding method for MIDI sequences (example). The input embedding in each timestep
+of the MeMIDI event sequence is the sum of the event embeddings, including tempo embedding, bar embedding,
+position embedding, token embedding, duration embedding, and velocity embedding in this timestep.
+A.2
+Additional Supplementary Tables
+Table S1: Subjective evaluation scores of generated melodies based on different melodic skeleton settings in
+Experiment 1 (mean ± standard deviation).
+No.
+Settings
+Rhythm
+Richness
+Structure
+Expectation
+Overall
+1
+Downbeat
+2.75 ± 0.96
+2.50 ± 0.98
+2.63 ± 1.13
+2.45 ± 1.03
+2.52 ± 1.03
+2
+Long Note
+2.70 ± 1.03
+2.57 ± 1.17
+2.77 ± 1.21
+2.47 ± 1.27
+2.67 ± 1.26
+3
+Rhythm
+3.02 ± 1.01
+3.10 ± 0.89
+2.88 ± 1.02
+2.82 ± 0.81
+2.93 ± 0.97
+4
+Tonic
+2.95 ± 1.11
+2.80 ± 1.09
+2.87 ± 0.95
+2.65 ± 1.01
+2.80 ± 1.06
+5
+Intersection
+2.60 ± 0.85
+2.52 ± 0.87
+2.53 ± 0.83
+2.28 ± 0.98
+2.43 ± 0.83
+6
+Union
+2.95 ± 0.95
+2.57 ± 1.05
+2.62 ± 1.09
+2.43 ± 1.05
+2.60 ± 1.04
+7
+Random25%
+2.80 ± 1.11
+2.78 ± 0.98
+2.67 ± 1.07
+2.62 ± 0.95
+2.65 ± 1.12
+8
+Random50%
+2.92 ± 0.98
+2.82 ± 0.89
+2.70 ± 1.01
+2.68 ± 1.03
+2.73 ± 0.95
+9
+Random75%
+2.82 ± 0.95
+2.58 ± 1.12
+2.50 ± 1.11
+2.45 ± 1.04
+2.53 ± 1.08
+Table S2: One-tailed t-test results between WuYun-RS and other music generation models on the five evaluation
+metrics in experiment 2.
+Model
+Rhythm
+Richness
+Structure
+Expectation
+Overall
+MT
+3.43 × 10−7
+5.37 × 10−9
+2.44 × 10−12
+6.50 × 10−6
+2.21 × 10−10
+PMT
+2.18 × 10−8
+1.21 × 10−9
+1.20 × 10−6
+2.55 × 10−6
+1.13 × 10−7
+MeMIDI
+1.88 × 10−6
+3.09 × 10−6
+6.86 × 10−7
+2.31 × 10−5
+6.29 × 10−7
+CWT
+3.31 × 10−4
+8.47 × 10−4
+3.02 × 10−4
+2.03 × 10−3
+1.69 × 10−3
+Melons
+4.60 × 10−3
+1.54 × 10−3
+8.02 × 10−4
+2.12 × 10−3
+7.41 × 10−3
+WuYun-RRS
+0.26
+0.36
+0.42
+0.42
+0.41
+MT, PMT, and CWT stand for Music Transformer, Pop Music Transformer, and Compound Word Trans-
+former, respectively.
+19
+

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+Performance Analysis of Superconductor-constriction-Superconductor Transmon
+Qubits
+Mingzhao Liu∗ and Charles T. Black†
+Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA
+This work presents a computational analysis of a superconducting transmon qubit design, in which
+the superconductor-insulator-superconductor (SIS) Josephson junction is replaced by a co-planar,
+superconductor-constriction-superconductor (ScS) junction.
+For short junctions having a Kulik-
+Omelyanchuk current-phase relationship, we find that the ScS transmon has an improved charge
+dispersion compared to the SIS transmon, with a tradeoff of 50% smaller anharmonicity. These
+calculations provide a framework for estimating the superconductor material properties and junction
+dimensions needed to provide proper ScS transmon operation at typical gigahertz frequencies.
+I.
+INTRODUCTION
+The transmon has become an enabling superconduct-
+ing qubit device architecture, with primary advantages of
+immunity to charge noise and longer coherence lifetimes
+achieved by designing the device to have Josephson en-
+ergy far exceeding the charging energy. Similar to other
+superconducting qubit architectures, the transmon core
+consists of one or more Josephson junctions (JJs), which
+are exclusively superconductor-insulator-superconductor
+tunnel junctions (SIS) — typically a thin film sand-
+wich structure of aluminum/aluminum oxide/aluminum
+(Al/AlOx/Al), in which AlOx is the tunnel barrier (Fig-
+ure 1a).
+Fabrication of Al/AlOx/Al SIS JJs typically involves
+physical vapor deposition of the top and bottom Al lay-
+ers from two different angles relative to the substrate,
+through a common mask [1]. After depositing the first
+Al layer, the sample is exposed to a controlled level of
+oxygen to form the thin AlOx barrier. This ingenious
+fabrication method has been refined over many years but
+will nevertheless be highly challenging to implement at
+the manufacturing scale required for larger-scale quan-
+tum computers. The exponential dependence of the JJ
+critical supercurrent (Ic) on tunnel barrier width also re-
+sults in a typical few percent variation in Ic across devices
+Al
+Al
+AlOx
+Superconductor
+Constriction
+a
+b
+Figure 1. (a) Schematic of an Al/AlOx/Al superconductor-
+insulator-superconductor (SIS) Josephson junction. For clar-
+ity, the native oxide covering both Al electrodes is omitted.
+(b) Schematic of a co-planar superconductor-constriction-
+superconductor (ScS) Josephson junction.
+∗ mzliu@bnl.gov
+† ctblack@bnl.gov
+fabricated within a few centimeters, even when oxidation
+conditions are tightly controlled.[2–5] Since the Joseph-
+son energy is directly proportional to Ic, this variation
+presents an additional design and manufacturing chal-
+lenge.
+In a transmon, the SIS JJ is shunted by a large capac-
+itor to minimize the charging energy and thus provide
+immunity to charge noise. Further, the qubit is coupled
+to a high-Q microwave resonator for readout. Typically,
+the shunting capacitor and the resonator are fabricated
+separately from the SIS JJ, using a superconductor with
+higher Tc and better chemical robustness compared to Al
+(e.g., niobium (Tc = 9.2 K)[6], tantalum (Tc = 4.4 K)[7],
+and titanium nitride (Tc = 5.6 K)[8]). The improved ro-
+bustness allows post-fabrication chemical treatments to
+remove surface contaminants, which contribute to TLS
+loss. However, most of these treatments are not possible
+after Al/AlOx/Al junction fabrication, due to the junc-
+tion’s fragile nature [9].
+In this work we analyze the performance impact
+of replacing the transmon SIS tunnel junction with
+a co-planar superconductor-constriction-superconductor
+(ScS) Josephson junction. A ScS JJ is comprised of two
+superconductors separated by a thin neck of the same
+superconductor (Figure 1b), with the constriction estab-
+lishing the superconducting phase difference that enables
+Josephson behavior. ScS JJs are co-planar (unlike SIS
+tunnel junctions) and can be fabricated using conven-
+tional lithography and metallization. Here, we follow the
+formalism established by Koch et al.
+in [10] to deter-
+mine the electrical properties of ScS transmons, which
+are shown to be different from SIS transmons, stemming
+from a different ScS JJ current-phase relationship (CPR
+or CΦR) compared to that of a SIS JJ.[11–13]. Compar-
+ing the two device architectures, we show that the ScS
+transmon has 50% less anharmonicity than the SIS trans-
+mon, for devices with the same Josephson energy and
+capacitive energy. However, the smaller anharmonicity
+is accompanied by a significantly smaller charge disper-
+sion, giving the ScS transmon stronger immunity against
+charge noise.
+arXiv:2301.04276v1  [cond-mat.supr-con]  11 Jan 2023
+
+2
+II.
+RESULTS AND DISCUSSION
+II.1.
+Current-phase relation of a short ScS junction
+Consider a ScS Josephson junction comprised of two
+large superconductors connected by a diffusive quasi-one-
+dimensional wire with length d ≪ √ξ0l and width w ≪
+d, where ξ0 is the Pippard superconducting coherence
+length, and l ≪ ξ0 is the dirty-limit electron mean free
+path. In this case, Kulik and Omelyanchuk showed that
+the CPR for the ScS junction (KO-1) at T = 0 K is:
+IScS(ϕ) = π∆
+eRn
+cos ϕ
+2 tanh−1 �
+sin ϕ
+2
+�
+,
+(1)
+in which ∆ is the superconducting energy gap and Rn
+is the normal state resistance of the junction.[11, 12]
+The junction critical current Ic,ScS = 0.662π∆/(eRn) is
+achieved at ϕ = (2k ± 0.627)π to satisfy dI(ϕ)/dϕ ∝
+1 − sin(ϕ/2) tanh−1 [sin(ϕ/2)] = 0. Given the Maclau-
+rin series tanh−1(x) = x + x3/3 + O(x5), Eq.
+1 may
+be rewritten to a form that resembles the CPR of a SIS
+Josephson junction, as
+IScS(ϕ) = 0.755Ic,ScS sin ϕ
+�
+1 + 1
+3 sin2 ϕ
+2 + O
+�
+sin4 ϕ
+2
+� �
+,
+which shows that the CPR of a ScS junction distorts
+from the conventional sinusoidal form, but still bears odd
+parity and a 2π periodicity (Figure 2a).
+II.2.
+Josephson energy of a ScS transmon
+The potential energy of a Josephson junction is given
+by the integral
+EJ(ϕ) =
+�
+IJV dt =
+�
+IJ
+Φ0
+2π
+dϕ
+dt dt =
+�
+IJ
+Φ0
+2π dϕ.
+(2)
+−2
+−1
+0
+1
+2
+φ/π
+−1.0
+−0.5
+0.0
+0.5
+1.0
+I(φ)/Ic
+ScS
+SIS
+−1.0
+−0.5
+0.0
+0.5
+1.0
+φ/π
+0
+1
+2
+3
+EJ(φ)/EJ
+ScS
+SIS
+φ2/2
+a
+b
+Figure 2. (a) The CPR of a ScS Josephson junction in the
+KO-1 limit (solid red) is distorted from the sinusoidal form
+of a SIS junction (dashed black). (b) The Josephson energy
+of a ScS transmon (solid red) deviates from the cosine form
+of a SIS transmon (dashed black) and has 50% smaller an-
+harmonicity at its lowest order (ϕ4). A harmonic parabola,
+ϕ2/2, is displayed (dotted cyan) for reference.
+−1.0
+−0.5
+0.0
+0.5
+1.0
+φ/π
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+EJ(φ)/EJ
+−1.0
+−0.5
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+φ/π
+EJ(φ)/EJ
+a
+b
+SIS
+ScS
+0.3031
+0.8800
+1.3993
+1.7751
+0.3096
+0.9145
+1.4896
+2.0077
+Figure 3. The eigenenergies (blue lines and numbers) and the
+probability densities (∥Ψ∥2) of the first 4 eigenstates of (a) a
+SIS transmon and (b) a ScS transmon, both with EJ/EC = 20
+and ng = 1/2. The corresponding potential energies, normal-
+ized by EJ, are plotted in red lines for both transmons.
+For a KO-1 junction defined by Eq. 1, the integral in Eq.
+2 leads to
+EJ,ScS(ϕ) = ∆Φ0
+2eRn
+�
+ln
+�
+cos2 ϕ
+2
+�
++ 2 sin ϕ
+2 tanh−1 �
+sin ϕ
+2
+� �
+.
+(3)
+Although this form appears very different from the po-
+tential energy of a SIS junction, EJ,SIS(ϕ) = EJ,SIS(1 −
+cos ϕ), with EJ,SIS = Ic,SISΦ0/2π, Maclaurin expansions
+of EJ,ScS and EJ,SIS make their similarities apparent:
+EJ,ScS(ϕ) = ∆Φ0
+4eRn
+�1
+2ϕ2 − 1
+48ϕ4 + O(ϕ6)
+�
+EJ,SIS(ϕ) = EJ,SIS
+�1
+2ϕ2 − 1
+24ϕ4 + O(ϕ6)
+�
+.
+(4)
+Comparing the coefficients of the harmonic (ϕ2) term,
+we observe that the Josephson energy of a ScS transmon
+can be defined as:
+EJ,ScS = ∆Φ0
+4eRn
+= 0.755Ic,ScSΦ0/(2π) ,
+(5)
+where
+the
+last
+equality
+recognizes
+that
+Ic,ScS
+=
+0.662π∆/(eRn). Eq. 5 shows that both potential ener-
+gies contain anharmonicity led by a ϕ4 term, from which
+we estimate that the anharmonicity of a ScS transmon
+is about one half that of a SIS transmon, for devices
+with the same EJ. This difference is clear when compar-
+ing normalized EJ(ϕ) of ScS and SIS transmons with a
+harmonic parabolic potential ϕ2/2 (Figure 2b). A more
+precise evaluation of the anharmonicity is given in II.3,
+by computing the ScS transmon eigenenergies.
+II.3.
+Eigenenergies and eigenstates of a ScS
+transmon
+A conventional SIS transmon has a Hamiltonian of the
+form:
+ˆHSIS = 4Ec(ˆn − ng)2 + EJ(1 − cos ˆϕ),
+(6)
+
+3
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+E0m/EC
+0
+20
+40
+60
+80
+100
+−1
+0
+1
+2
+3
+4
+5
+(E12 - E01)/EC
+ScS
+SIS
+a
+b
+EJ/EC
+EJ/EC
+ScS
+SIS
+m = 0
+1
+2
+3
+0
+20
+40
+60
+80
+100
+100
+τp (ns)
+SIS
+ScS
+EJ/EC
+10-1
+10-2
+c
+Figure 4. (a) Transition energy E0m = Em − E0 at ng = 1/2 and (b) oscillator anharmonicity (E12 − E01) at ng = 1/2, as
+functions of EJ/EC for ScS transmon (solid lines) and SIS transmon (dashed lines). (c) The minimal pulse duration (τp) of
+ScS (solid line) and SIS transmons (dashed line) vs. EJ/EC, all operated at ω01 = 2π × 10 GHz.
+where ng is the offset charge. The wave equation for a
+SIS transmon can be solved analytically.[10]
+In the ScS transmon, the potential energy is given by
+Eq. 3, so that the Hamiltonian becomes,
+ˆHScS = 4EC(ˆn − ng)2 + EJ
+�
+2 ln
+�
+cos2 ˆϕ
+2
+�
++ 4 sin ˆϕ
+2 tanh−1
+�
+sin ˆϕ
+2
+� �
+.
+(7)
+The wave equation of a ScS transmon can be solved nu-
+merically using the finite difference method, in which the
+Hamiltonian is expressed in a discretized space of phase
+ϕ ∈ [−π, π), with the periodic boundary condition ap-
+plied to both ends. The validity of the computation is
+confirmed by comparing a similar numerical solution of
+the wave equation for a SIS transmon with the analytical
+solutions presented by Koch et al.[10] Figure 3 compares
+the first 4 eigenstates of a SIS transmon and a ScS trans-
+mon, both with EJ/EC = 20 and ng = 1/2. Although
+the lower level eigenenergies and eigenfunctions are sim-
+ilar, the differences become more apparent at higher en-
+ergies. This trend is more clearly observed for the transi-
+tion energies E0m = Em − E0 calculated for both trans-
+mon types, across a range of EJ/EC from 1 and 100
+(Figure 4a). This difference reflects the smaller anhar-
+monicity of the ScS transmon, compared to the SIS trans-
+mon. By treating the leading anharmonic term (−ϕ4/24)
+in Eq.
+4 as a perturbation to the harmonic potential
+and applying the first-order perturbation theory, the mth
+eigenenergy of a SIS transmon is approximated by [10]
+Em,SIS ≈ ℏωp
+�
+m + 1
+2
+�
+− EC
+4
+�
+2m2 + 2m + 1
+�
+,
+(8)
+in which ℏωp = √8EJEC is the Josephson plasma energy.
+The transition energy between the (m − 1)th and mth
+levels is therefore
+Em−1,m,SIS ≈ ℏωp − mEC.
+(9)
+From Eq.
+9, we find that the anharmonicity of SIS
+transmon αSIS ≡ E12,SIS−E01,SIS is approximately −EC.
+By applying the same first-order perturbation theory cal-
+culation but realizing that the perturbation term is half
+as a SIS transmon (Eq. 4), the mth eigenenergy of a ScS
+transmon can be approximated by
+Em,ScS ≈ ℏωp
+�
+m + 1
+2
+�
+− EC
+8
+�
+2m2 + 2m + 1
+�
+,
+(10)
+so that its anharmonicity, αScS, is approximately −EC/2,
+or half the anharmonicity of a SIS transmon. We can vi-
+sualize this finding in a plot of numerical results, looking
+at transmons with EJ/EC ≥ 20 (Figure 4b).
+The smaller anharmonicity of a ScS transmon means
+that the transitions E01 and E12 lie closer in energy, so
+that a longer RF pulse is needed to correctly excite the
+desired transition E01. The minimal pulse duration can
+be estimated as τp ≈ ℏ|α|−1.
+As shown in Figure 4c,
+despite its lower anharmonicity, τp of the ScS transmon
+remains below 1 ns even for EJ/EC = 100, when the
+qubit operates at 10 GHz. Because typical qubit pulse
+durations are ∼10 ns, we may conclude that the lower
+anharmonicity will not inhibit normal operation of a ScS
+transmon.
+II.4.
+Charge dispersion of a ScS transmon
+A primary benefit of the transmon architecture is its
+relative immunity to charge noise, when designed to op-
+erate in the regime of EJ ≫ EC. In a SIS transmon, the
+charge dispersion of the mth level decreases exponentially
+with
+�
+8EJ/EC, following [10]
+ϵm ≡ Em(ng = 1/2) − Em(ng = 0)
+≈ EC
+24m+5
+(−1)mm!
+�
+2
+π
+� EJ
+2EC
+� m
+2 + 3
+4
+e−√
+8EJ/EC.
+(11)
+Intuitively, the charge dispersion is related to the
+tunneling probability between neighboring potential en-
+ergy valleys (Figure 3), e.g., when ϕ makes a full 2π
+
+4
+−2
+−1
+0
+1
+2
+ng
+0.5
+1.0
+1.5
+2.0
+Em/EJ
+SIS
+ScS
+SIS
+ScS
+|ϵm|/E01(ng = 1/2)
+101
+10-3
+10-7
+10-1
+10-5
+10-9
+10-11
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+(EJ/EC)1/2
+10
+20
+30
+40
+50 60 70 80 90 100
+EJ/EC
+a
+b
+m = 0
+1
+2
+m = 0
+1
+2
+0
+20
+40
+60
+80
+100
+SIS
+ScS
+T2 (ns)
+EJ/EC
+100
+10-2
+104
+102
+106
+108
+c
+Figure 5. (a) The eigenenergies Em of the lowest 3 eigenstates (m = 0, 1, 2) of a ScS transmon (solid line) and a SIS transmon
+(dashed lines), both with EJ/EC = 10, as functions of the offset charge ng. (b) The charge dispersion ϵm of the lowest 3
+eigenstates of a ScS transmon (solid line) and a SIS transmon (dashed lines), as functions of EJ/EC. (c) The dephasing time
+T2 of ScS (solid line) and SIS transmons (dashed line) vs. EJ/EC, all operated at ω01 = 2π × 10 GHz.
+rotation.[10] By this reasoning, we may expect the higher
+barrier height of a ScS transmon (∼ 2.8EJ vs.
+2EJ)
+to better suppress the tunneling probability and provide
+lower charge dispersion, compared to a SIS transmon.
+Figure 5a plots the first three eigenenergies Em (m =
+0, 1, 2) versus the effective offset charge ng for both SIS
+(dashed) and ScS (solid) transmons, with EJ/EC = 10.
+Clearly, the ScS transmon eigenenergies are more weakly
+perturbed by ng. Calculations of the charge dispersion,
+ϵm = Em(ng = 1/2) − Em(ng = 0), across a wide range
+of 1 ≤ EJ/EC ≤ 100 show that suppression of charge dis-
+persion in the ScS transmon becomes more effective for
+larger EJ/EC ratios (Figure 5b). When EJ/EC = 100,
+the charge dispersion of the first excited state of a ScS
+transmon, ϵ1,ScS, is over one order of magnitude less than
+the corresponding SIS transmon. It is noted that com-
+putation on SIS transmon matches the analytical result
+very well[10], again demonstrating the high numerical
+precision of our finite difference computation.
+Never-
+theless, the computational error becomes significant as
+the normalized charge dispersion, |ϵm|/E01, approaches
+10−11 and smaller. This is due to the accumulation of
+floating-point error that eventually shows up for evaluat-
+ing the vanishing difference between the two eigenener-
+gies at ng = 0 and 1/2.
+In Figure 5b, the y−axis is presented in the logarith-
+mic scale and the x−axis is presented in the scale of
+�
+EJ/EC, so that all curves take a linear form approach-
+ing large EJ/EC values. For the SIS transmon, the slope
+matches the expected exp(−
+�
+8EJ/EC) dependence in
+Eq. 11. For the ScS transmon, the slope is larger, and is
+best described by:
+ϵm ∝ exp
+�
+−
+�
+1.16 × 8EJ/EC
+�
+.
+The improved charge dispersion makes the ScS trans-
+mon both less sensitive to charge noise and, in turn, gives
+it a longer dephasing time T2.
+For dephasing caused
+by slow charge fluctuations of large amplitude, Koch et
+al[10] has found an upper limit of T2 given by
+T2 ≈
+4ℏ
+e2π|ϵ1|.
+Using this relation, we compare T2 for both SIS and ScS
+transmons for EJ/EC between 1 and 100 (Figure 5c).
+The ScS transmon improves T2 across the entire range of
+EJ/EC and especially at higher ratios. At EJ/EC = 100,
+the SIS transmon has a T2 ceiling of about 3 ms, com-
+pared to about 50 ms for the ScS transmon, an over 10
+fold increase. At present, because the T1 lifetime of SIS
+transmon qubits is still beyond 1 ms and not limited by
+the charge noise, this benefit of the ScS transmon archi-
+tecture will have little performance benefit. However, be-
+cause we expect the lifetimes of superconducting qubits
+to continue improving (Schoelkopf’s Law) [14], we an-
+ticipate a point when charge noise dephasing becomes a
+bottleneck, and the ScS transmon architecture can offer
+effective mitigation.
+II.5.
+ScS transmon design parameters
+The operational behavior of a ScS transmon is deter-
+mined by its EJ and EC, which define the operating fre-
+quency ω01, the relative immunity to charge noise (ϵ1),
+and the minimum excitation pulse duration (τp).
+Be-
+cause these three quantities are determined by EJ and
+EC, they are not independent. We can visualize this in-
+terdependence with three sets contour lines plotted in the
+plane of EJ versus EC in Figure 6. These contours rep-
+resent: (1) a transmon operating frequency (ω01/(2π))
+between 1 and 10 GHz (set of red, descending diagonal
+lines), (2) ratios of EJ/EC from 10, 100, and 1000 (set
+of blue, ascending diagonal lines), and the minimum ex-
+citation pulse duration τp between 0.32 and 10 ns (set of
+dashed, predominantly vertical lines). Selecting two of
+these defines the third one. For example, a ScS trans-
+mon designed to operate at ω01/(2π) = 5 GHz and with
+
+5
+EC/(2�ħ) (GHz)
+102
+EJ/(2�ħ) (GHz)
+101
+100
+10-2
+10-1
+100
+EJ/EC=103
+EJ/EC=102
+EJ/EC=101
+ω01/(2�) = 1 GHz
+2 GHz
+3 GHz
+4 GHz
+5 GHz
+6 GHz
+7 GHz
+8 GHz
+9 GHz
+10 GHz
+τp = 10 ns
+7.94 ns
+6.31 ns
+5.01 ns
+1.58 ns
+1.26 ns
+1.00 ns
+0.79 ns
+0.63 ns
+0.50 ns
+0.40 ns
+3.98 ns
+0.32 ns
+104
+105
+2
+3
+4
+5
+6
+7
+8
+9
+2
+3
+4
+5
+6
+7
+8
+9
+103
+2
+3
+4
+5
+6
+7
+8
+9
+2
+3
+4
+5
+6
+7
+8
+9
+3.16 ns
+2.51 ns
+2.00 ns
+Rn/Tc (Ω K-1)
+102
+103
+2x101
+2
+3
+4
+5
+6
+7 8 9
+2
+3
+4
+5
+6
+7 8 9
+2
+3
+4
+5
+6
+7
+8
+9
+3
+4
+5
+6
+7
+8
+9
+CΣ (fF)
+Figure 6. A graphical guide for designing ScS transmon with required EJ and EC to match desired transmon frequency ω01
+and minimum pulse duration τp. The red lines are contours lines for transmon frequencies set at values between 1 and 10 GHz.
+The dashed black lines are contours lines for τp set at a few values between 0.32 and 10 ns. The blue lines are contours lines
+for EJ/EC ratios set at 10, 100, and 1000. A second x-axis that is parallel to EC is presented for CΣ, following CΣ = e2/2EC.
+Simlarly, a second y-axis that is parallel to EJ is presented for Rn/Tc, following Rn/Tc = 1.76kBΦ0/(4eEJ).
+a readout pulse of τp = 4 ns (green dot in Figure 6) will
+have a EJ/EC ratio of about 600.
+Instead, a shorter
+excitation pulse of τp = 1 ns (purple dot in Figure 6)
+requires a tradeoff of smaller EJ/EC ≈ 40, and thus less
+immunity against charge noise.
+Importantly, EJ and EC of a ScS transmon are set by
+the physical device dimensions and fundamental prop-
+erties of the materials composing it. EJ is determined
+by the superconducting energy gap of the material (∆)
+and the normal state resistance of the junction (Rn) (Eq.
+5).
+For a BCS superconductor where ∆ = 1.76kBTc,
+we can express EJ in terms of the material properties
+Rn/Tc = 1.76kBΦ0/(4eEJ,ScS), which is shown as the
+second (right) y-axis in Figure 6. Similarly, because EC
+is set by the total capacitance (CΣ = e2/2EC,ScS) which
+depends on device geometry and dielectric properties, we
+can express EC as a capacitance, shown as a second (top)
+x-axis in Figure 6.
+Returning to the example, we can now see from Figure
+6 that designing a ScS transmon with ω01/(2π) = 5 GHz,
+τp = 4 ns, and EJ/EC ratio of about 600 (green dot) re-
+quires a junction with Rn/Tc ≈ 3 kΩ · K−1 and capacitor
+with CΣ ≈ 250 fF. The properties can be realized by a
+constriction junction fabricated from a thin film super-
+conductor that has both a relatively high normal state
+resistivity and a long superconducting coherence length.
+As one example, a 10-nm-thick PtSi film was reported
+has normal-state sheet resistance Rs = 67 Ω/□, super-
+conducting Tc = 0.63 K, and Pippard coherence length
+ξ = 440 nm.[15] Using these material parameters, we can
+meet the ScS transmon design criteria using a constric-
+tion junction with physical length of 440 nm and width of
+
+6
+16 nm. The qubit capacitor physical dimensions should
+be designed for CΣ ≈ 250 fF, according to Figure 6. If one
+instead desires the shorter readout pulse time of τp = 1
+ns (purple dot), the constriction must have Rn/Tc ≈ 10
+kΩ · K−1 and CΣ ≈ 70 fF. For the same PtSi supercon-
+ductor, these values can be met with physical length of
+440 nm and width of 5 nm. which are more challenging
+dimensions to fabricate.
+III.
+CONCLUSION
+In summary, we have demonstrated through computa-
+tion that a short ScS Josephson junction can be used as
+a drop-in replacement for the SIS tunnel junction in a
+transmon qubit. In the transmon regime (EJ ≫ EC), a
+ScS transmon has 50% smaller anharmonicity than a SIS
+transmon, but is compensated by its appreciably lower
+charge dispersion that provides a significantly higher T2
+ceiling. Using this analysis, we estimate that high per-
+formance ScS transmons can be achieved with constric-
+tions having a normal state resistance of a few kiloohms,
+which can be made from a thin nanobridge formed in low
+Tc superconductors using conventional, high-resolution
+nanofabrication techniques.
+The ScS transmon design
+allows all components, including constriction junction,
+capacitor, and resonator, to be fabricated in a single
+lithography step. This is a significant simplification com-
+pared to multistep SIS transmon fabrication, and also
+provides an robust architecture amenable to device post-
+processing, cleaning, and encapsulation.
+ACKNOWLEDGMENTS
+This material is based upon work supported by the
+U.S. Department of Energy, Office of Science, National
+Quantum Information Science Research Centers, Co-
+design Center for Quantum Advantage (C2QA) under
+contract number DE-SC0012704.
+This research used
+computational resources of the Center for Functional
+Nanomaterials (CFN), which is a U.S. Department of
+Energy Office of Science User Facility, at Brookhaven Na-
+tional Laboratory under Contract No. DE-SC0012704.
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+

PdE3T4oBgHgl3EQfCglF/content/tmp_files/load_file.txt

@@ -0,0 +1,477 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf,len=476
+page_content='Performance Analysis of Superconductor-constriction-Superconductor Transmon  Qubits  Mingzhao Liu∗ and Charles T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Black†  Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA  This work presents a computational analysis of a superconducting transmon qubit design, in which  the superconductor-insulator-superconductor (SIS) Josephson junction is replaced by a co-planar,  superconductor-constriction-superconductor (ScS) junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  For short junctions having a Kulik-  Omelyanchuk current-phase relationship, we find that the ScS transmon has an improved charge  dispersion compared to the SIS transmon, with a tradeoff of 50% smaller anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' These  calculations provide a framework for estimating the superconductor material properties and junction  dimensions needed to provide proper ScS transmon operation at typical gigahertz frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  INTRODUCTION  The transmon has become an enabling superconduct-  ing qubit device architecture, with primary advantages of  immunity to charge noise and longer coherence lifetimes  achieved by designing the device to have Josephson en-  ergy far exceeding the charging energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Similar to other  superconducting qubit architectures, the transmon core  consists of one or more Josephson junctions (JJs), which  are exclusively superconductor-insulator-superconductor  tunnel junctions (SIS) — typically a thin film sand-  wich structure of aluminum/aluminum oxide/aluminum  (Al/AlOx/Al), in which AlOx is the tunnel barrier (Fig-  ure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Fabrication of Al/AlOx/Al SIS JJs typically involves  physical vapor deposition of the top and bottom Al lay-  ers from two different angles relative to the substrate,  through a common mask [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' After depositing the first  Al layer, the sample is exposed to a controlled level of  oxygen to form the thin AlOx barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This ingenious  fabrication method has been refined over many years but  will nevertheless be highly challenging to implement at  the manufacturing scale required for larger-scale quan-  tum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The exponential dependence of the JJ  critical supercurrent (Ic) on tunnel barrier width also re-  sults in a typical few percent variation in Ic across devices  Al  Al  AlOx  Superconductor  Constriction  a  b  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (a) Schematic of an Al/AlOx/Al superconductor-  insulator-superconductor (SIS) Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' For clar-  ity, the native oxide covering both Al electrodes is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (b) Schematic of a co-planar superconductor-constriction-  superconductor (ScS) Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  ∗ mzliu@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='gov  † ctblack@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='gov  fabricated within a few centimeters, even when oxidation  conditions are tightly controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [2–5] Since the Joseph-  son energy is directly proportional to Ic, this variation  presents an additional design and manufacturing chal-  lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  In a transmon, the SIS JJ is shunted by a large capac-  itor to minimize the charging energy and thus provide  immunity to charge noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Further, the qubit is coupled  to a high-Q microwave resonator for readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Typically,  the shunting capacitor and the resonator are fabricated  separately from the SIS JJ, using a superconductor with  higher Tc and better chemical robustness compared to Al  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=', niobium (Tc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='2 K)[6], tantalum (Tc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='4 K)[7],  and titanium nitride (Tc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='6 K)[8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The improved ro-  bustness allows post-fabrication chemical treatments to  remove surface contaminants, which contribute to TLS  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' However, most of these treatments are not possible  after Al/AlOx/Al junction fabrication, due to the junc-  tion’s fragile nature [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  In this work we analyze the performance impact  of replacing the transmon SIS tunnel junction with  a co-planar superconductor-constriction-superconductor  (ScS) Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' A ScS JJ is comprised of two  superconductors separated by a thin neck of the same  superconductor (Figure 1b), with the constriction estab-  lishing the superconducting phase difference that enables  Josephson behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' ScS JJs are co-planar (unlike SIS  tunnel junctions) and can be fabricated using conven-  tional lithography and metallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Here, we follow the  formalism established by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  in [10] to deter-  mine the electrical properties of ScS transmons, which  are shown to be different from SIS transmons, stemming  from a different ScS JJ current-phase relationship (CPR  or CΦR) compared to that of a SIS JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='[11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Compar-  ing the two device architectures, we show that the ScS  transmon has 50% less anharmonicity than the SIS trans-  mon, for devices with the same Josephson energy and  capacitive energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' However, the smaller anharmonicity  is accompanied by a significantly smaller charge disper-  sion, giving the ScS transmon stronger immunity against  charge noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='04276v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='supr-con]  11 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Current-phase relation of a short ScS junction  Consider a ScS Josephson junction comprised of two  large superconductors connected by a diffusive quasi-one-  dimensional wire with length d ≪ √ξ0l and width w ≪  d, where ξ0 is the Pippard superconducting coherence  length, and l ≪ ξ0 is the dirty-limit electron mean free  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' In this case, Kulik and Omelyanchuk showed that  the CPR for the ScS junction (KO-1) at T = 0 K is:  IScS(ϕ) = π∆  eRn  cos ϕ  2 tanh−1 �  sin ϕ  2  �  ,  (1)  in which ∆ is the superconducting energy gap and Rn  is the normal state resistance of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [11, 12]  The junction critical current Ic,ScS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='662π∆/(eRn) is  achieved at ϕ = (2k ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='627)π to satisfy dI(ϕ)/dϕ ∝  1 − sin(ϕ/2) tanh−1 [sin(ϕ/2)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Given the Maclau-  rin series tanh−1(x) = x + x3/3 + O(x5), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  1 may  be rewritten to a form that resembles the CPR of a SIS  Josephson junction, as  IScS(ϕ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='755Ic,ScS sin ϕ  �  1 + 1  3 sin2 ϕ  2 + O  �  sin4 ϕ  2  � �  ,  which shows that the CPR of a ScS junction distorts  from the conventional sinusoidal form, but still bears odd  parity and a 2π periodicity (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Josephson energy of a ScS transmon  The potential energy of a Josephson junction is given  by the integral  EJ(ϕ) =  �  IJV dt =  �  IJ  Φ0  2π  dϕ  dt dt =  �  IJ  Φ0  2π dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (2)  −2  −1  0  1  2  φ/π  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  I(φ)/Ic  ScS  SIS  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  φ/π  0  1  2  3  EJ(φ)/EJ  ScS  SIS  φ2/2  a  b  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (a) The CPR of a ScS Josephson junction in the  KO-1 limit (solid red) is distorted from the sinusoidal form  of a SIS junction (dashed black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (b) The Josephson energy  of a ScS transmon (solid red) deviates from the cosine form  of a SIS transmon (dashed black) and has 50% smaller an-  harmonicity at its lowest order (ϕ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' A harmonic parabola,  ϕ2/2, is displayed (dotted cyan) for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  φ/π  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  EJ(φ)/EJ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  φ/π  EJ(φ)/EJ  a  b  SIS  ScS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='3031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='8800  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='3993  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='7751  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='3096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='9145  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='4896  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0077  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The eigenenergies (blue lines and numbers) and the  probability densities (∥Ψ∥2) of the first 4 eigenstates of (a) a  SIS transmon and (b) a ScS transmon, both with EJ/EC = 20  and ng = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The corresponding potential energies, normal-  ized by EJ, are plotted in red lines for both transmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  For a KO-1 junction defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' 1, the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  2 leads to  EJ,ScS(ϕ) = ∆Φ0  2eRn  �  ln  �  cos2 ϕ  2  �  + 2 sin ϕ  2 tanh−1 �  sin ϕ  2  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (3)  Although this form appears very different from the po-  tential energy of a SIS junction, EJ,SIS(ϕ) = EJ,SIS(1 −  cos ϕ), with EJ,SIS = Ic,SISΦ0/2π, Maclaurin expansions  of EJ,ScS and EJ,SIS make their similarities apparent:  EJ,ScS(ϕ) = ∆Φ0  4eRn  �1  2ϕ2 − 1  48ϕ4 + O(ϕ6)  �  EJ,SIS(ϕ) = EJ,SIS  �1  2ϕ2 − 1  24ϕ4 + O(ϕ6)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (4)  Comparing the coefficients of the harmonic (ϕ2) term,  we observe that the Josephson energy of a ScS transmon  can be defined as:  EJ,ScS = ∆Φ0  4eRn  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='755Ic,ScSΦ0/(2π) ,  (5)  where  the  last  equality  recognizes  that  Ic,ScS  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='662π∆/(eRn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' 5 shows that both potential ener-  gies contain anharmonicity led by a ϕ4 term, from which  we estimate that the anharmonicity of a ScS transmon  is about one half that of a SIS transmon, for devices  with the same EJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This difference is clear when compar-  ing normalized EJ(ϕ) of ScS and SIS transmons with a  harmonic parabolic potential ϕ2/2 (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' A more  precise evaluation of the anharmonicity is given in II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='3,  by computing the ScS transmon eigenenergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Eigenenergies and eigenstates of a ScS  transmon  A conventional SIS transmon has a Hamiltonian of the  form:  ˆHSIS = 4Ec(ˆn − ng)2 + EJ(1 − cos ˆϕ),  (6)  3  0  20  40  60  80  100  0  20  40  60  80  E0m/EC  0  20  40  60  80  100  −1  0  1  2  3  4  5  (E12 - E01)/EC  ScS  SIS  a  b  EJ/EC  EJ/EC  ScS  SIS  m = 0  1  2  3  0  20  40  60  80  100  100  τp (ns)  SIS  ScS  EJ/EC  10-1  10-2  c  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (a) Transition energy E0m = Em − E0 at ng = 1/2 and (b) oscillator anharmonicity (E12 − E01) at ng = 1/2, as  functions of EJ/EC for ScS transmon (solid lines) and SIS transmon (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (c) The minimal pulse duration (τp) of  ScS (solid line) and SIS transmons (dashed line) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' EJ/EC, all operated at ω01 = 2π × 10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  where ng is the offset charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The wave equation for a  SIS transmon can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [10]  In the ScS transmon, the potential energy is given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' 3, so that the Hamiltonian becomes,  ˆHScS = 4EC(ˆn − ng)2 + EJ  �  2 ln  �  cos2 ˆϕ  2  �  + 4 sin ˆϕ  2 tanh−1  �  sin ˆϕ  2  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (7)  The wave equation of a ScS transmon can be solved nu-  merically using the finite difference method, in which the  Hamiltonian is expressed in a discretized space of phase  ϕ ∈ [−π, π), with the periodic boundary condition ap-  plied to both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The validity of the computation is  confirmed by comparing a similar numerical solution of  the wave equation for a SIS transmon with the analytical  solutions presented by Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [10] Figure 3 compares  the first 4 eigenstates of a SIS transmon and a ScS trans-  mon, both with EJ/EC = 20 and ng = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Although  the lower level eigenenergies and eigenfunctions are sim-  ilar, the differences become more apparent at higher en-  ergies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This trend is more clearly observed for the transi-  tion energies E0m = Em − E0 calculated for both trans-  mon types, across a range of EJ/EC from 1 and 100  (Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This difference reflects the smaller anhar-  monicity of the ScS transmon, compared to the SIS trans-  mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' By treating the leading anharmonic term (−ϕ4/24)  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  4 as a perturbation to the harmonic potential  and applying the first-order perturbation theory, the mth  eigenenergy of a SIS transmon is approximated by [10]  Em,SIS ≈ ℏωp  �  m + 1  2  �  − EC  4  �  2m2 + 2m + 1  �  ,  (8)  in which ℏωp = √8EJEC is the Josephson plasma energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The transition energy between the (m − 1)th and mth  levels is therefore  Em−1,m,SIS ≈ ℏωp − mEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (9)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  9, we find that the anharmonicity of SIS  transmon αSIS ≡ E12,SIS−E01,SIS is approximately −EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  By applying the same first-order perturbation theory cal-  culation but realizing that the perturbation term is half  as a SIS transmon (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' 4), the mth eigenenergy of a ScS  transmon can be approximated by  Em,ScS ≈ ℏωp  �  m + 1  2  �  − EC  8  �  2m2 + 2m + 1  �  ,  (10)  so that its anharmonicity, αScS, is approximately −EC/2,  or half the anharmonicity of a SIS transmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' We can vi-  sualize this finding in a plot of numerical results, looking  at transmons with EJ/EC ≥ 20 (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The smaller anharmonicity of a ScS transmon means  that the transitions E01 and E12 lie closer in energy, so  that a longer RF pulse is needed to correctly excite the  desired transition E01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The minimal pulse duration can  be estimated as τp ≈ ℏ|α|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  As shown in Figure 4c,  despite its lower anharmonicity, τp of the ScS transmon  remains below 1 ns even for EJ/EC = 100, when the  qubit operates at 10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Because typical qubit pulse  durations are ∼10 ns, we may conclude that the lower  anharmonicity will not inhibit normal operation of a ScS  transmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Charge dispersion of a ScS transmon  A primary benefit of the transmon architecture is its  relative immunity to charge noise, when designed to op-  erate in the regime of EJ ≫ EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' In a SIS transmon, the  charge dispersion of the mth level decreases exponentially  with  �  8EJ/EC, following [10]  ϵm ≡ Em(ng = 1/2) − Em(ng = 0)  ≈ EC  24m+5  (−1)mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  �  2  π  � EJ  2EC  � m  2 + 3  4  e−√  8EJ/EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  (11)  Intuitively, the charge dispersion is related to the  tunneling probability between neighboring potential en-  ergy valleys (Figure 3), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=', when ϕ makes a full 2π  4  −2  −1  0  1  2  ng  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='0  Em/EJ  SIS  ScS  SIS  ScS  |ϵm|/E01(ng = 1/2)  101  10-3  10-7  10-1  10-5  10-9  10-11  1  2  3  4  5  6  7  8  9  10  (EJ/EC)1/2  10  20  30  40  50 60 70 80 90 100  EJ/EC  a  b  m = 0  1  2  m = 0  1  2  0  20  40  60  80  100  SIS  ScS  T2 (ns)  EJ/EC  100  10-2  104  102  106  108  c  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (a) The eigenenergies Em of the lowest 3 eigenstates (m = 0, 1, 2) of a ScS transmon (solid line) and a SIS transmon  (dashed lines), both with EJ/EC = 10, as functions of the offset charge ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (b) The charge dispersion ϵm of the lowest 3  eigenstates of a ScS transmon (solid line) and a SIS transmon (dashed lines), as functions of EJ/EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' (c) The dephasing time  T2 of ScS (solid line) and SIS transmons (dashed line) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' EJ/EC, all operated at ω01 = 2π × 10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [10] By this reasoning, we may expect the higher  barrier height of a ScS transmon (∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='8EJ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  2EJ)  to better suppress the tunneling probability and provide  lower charge dispersion, compared to a SIS transmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Figure 5a plots the first three eigenenergies Em (m =  0, 1, 2) versus the effective offset charge ng for both SIS  (dashed) and ScS (solid) transmons, with EJ/EC = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Clearly, the ScS transmon eigenenergies are more weakly  perturbed by ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Calculations of the charge dispersion,  ϵm = Em(ng = 1/2) − Em(ng = 0), across a wide range  of 1 ≤ EJ/EC ≤ 100 show that suppression of charge dis-  persion in the ScS transmon becomes more effective for  larger EJ/EC ratios (Figure 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' When EJ/EC = 100,  the charge dispersion of the first excited state of a ScS  transmon, ϵ1,ScS, is over one order of magnitude less than  the corresponding SIS transmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' It is noted that com-  putation on SIS transmon matches the analytical result  very well[10], again demonstrating the high numerical  precision of our finite difference computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Never-  theless, the computational error becomes significant as  the normalized charge dispersion, |ϵm|/E01, approaches  10−11 and smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This is due to the accumulation of  floating-point error that eventually shows up for evaluat-  ing the vanishing difference between the two eigenener-  gies at ng = 0 and 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  In Figure 5b, the y−axis is presented in the logarith-  mic scale and the x−axis is presented in the scale of  �  EJ/EC, so that all curves take a linear form approach-  ing large EJ/EC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' For the SIS transmon, the slope  matches the expected exp(−  �  8EJ/EC) dependence in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' For the ScS transmon, the slope is larger, and is  best described by:  ϵm ∝ exp  �  −  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='16 × 8EJ/EC  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The improved charge dispersion makes the ScS trans-  mon both less sensitive to charge noise and, in turn, gives  it a longer dephasing time T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  For dephasing caused  by slow charge fluctuations of large amplitude, Koch et  al[10] has found an upper limit of T2 given by  T2 ≈  4ℏ  e2π|ϵ1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Using this relation, we compare T2 for both SIS and ScS  transmons for EJ/EC between 1 and 100 (Figure 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The ScS transmon improves T2 across the entire range of  EJ/EC and especially at higher ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' At EJ/EC = 100,  the SIS transmon has a T2 ceiling of about 3 ms, com-  pared to about 50 ms for the ScS transmon, an over 10  fold increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' At present, because the T1 lifetime of SIS  transmon qubits is still beyond 1 ms and not limited by  the charge noise, this benefit of the ScS transmon archi-  tecture will have little performance benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' However, be-  cause we expect the lifetimes of superconducting qubits  to continue improving (Schoelkopf’s Law) [14], we an-  ticipate a point when charge noise dephasing becomes a  bottleneck, and the ScS transmon architecture can offer  effective mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  ScS transmon design parameters  The operational behavior of a ScS transmon is deter-  mined by its EJ and EC, which define the operating fre-  quency ω01, the relative immunity to charge noise (ϵ1),  and the minimum excitation pulse duration (τp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Be-  cause these three quantities are determined by EJ and  EC, they are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' We can visualize this in-  terdependence with three sets contour lines plotted in the  plane of EJ versus EC in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' These contours rep-  resent: (1) a transmon operating frequency (ω01/(2π))  between 1 and 10 GHz (set of red, descending diagonal  lines), (2) ratios of EJ/EC from 10, 100, and 1000 (set  of blue, ascending diagonal lines), and the minimum ex-  citation pulse duration τp between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='32 and 10 ns (set of  dashed, predominantly vertical lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Selecting two of  these defines the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' For example, a ScS trans-  mon designed to operate at ω01/(2π) = 5 GHz and with  5  EC/(2�ħ) (GHz)  102  EJ/(2�ħ) (GHz)  101  100  10-2  10-1  100  EJ/EC=103  EJ/EC=102  EJ/EC=101  ω01/(2�) = 1 GHz  2 GHz  3 GHz  4 GHz  5 GHz  6 GHz  7 GHz  8 GHz  9 GHz  10 GHz  τp = 10 ns  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='94 ns  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='31 ns  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='01 ns  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='58 ns  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='26 ns  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='00 ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='79 ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='63 ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='50 ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='40 ns  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='98 ns  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='32 ns  104  105  2  3  4  5  6  7  8  9  2  3  4  5  6  7  8  9  103  2  3  4  5  6  7  8  9  2  3  4  5  6  7  8  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='16 ns  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='51 ns  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='00 ns  Rn/Tc (Ω K-1)  102  103  2x101  2  3  4  5  6  7 8 9  2  3  4  5  6  7 8 9  2  3  4  5  6  7  8  9  3  4  5  6  7  8  9  CΣ (fF)  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' A graphical guide for designing ScS transmon with required EJ and EC to match desired transmon frequency ω01  and minimum pulse duration τp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The red lines are contours lines for transmon frequencies set at values between 1 and 10 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The dashed black lines are contours lines for τp set at a few values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='32 and 10 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The blue lines are contours lines  for EJ/EC ratios set at 10, 100, and 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' A second x-axis that is parallel to EC is presented for CΣ, following CΣ = e2/2EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Simlarly, a second y-axis that is parallel to EJ is presented for Rn/Tc, following Rn/Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='76kBΦ0/(4eEJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  a readout pulse of τp = 4 ns (green dot in Figure 6) will  have a EJ/EC ratio of about 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Instead, a shorter  excitation pulse of τp = 1 ns (purple dot in Figure 6)  requires a tradeoff of smaller EJ/EC ≈ 40, and thus less  immunity against charge noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Importantly, EJ and EC of a ScS transmon are set by  the physical device dimensions and fundamental prop-  erties of the materials composing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' EJ is determined  by the superconducting energy gap of the material (∆)  and the normal state resistance of the junction (Rn) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  For a BCS superconductor where ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='76kBTc,  we can express EJ in terms of the material properties  Rn/Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='76kBΦ0/(4eEJ,ScS), which is shown as the  second (right) y-axis in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Similarly, because EC  is set by the total capacitance (CΣ = e2/2EC,ScS) which  depends on device geometry and dielectric properties, we  can express EC as a capacitance, shown as a second (top)  x-axis in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  Returning to the example, we can now see from Figure  6 that designing a ScS transmon with ω01/(2π) = 5 GHz,  τp = 4 ns, and EJ/EC ratio of about 600 (green dot) re-  quires a junction with Rn/Tc ≈ 3 kΩ · K−1 and capacitor  with CΣ ≈ 250 fF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The properties can be realized by a  constriction junction fabricated from a thin film super-  conductor that has both a relatively high normal state  resistivity and a long superconducting coherence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  As one example, a 10-nm-thick PtSi film was reported  has normal-state sheet resistance Rs = 67 Ω/□, super-  conducting Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='63 K, and Pippard coherence length  ξ = 440 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' [15] Using these material parameters, we can  meet the ScS transmon design criteria using a constric-  tion junction with physical length of 440 nm and width of  6  16 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' The qubit capacitor physical dimensions should  be designed for CΣ ≈ 250 fF, according to Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' If one  instead desires the shorter readout pulse time of τp = 1  ns (purple dot), the constriction must have Rn/Tc ≈ 10  kΩ · K−1 and CΣ ≈ 70 fF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' For the same PtSi supercon-  ductor, these values can be met with physical length of  440 nm and width of 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' which are more challenging  dimensions to fabricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  CONCLUSION  In summary, we have demonstrated through computa-  tion that a short ScS Josephson junction can be used as  a drop-in replacement for the SIS tunnel junction in a  transmon qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' In the transmon regime (EJ ≫ EC), a  ScS transmon has 50% smaller anharmonicity than a SIS  transmon, but is compensated by its appreciably lower  charge dispersion that provides a significantly higher T2  ceiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Using this analysis, we estimate that high per-  formance ScS transmons can be achieved with constric-  tions having a normal state resistance of a few kiloohms,  which can be made from a thin nanobridge formed in low  Tc superconductors using conventional, high-resolution  nanofabrication techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  The ScS transmon design  allows all components, including constriction junction,  capacitor, and resonator, to be fabricated in a single  lithography step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' This is a significant simplification com-  pared to multistep SIS transmon fabrication, and also  provides an robust architecture amenable to device post-  processing, cleaning, and encapsulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This material is based upon work supported by the  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Department of Energy, Office of Science, National  Quantum Information Science Research Centers, Co-  design Center for Quantum Advantage (C2QA) under  contract number DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='  This research used  computational resources of the Center for Functional  Nanomaterials (CFN), which is a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' Department of  Energy Office of Science User Facility, at Brookhaven Na-  tional Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
+page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE3T4oBgHgl3EQfCglF/content/2301.04276v1.pdf'}
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@@ -0,0 +1,1274 @@
+arXiv:2301.11834v1  [cond-mat.soft]  27 Jan 2023
+Gas Diffusion in Cement Pastes: An Analysis using a
+Fluctuating Diffusivity Model
+Fumiaki Nakai1, Takato Ishida1,∗
+Department of Materials Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa,
+Nagoya 464-8603, Japan
+Abstract
+This work propose an application of the concept of fluctuating diffusivity to the dif-
+fusion of gas molecules in cementitious materials, particularly through a two-state
+fluctuating diffusivity (2SFD) model. The 2SFD model is utilized to investigate the
+diffusion of oxygen in cement pastes. The analysis provides a reasonable description
+of the diffusion coefficient of oxygen in cement pastes, and highlights the presence of
+non-Gaussian diffusion, which can be attributed to the heterogeneous microstructure.
+The presence of non-Gaussianity in the probability density of the molecule’s displace-
+ment, characterized by heavier tails than those of the Gaussian distribution, may have
+a significant impact on the durability assessments of concrete structures.
+1. Introduction
+Since the invention of Portland cement by Joseph Aspdin in 1824, cementitious ma-
+terials have been widely utilized in the construction of infrastructure. In recent decades,
+there has been a growing emphasis on assessing the long-term performance of rein-
+forced concrete structures, with a focus on reducing carbon emissions and preserving
+resources. The durability of concrete structures can be compromised by the penetration
+of aggressive lightweight molecules (causing chemical degradation [1] such as carbon-
+ation [2, 3, 4], corrosion [5, 6], sulphate attack [7], calcium leaching [8, 9]), making
+the examination of transport phenomena in cementitious materials a vital subject in the
+field of cement and concrete research. It is an undeniable fact that cementitious ma-
+terials are inherently porous in nature, possessing pores of various scales. Diffusion,
+the primary mode of mass transport, has been comprehended by devising effective dif-
+fusion coefficients which properly reflect the characteristics of the pore network struc-
+ture (tortuosity, connectivity, constrictivity, formation factor [10, 11, 12, 13, 14, 15])
+and by utilizing them to solve the diffusion equation. It is obvious that the probabilis-
+tic displacement distribution is Gaussian when the conventional diffusion equation is
+∗Corresponding author
+Email addresses: nakai.fumiaki.c7@s.mail.nagoya-u.ac.jp (Fumiaki Nakai),
+ishida@mp.pse.nagoya-u.ac.jp (Takato Ishida )
+
+resolved [16, 17]. However, recent research in the field of theoretical physics has high-
+lighted the existence of cases in which the displacement distribution deviates from a
+Gaussian distribution, depending on the spatio-temporal scale of interest. Such non-
+Gaussianity may have a significant impact on the long-term reliability probability as-
+sessment of reinforced concrete structures. We have effectively formulated the concept
+in a form that is applicable to diffusion in cementitious materials.
+A microstructure of cementitious materials inherently exhibits a heterogeneous
+composition, which can result in the non-Gaussian diffusion of gases. To effectively
+describe the diffusion in heterogeneous material, the concept of fluctuating diffusivity
+(FD) [18, 19, 20, 21, 22, 23, 24] has been demonstrated to be useful, as evidenced by
+the studies for the glass forming liquid [25], colloidal suspensions [26, 27], and bio-
+logical systems [28, 29]. The diffusion of a free molecule with fluctuating diffusivity
+is described by the equation
+∂G(x; t)
+∂t
+= D(t)∇2G(x; t)
+(1)
+where t denotes the time, x represents the displacement vector of the particle, G(x; t)
+is the probability density of x at time t, and D(t) represents the fluctuating diffusivity
+and is subject to a stochastic process. By providing a simple and physically reasonable
+rule for D(t), it is possible to theoretically analyze the dynamics of the diffusing par-
+ticles. The Fluctuating diffusivity is based on the idea that the diffusion environment
+experienced by the particle changes in time, either as a result of a temporal alteration
+in the environment or due to the migration of particles to a distinct milieu. Upon ini-
+tial inspection, one may think that the fluctuating diffusivity approach, expressed as
+Eq. (1), is similar to the time-dependent diffusivity models taking into account the
+long-term effects of changing diffusion media, such as prolonged hydration reactions
+and accumulated damages [30, 31, 32]. However, it is important to note that these
+two approaches are fundamentally distinct in terms of their concepts and underlying
+motivations. The fluctuating diffusivity approach posits that the diffusion coefficient
+changes stochastically over time, reflecting the temporal and spatial heterogeneity of
+the matrix. In contrast, the time-dependent diffusion coefficient varies deterministi-
+cally, reflecting the time evolution of internal microstructures caused by the long-term
+effects. In this paper, the latter approach, which is characterized by the deterministic
+variation of the diffusion coefficient, is referred to as deterministic drifting diffusiv-
+ity (DDD), and is distinguished from the fluctuating diffusivity. It is undeniable that
+the extensive research conducted on DDD has greatly enhanced our understanding of
+transport phenomena in cementitious materials and continues to be applied effectively
+in current studies. It is important to note that fluctuating diffusivity does not aim to
+replace or update DDD, but rather it takes a distinct physical perspective. In fact, the
+target timescale is significantly different between the fluctuating diffusivity and DDD
+approaches. Typically, the FD analyzes the particle diffusion on a timescale where the
+particle diffuses over the characteristic length of the heterogeneous environment, while
+the DDD approach focuses on the timescale where the state of the diffusion medium
+changes over a prolonged period. Here, it is important to note that some studies have
+employed DDD approach [30], which does not treat temporal and spatial fluctuations
+and is inadequate in describing diffusion in heterogeneous environments. The appli-
+2
+
+cation of the fluctuating diffusivity framework allows for an effective analysis of the
+phenomena of small molecule diffusion in cementitious materials, where the diffu-
+sivity may fluctuate spatio-temporally in response to the heterogeneous nature of the
+diffusion medium. In the context of diffusion in cementitious materials, it should be ef-
+fortless for researchers in the field of cement materials to envision diffusion phenomena
+that fall within the scope of such a framework, such as gas diffusion in a depercolated
+capillary pore network, cases of diffusion coupling with adsorption on the pore wall or
+dissolution in the pore solution. Additionally, phenomena such as the consumption of
+CO2 by carbonation and the immobilization of chloride ions through Friedel’s salt and
+calcium oxychlorides formation [33, 34], may also fall within the scope of this frame-
+work if these phenomena are regarded as trapping states with quite long time constants.
+When the timescale of observation is comparable to a timescale where the molecules
+diffuse over the characteristic length of the heterogeneous environment, non-Gaussian
+behavior of the displacement distribution is exhibited, i.e., the tails of the displacement
+distribution tend to be heavy).
+Let us herein present several sophisticated approaches for investigating diffusion
+in cementitious materials. There are two primary existing methods for understanding
+the diffusion phenomena of small molecules in cementitious materials: (i) numerical
+diffusion simulations on virtual microstructures that replicate the microstructural char-
+acteristics of cementitious materials, and (ii) empirical or semi-empirical modeling of
+effective diffusion coefficients through a process of homogenization. In recent years,
+the former approach of numerical diffusion simulations on virtual microstructures has
+made significant progress, successfully simulating the diffusion of various diffusants
+in cementitious materials of various types and compositions, both with and without
+interfacial transition zones (ITZs) [35, 36, 6, 37, 38, 39, 40, 41, 42, 43]. A particularly
+successful recent approach within this model has been the implementation of numerical
+diffusion models, such as those based on the Lattice Bolzmann method [37, 40, 41, 42],
+random walk method [6, 39, 44, 43], and finite element method [35], utilizing virtual
+3D microstructures generated by hydration models. Several hydration models have
+been previously proposed, such as CHEMHYD3D [45, 46, 47], HYMOSTRUC3D
+[48], THAMES [49, 50], DuCOM [51], IPKM [52], µic [53], which are widely used
+in the field of cement and concrete research. In such microstructure-guided diffusion
+models, the CHEMHYD3D model (a voxel-based approach) devised by Bentz and Gar-
+boczi [45, 46, 47] and the HYMOSTRUC3D (a vector-based approach) developed by
+van Breugel [48], are commonly utilized [35, 54, 40, 41]. Both CHEMHYD3D and
+HYMOSTRUC3D are founded upon Jennings’s colloidal model of Calcium-Silicate-
+Hydrates (CSH) morphology [55]. Recently, advancements in the force field of molec-
+ular dynamics in cementitious materials is becoming quite well-developed [56, 57, 58].
+Zhang et al. reported the modeling of diffusion simulations using the random walk
+method on structures generated by molecular dynamics [43]. The latter approach en-
+tails describing mass diffusion phenomena through empirically or semi-empirically
+modeling the effective diffusion coefficient in heterogeneous media and solving the
+standard diffusion equations utilizing that effective diffusion coefficient. The effec-
+tive diffusion coefficients are modeled in accordance with homogenization procedures
+commonly utilized in the field of composite materials, and are inferred to be in agree-
+ment with experimental observations and structural insights garnered from hydration
+3
+
+models [59, 60, 61, 62, 63, 14, 64, 65]. In the realm of finite element-based analysis
+utilizing representative elementary volume (REV) meshes (where the discretizing mesh
+size is generally greater than the discretization scale in microstructure-guided models),
+the identical homogenization procedure is applied to assign an effective diffusion co-
+efficient to each REV mesh [66, 15]. The empirical relationship linking the parameters
+of capillary pore and the effective diffusion coefficient is well organized in a critical
+review article by Patel et al [11]. When the porosity is known, the primary strategy is
+to attempt to express the effective diffusion coefficient through Archie’s law [67], and
+when porosity data is unavailable, the effective diffusion coefficient is frequently de-
+rived via the Powers model [68], which can link the hydration degree and water-cement
+ratio (w/c) to the capillary porosity. Yamaguchi et al. refined the empirical relationship
+by assessing the accessible capillary pores, and demonstrated that the modified model
+is efficacious in describing the effective diffusion coefficient of tritiated water [69].
+Furthermore, the empirical effective diffusive coefficient has been adapted to include
+semi-empirical parameters that characterize the morphology of the pore network (tor-
+tuosity, connectivity, constrictivity, formation factor) [10, 11, 12, 13, 14, 15]. There has
+been extensive research aimed at relating these parameters to the actual pore topology
+obtained from imaging techniques, rather than simply adjusting bulk diffusion coef-
+ficients to effective diffusion coefficients [70, 71, 10, 66, 72, 13, 15]. Recently, an
+attempt has been reported to construct a regression model for the diffusion of chloride
+ions in concrete using machine learning techniques [73]. It is important to note that
+none of the models presented in this paragraph, which express diffusion coefficients,
+can be considered universally applicable. For instance, the microstructure-based dif-
+fusion model in dry cement paste established by Liu et al., despite taking into account
+various factors related multi-scale properties, cannot perfectly explain the diffusion co-
+efficient in low w/c mixing cement pastes [40]. This discrepancy may be attributed to
+the structural fluctuations of the generated virtual microstructures, which have a greater
+impact on the apparent diffusivity in the regime of low w/c regime. Additionally, the
+empirical model also appears to exhibit a somewhat greater discrepancy between its
+predicted diffusion coefficients and those observed in the low w/c regime [11]. In this
+work, we introduce an up-to-date concept of theoretical physics, “fluctuating diffusiv-
+ity”, to the cement and concrete field. The proposed framework enables the incor-
+poration of morphological features of heterogeneous medias and the consideration of
+several types of diffusion as stochastic processes, without the requirement for detailed
+structural information or multiple empirical parameters.
+The paper is structured as follows: In Section 2.1, we present a comprehensive for-
+mulation of the fluctuating diffusivity using a general discretized state. In Section 2.2,
+we delve into a simplified two-state fluctuating diffusivity (2SFD) model, following
+the work by Uneyama et al [21] and Miyaguchi et al [74]. We analytically calculate
+the self-part of the intermediate scattering function and the second and fourth moments
+of the probability density of particle displacement, which are integral components for
+discussing the probability density of the displacement within the model. In Section 2.3,
+we apply the 2SFD model to a fundamental system, specifically the diffusion of O2 in
+cement pastes under standard temperatures and pressures as a preliminary test case.
+The subsequent Section 3 discusses the distinctions of the proposed model in com-
+parison to existing models, its scope of applicability and limitations, its potential for
+4
+
+generalization to cementitious systems, and the potential impact of the derived diffuse
+displacement distribution on the long-term durability assessment of future structures.
+The conclusion is provided in section 4.
+2. Theory
+2.1. Fluctuating diffusivity with n-states
+The fluctuating diffusivity can be represented by the diffusion equation, which in-
+cludes a fluctuating diffusivity term, D(t), as
+∂G(x; t)
+∂t
+= D(t)∇2G(x; t)
+(2)
+where x represents the tracer position, t denotes the time, G(x; t) is the probabil-
+ity density of x for a given t, and D(t) is the time-dependent fluctuating diffusivity.
+While this work analyzes the 2SFD model in the following subsections, the calcula-
+tion method is not restricted to the two-state. Thus, we here calculate for the general
+n-states case as
+D(t) = D⊤ξ(t)
+(3)
+where D⊤ = (D1, D2, · · · , Dn) is the vector of the diffusion coefficients and its
+component Di denotes the diffusion coefficient of the i-th state. ξ(t) indicates the state
+of the diffusivity at time t; ξi = 1 and the other components are zero.
+We here describe the probability density vector where the particle is in i-state at
+time t as P (t), and its stochastic process is described as:
+∂P (t)
+∂t
+= RP (t)
+(4)
+where R represents the transition matrix. From this expression, we can formally ex-
+press the probability density of P (t+ ∆) with the infinitesimal time step ∆ for a given
+P (t) as
+P (t + ∆) = exp (∆R) P (t)
+(5)
+From this expression, the transition probability where the state changes from ξ(t) to
+ξ(t + ∆) is
+P(ξ(t + ∆); ξ(t)) = ξ⊤(t + ∆) exp (∆R) ξ(t)
+(6)
+To proceed with the calculation of Eq. (2), the intermediate scattering function:
+F(k, t) =
+�
+e−ik·rG(x; t) is useful. By taking the Fourier-transform of Eq. 2, we
+obtain the differential equation with F(k, t) as follows.
+∂F(k; t)
+∂t
+= −D(t)k2F(k; t).
+(7)
+This differential equation is formally solved as [22, 24]
+F(k; t) =
+�
+exp
+�
+−k2
+� t
+0
+D(t′)dt′
+��
+D
+(8)
+5
+
+where ⟨· · · ⟩D denotes the ensemble average for D(t). Formally, Eq. (8) can be de-
+scribed as a discretized form as
+F(k; t) =
+�
+ξ(j∆t)
+exp
+
+−
+t/∆−1
+�
+j=0
+∆k2D⊤ξ(j∆)
+
+ ×
+t/∆−1
+�
+j=0
+[P(ξ((j + 1)∆); ξ(j∆))] ξ⊤(0)P (0)
+=
+�
+ξ(j∆)
+t/∆−1
+�
+j=0
+�
+exp
+�
+−∆k2D⊤ξ(j∆)
+�
+×
+ξ⊤((j + 1)∆) exp (∆R) ξ(j∆)
+�
+ξ⊤(0)P (0)
+(9)
+This equation is akin to that of the partition function of the Ising model under an exter-
+nal field. Then, we define the transfer matrix as
+ξ⊤T ξ(t) =ξ⊤(t + ∆) exp
+�
+∆R − ∆k2D⊤ [ξ(t + ∆) + ξ(t)]
+2
+�
+ξ(t)
+(10)
+Since ∆ is an infinitesimal quantity, the elements of the transfer matrix can be ex-
+pressed as:
+Tij = exp(∆R)ij exp(−∆k2Djδij) = δij + ∆(Rij − k2Djδij)
+(11)
+For the sake of brevity, we also define the matrix Qij as:
+Tij = δij + ∆Qij
+(12)
+By utilizing the transfer matrix, Eq. (9) can be reduced to
+F(k; t) =
+�
+ξ(j∆)
+e∆k2D⊤ξ(t)/2×
+t/∆−1
+�
+j=0
+ξ⊤((j + 1)∆)T ξ(j∆)e−∆k2D⊤ξ(0)/2ξ⊤(0)P (0)
+=
+�
+ξ(j∆)
+t/∆−1
+�
+j=0
+ξ⊤((j + 1)∆)T ξ(j∆)ξ⊤(0)P (0)
+=
+�
+ξ(t)
+ξ⊤(t)T t/∆P (0) =
+�
+ξ(t)
+ξ⊤(t)etQP (0)
+(13)
+This equation can be calculated when the initial probability density P (0), the i-th state
+diffusivity coefficient from Eq.(3), and the transition probability R from Eq.(4) are
+provided.
+6
+
+2.2. 2SFD model
+We here consider the two-state fluctuating diffusivity (2SFD) model following the
+literature by Uneyama et al [21] and Miyaguchi et al [74], which serves as a mathemati-
+cally tractable model. The diffusivity of the particle in the 2SFD model is characterized
+by distinct variables, D⊤ = (Df, Ds), and the transition probability matrix, R, which
+is represented as
+R =
+�−rf
+rs
+rf
+−rs
+�
+(14)
+In the equilibrium state, the initial probability density is given by
+P (0) =
+1
+rf + rs
+�
+rs
+rf
+�
+(15)
+Then, the matrix Q in Eq. (13) is presented as
+Q =
+�
+−rf − k2Df
+rs
+rf
+−rs − k2Ds
+�
+(16)
+For this Q, the eigenvalues and the corresponding eigenvectors are respectively given
+by:
+λ± = −rf + k2Df + rs + k2Ds ±
+�
+(rf + k2Df − rs − k2Ds)2 + 4rfrs
+2
+(17)
+v± =
+�
+−
+rf+k2Df −rs−k2Ds±√
+(rf+k2Df −rs−k2Ds)2+4rf rs
+2rf
+1
+�
+(18)
+Using λ± and v±, matrix Q can be described as
+Q = (v+, v−)
+�
+λ+
+0
+0
+λ−
+�
+(v+, v−)−1
+(19)
+Combining Eq. (13) and (19), we obtain
+F(k; t) =(1, 1)(v+, v−)
+�eλ+t
+0
+0
+eλ−t
+�
+(v+, v−)−1
+1
+rf + rs
+�rs
+rf
+�
+=χ+eλ+t + χ−eλ−t
+(20)
+where we defined χ± as
+χ± = 1
+2
+�
+1 ± (k2Df − k2Ds)(rf − rs) + (rs + rf)2
+(λ+ − λ−)(rf + rs)
+�
+(21)
+Eq. (20) includes all information for the probability density function G(x, t). From
+Eq. (20), we can calculate all moments of the probability density such as second and
+fourth moments (⟨x2(t)⟩ and ⟨x4(t)⟩), respectively, where the bracket ⟨· · · ⟩ denotes
+the statistical average. The utilization of higher moments serves to quantify the devi-
+ation of G(r; t) from the Gaussian distribution, as will be discussed subsequently. As
+7
+
+per the definition of the self-part of the intermediate scattering function, these moments
+are formally obtained in the isotropic system as
+⟨x2(t)⟩ = − ∂2
+∂k2 F(k, t)|k=0
+(22)
+⟨x4(t)⟩ = ∂2
+∂k2
+∂2F(k, t)
+∂k2
+|k=0
+(23)
+To assign Eq. (20) to Eq. (22), we obtain
+⟨x2(t)⟩ = 6Dfrs + Dsrf
+rf + rs
+t,
+(24)
+Using this relation, the average diffusion coefficient D can be determined through the
+relation ⟨x2(t)⟩ = 6Dt in a three-dimensional system as
+D = Dfrs + Dsrf
+rf + rs
+(25)
+This outcome indicates that the average diffusion coefficient in the present 2SFD model
+is the weighted average of Df and Ds with the transition rates rf and rs. Furthermore,
+by utilizing Eq.(23), we can obtain an analytical expression for the fourth moment of
+G(r; t) as
+⟨x4(t)⟩ =120
+�(Dfrs + Dsrf)2
+2(rf + rs)2
+t2−
+(Df − Ds)2rfrs
+(rf + rs)4
+�
+1 − (rf + rs)t − e−(rf+rs)t��
+,
+(26)
+which is used later.
+2.3. Application of 2SFD model to gas O2 in cement paste
+In this study, we address the fundamental problem of O2 diffusion, which is known
+to be one of the basic aggressive gases that can affect the long-term performance of
+reinforced concrete structures [75]. The diffusion of oxygen in dry cement paste (i.e.,
+the absence of free water in capillary pores), is chosen as the primary case study. This
+system was selected as it presents a relatively simple diffusion medium of cementi-
+tious materials, yet offers somewhat heterogeneity. We here focus on the O2 diffusion
+in dry cement paste consisting of the capillary pore phase and the colloidal CSH [55]
+phase under ambient temperature and pressure conditions T = 298 K, P = 1 atm.
+As depicted in Figure 1, the colloidal CSH consists of two different density phases in
+proximity to the surface and the hydration front, which are classified as LD-CSH (low-
+density CSH) and HD-CSH (high-density CSH), respectively [76, 77]. For simplicity,
+this study treats the capillary pore phase and the LD-CSH phase are considered as
+diffusive, while the HD-CSH and unhydrated clinker regions as non-diffusive phases.
+Given the non-negligible difference in density between the LD-CSH and HD-CSH, we
+tentatively assumed that O2 molecules cannot be able to penetrate into the HD-CSH
+8
+
+Capillary pore
+: O  molecule
+2
+(Fast diffusive phase; Df )
+(Slow diffusion phase; Ds )
+LD CSH
+HD CSH & clinker
+(Non-diffusive)
+Figure 1: Schematic diagram of O2 diffusion in a cement paste, consisting of three phases: capillary
+pore, low-density CSH (diffusive) phase and non-diffusive phase (high-density CSH and unhydrated cement
+clinker)
+Trapping at CSH phase
+Diffusion in capillary pores
+fast
+slow
+Figure 2: Schematic illustration of transitional process of diffusivity D(t). Df corresponds diffusivity of
+the fast diffusion in capillary pores, and Ds corresponds diffusivity of the slow diffusion at CSH phase.
+phase through the LD-CSH phase. This study regards the diffusion in the capillary
+void as a rapid diffusion process (diffusion coefficient Df), comprising both molecu-
+lar diffusion and Knudsen diffusion, while diffusion in the LD-CSH is considered as
+a slow diffusion process (diffusion coefficient Ds). They are used as the inputs for
+the 2SFD model, as illustrated in Figure 2. Note that the following analyses derive
+all characteristic values of the heterogeneous diffusion media through physically rea-
+sonable estimations. In our system, at ambient temperature and pressure, the impact
+of surface diffusion on the overall diffusion characteristics is possibly negligible (the
+coverage of O2 molecule is approximately 0.01 or less, it could be estimated by the
+similar way in Ref. [40]).
+From this point on, the system setup is described in detail. The size of the colloidal
+CSH is assumed to be l = 50 nm, which is determined based on the size of the globule
+floc in the CM-II model proposed by Jennings [78]. In this study, the thickness of the
+LD-CSH on colloidal CSH, which is treated as the diffusive phase, is assumed as 10 nm
+from the surface, in accordance with the value utilized in the previous microstructure-
+guided model [40]. The porosity is represented by φ, and the number density of the
+colloidal CSH is denoted by ρ. For simplicity, we assume that the colloidal CSH is
+spherical, and then the relation between φ and ρ is described as
+1 − φ = ρπl3
+6
+(27)
+With the parameters specified above, we describe the four input parameters, namely
+9
+
+Df, Ds, rf, and rs, in the 2SFD model. In the present model, the diffusion coefficient
+of the fast state, Df, can be considered as the harmonic average of the molecular and
+Knudsen diffusion coefficients, DM and DK, as follows:
+Df =
+DMDK
+DM + DK
+(28)
+In the ordinary pressure and temperature conditions, DM is estimated as [79]
+DM = 3kBT
+8Pσ2
+�
+kBT
+πm
+(29)
+where kB denotes the Boltzmann constant. σ and m represent the diameter and mass
+of the Oxygen, respectively. They are effectively given as σ = 3.46×10−10m [80, 81]
+and m = 5.31 × 10−26kg. From these variables, DM is estimated as DM = 1.99 ×
+10−5m2s−1. In a complex system such as cement materials, the estimation of DK
+is difficult. We roughly estimate DK by approximating the target cement system as a
+Lorentz gas, i.e. a single mobile particle in fixed spherical obstacles. An analogous
+postulation was utilized in the research examining gas diffusion in cement paste by Liu
+et al [40]. Under this assumption, the diffusion coefficient is determined as [82]
+DK = ¯v2τ
+3
+(30)
+where ¯v denotes the mean speed of the Oxygen, given as ¯v =
+�
+8kBT/πm, and τ
+represents the mean free time. The estimation of τ is a challenging task, however,
+it has been roughly estimated from the mean pore size [40]. In this study, a rough
+approximation of τ is made by considering the gas kinetics. When the colloidal CSH
+is dilute, the mean free time can be expressed as τ = 4/ρπl2¯v, where it is assumed
+that the interaction distance between O2 and the colloidal CSH is approximated as (l +
+σ)/2 ≃ l/2. This estimated τ is not adequate for the low porosity regime, for instance,
+τ should be 0 for φ = 0. To account for the case of small φ, a phenomenological
+description of τ as depicted in previous literature [83] is employed:
+τ =
+�
+1 − ρπl3
+6
+�
+4
+ρπl2¯v
+(31)
+Combining Eqs. (27), (30), and (31) we obtain
+DK =
+4lφ
+9(1 − φ)
+�
+2kBT
+πm
+(32)
+In this expression, DK becomes 0 for φ = 0 and diverges for φ = 1, this is in agree-
+ment with the intuitive representation of Knudsen diffusion. The slow diffusion state
+pertains to diffusion within the LD-CSH phase. The determination of diffusivity is
+not straightforward as the handling of diffusion within the LD-CSH phase is complex.
+Though this estimation remains an open problem, prior investigations suggest that there
+may exist two possible approaches, (i) consider it as surface diffusion and determining
+10
+
+the diffusion coefficient through Wu’s empirical equation [84] and the model of Chen
+and Yang [85], which are commonly employed in the context of shale gas, or (ii) by
+utilizing effective medium theory as demonstrated by Patel et al [61]. Here, we ten-
+tatively assume the slow diffusion coefficient as Ds = 10−8m2s−1. This value does
+not contradict with both estimations introduced above. The first approach necessitates
+the isosteric adsorption heat (∆H) as an input for Wu’s empirical equation [84]. If
+we adopt the isosteric adsorption heat of CO2 on the CSH surface, ∆H ∼ 10 kJ/mol
+is tentatively applied to O2 as the same procedure conducted by Liu et al. [40], the
+Ds would be of the order of 10−8m2s−1. Furthermore, Patel et al. also reported that
+the C-S-H diffusivity is three orders of magnitude lower than the bulk diffusivity for
+various diffusants [61]. Subsequently, the transition rates rf and rs are determined
+consistently with information on the pore structure. The rf corresponds to the transi-
+tion rate from the fast diffusion state at capillary pores to the slow diffusion state in
+the LD-CSH phase. In the dilute limit of the volume fraction of the CSH phase, the
+average capillary pore size, L, can be roughly approximated to be ρ−1/3. When the
+volume fraction of the CSH phase is not dilute, the effect of excluded volume must be
+taken into account, which can be phenomenologically estimated. As L approaches 0
+when the space is entirely occupied by the CSH phase and diverges when the CSH is
+absent, a possible relation between L and the CSH number density is
+L =
+�
+ρ
+(1 − ρπl3/6)
+�−1/3
+= l
+�
+πφ
+6(1 − φ)
+�1/3
+(33)
+If it is assumed that rf represents the rate of contact with the colloidal CSH from the
+capillary pore phase, rf can be estimated as
+Df = L2rf
+(34)
+The rs represents the transition rate from the slow diffusion state in the LD-CSH phase
+back to the fast diffusion state at capillary pores. Considering that the thickness of the
+LD-CSH phase is about lLD = 10nm and that it can escape from the surface by about
+10nm motion, the following estimation can be obtained.
+Ds = l2
+LDrs
+(35)
+By utilizing the relations of Df, Ds, rf, and rs as stated above, we can calculate the
+dynamics of the O2 in the 2SFD model.
+The representation of the trajectory may prove informative in providing an intu-
+itive understanding of the 2SFD model. Subsequently, utilizing a kinetic Monte Carlo
+scheme [86, 87] based on Equations (2) and (4), we numerically calculate the trajectory.
+Figure 3 illustrates a representative trajectory of an O2 molecule in the 2SFD model,
+depicted by the red curve with black dots indicating a time interval of (rf + rs)−1.
+For comparative purposes, the trajectory of an O2 molecule moving without fluctuat-
+ing diffusivity (diffusion coefficient kept constant as per Equation (25)) is presented
+by the yellow curve with black dots plotted at every time interval (rf + rs)−1. The
+red curve effectively captures the heterogeneous diffusivity, which can be interpreted
+as a reflection of the heterogeneous nature of the cement paste. In contrast, the yellow
+curve does not exhibit heterogeneity, of course.
+11
+
+x
+y
+2SFD model
+Constant diffusion coefficient
+100 nm
+Figure 3: The trajectory of O2 over the observed time duration 1000/(rf + rs) at a porosity φ = 0.5 is
+represented by the pink curve. For comparison, the trajectory of particle diffusion with a constant diffusion
+coefficient, as represented by equation (25), is depicted by the blue curve. Closed circle symbols are also
+displayed at the same time intervals of (rf + rs)−1.
+12
+
+From Equation (25), we depict the diffusion coefficient D as a function of porosity
+φ in Figure (4). For comparative purposes, data obtained in previous studies by Yio
+et al. [13], Boumaaza et al. [88], and Houst and Wittmann [89] are represented by
+blue, purple, and green symbols, respectively. Table 1 summarizes the detailed condi-
+tions of previous works that measured oxygen diffusion coefficients in cement pastes.
+In this study, the ideal comparison for the measured O2 diffusivity in cement pastes
+would have been to data obtained from completely dry cement pastes, as reported by
+Boumaaza et al. [88]. However, to the best of our knowledge, such data is quite scarce.
+Therefore, in order to provide a reasonable comparison, we have elected to include
+the results of previous studies that have measured O2 diffusivity in cement pastes un-
+der conditions of relatively low humidity, as suggested by the findings of Houst and
+Wittmann [89] that the effect of relative humidity on diffusivity is minimal below 55%.
+Specifically, we have included the results of Yio et al. [13], Houst and Wittmann [89]
+as comparable data for O2 diffusivity in cement pastes. However, we have not included
+the part of the results in Yio et al. where the hydration reaction was not fully completed
+in the comparison data. The size of the colloidal CSH changes as the hydration reaction
+progresses, which affects the estimated diffusion coefficient in this 2SFD model, as we
+will discuss later in the study (See discussion section). Our theoretical results exhibit
+a qualitative agreement with the data presented in these prior works. It is important to
+note that the four inputs, Df, Ds, rf, and rs, are derived from system parameters and
+can be determined through physical considerations.
+Porosity
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+1
+2
+3
+4
+[m / s]
+2
+Yio et al. (2019)
+Boumaaza et al. (2018)
+Houst and Wittmann (1994)
+2SFD model
+Figure 4: Diffusion coefficient of the molecule O2 against the porosity φ.
+Gas diffusion in cementitious materials is often characterized by the diffusion coef-
+ficient, however, it may be insufficient to fully explore the corrosion of reinforcement.
+The tail of the probability density of the displacement G(r; t) should also be taken into
+account. We herein analyze the probability density of the displacement for a single de-
+13
+
+Table 1: Detail information of previous gas diffusion datasets.
+Ref.
+w/c ratio
+Curing
+Drying method
+Porosity
+Conditions
+Yio et al. [13]
+0.30
+Cured at 100% RH,
+Kept in 55% RH, 293 K
+0.133
+0.5 ∼ 2.5 atm
+0.45
+293 K for 90 days
+0.194
+and room temperature
+Boumaaza
+0.50
+Cured at 100% RH
+Oven-dried
+0.492
+1 atm and 293 K
+et al. [88]
+for 1 day, 2 months and 8 month
+0.455
+0.417
+0.60
+0.483
+0.454
+Houst and
+0.40
+Immersed in lime water
+Oven-dried
+0.110
+1 atm, room temperature
+Wittmann [89]
+0.80
+for 6 months or more
+0.390
+and 47 % RH
+gree of freedom, x, G(x; t), which is derived from the inverse Fourier transform of the
+self-part of the intermediate scattering function as G(x; t) =
+�
+eikxxF(kx; t). F(kx; t)
+can be computed from Eq.(2) by substituting x with x; as a result, Eq.(20) where k is
+substituted with kx is obtained. As the analytical calculation of the inverse transfor-
+mation of F(kx; t) is difficult, we perform the numerical integration. Fig. 5 illustrates
+the probability density of the O2 displacement for various time durations t at a typical
+porosity φ = 0.5. We scale the horizontal and vertical axes by the standard deviation of
+the displacement
+√
+2Dt and display the Gaussian distribution function as a reference.
+In the short time scale t ≤ 10−8s, clear deviations of G(x; t) from the Gaussian distri-
+bution are observed. These deviations gradually diminish with increasing observation
+time t. t ∼ 10−8s is comparable to the timescales of the inverse of rf or rs. This result
+suggests that the Gaussian approximation for G(x; t) may not be appropriate for the
+timescale over which O2 diffuses the lengths of colloidal CSH or the capillary pore.
+Our result is reasonable since non-Gaussian distributions, in fact, have been frequently
+observed at the microscopic scale in various heterogeneous systems such as confined
+water in CSH [90], glass-forming liquids [25], or colloidal suspensions [26, 27].
+To characterize non-Gaussian diffusion, the non-Gaussian parameter α is often em-
+ployed [91, 92, 93] and defined in three-dimensional systems as [94]
+α(t) = 3⟨r4(t)⟩
+5⟨r2(t)⟩2 − 1
+(36)
+where brackets denote the statistical average. α is equal to zero when the stochastic
+process of displacement conforms to a Gaussian distribution; if the dynamics of the
+particle can be described by the conventional diffusion equation with constant diffusiv-
+ity, α is equal to zero. Empirically, non-Gaussianity cannot be neglected for α > 0.1.
+We have obtained the second moment ⟨r2(t)⟩ and the fourth moment ⟨r4(t)⟩ as repre-
+sented by Equations (24) and (26), respectively. Consequently, we can determine α as
+the following expression:
+α(t) =
+2(Df − Ds)2rfrs
+(Dfrs + Dsrf)2(rf + rs)2
+e−(rf+rs)t + (rf + rs)t − 1
+t2
+(37)
+Fig. 6 displays α against time t with various porosity φ.
+α exhibits strong non-
+Gaussianity for the small-time regime t ≪ 10−9, and it non-monotonically changes
+with increasing porosity φ. This result may be reasonable since the heterogeneity of
+14
+
+ 
+ 
+ 
+ 
+ 
+0
+1
+2
+3
+4
+-1
+-2
+-3
+-4
+10
+-1
+10
+ 0
+10
+-2
+10
+-3
+=10
+-9
+=10
+-8
+=10
+-7
+=10
+-6
+Gaussian
+Figure 5: Probability density of O2 displacement for various time duration t at the porosity φ = 0.5. The
+horizontal and vertical axes are normalized by the standard deviation
+√
+2Dt. For comparison, the Gaussian
+distribution is also presented with the black curve.
+the diffusivity will disappear for φ = 0 and φ = 1. Additionally, the non-Gaussian
+parameter α exhibits a decrease from 10−9s < t < 10−8, indicating that diffusion in
+the timescale of t < 10−8 cannot be described by a Gaussian process or a conventional
+diffusion equation with constant diffusivity.
+3. Discussion
+In this study, we employed the 2SFD model for O2 diffusion in cement pastes,
+which constitutes a stochastic diffusion model comprising parameters that can be phys-
+ically inferred from an abundance of experimental studies on gas diffusivity in cemen-
+titious materials, while incorporating some crucial aspects of microstructures. This
+model effectively addresses stochastic processes involving transitions between multi-
+ple diffuse states, and is capable of analytically determining the probabilistic displace-
+ment distribution including the non-Gaussian parameter. Therefore, we posit that it
+constitutes a highly flexible framework that can be easily modified as long as the tran-
+sition rates between multiple diffuse states including additional states can be effectively
+assessed.
+In the above analysis, we tentatively assumed a colloidal CSH dimension of 50 nm.
+This is possibly acceptable since the Jennings’s CSH morphological model of CM-II
+[78], suggests that the size of globule flocs within the ranges from 30 to 60 nm. The
+estimated net diffusion coefficient, as one of the outputs of the 2SFD model increases
+as the assumed colloidal CSH size increases, as shown in Figure 7. This behavior is
+consistent with the experimental results reported by Bentz et al. [95] that the diffusion
+15
+
+Time
+10
+ 0
+10
+ 1
+10
+ -11
+10
+ -10
+10
+ -9
+10
+ -8
+=0.005
+=0.250
+=0.450
+=0.650
+=0.850
+Figure 6: Non-Gaussian parameter α against time t with various porosity φ.
+coefficient increases with the size of the cement particles used in the cement paste in
+the high-porosity region, while remaining largely independent of cement particle size
+in the low-porosity region.
+In this model, we have demonstrated that the assumed size of the colloidal CSH
+not only has an impact on the diffusion coefficient, but also influences the higher-
+order moments of the probability distribution of displacement, such as the shape of
+the distribution function and the non-Gaussian parameter. Besides the size effect of the
+colloidal CSH, the examination of the influence of the shape of the assumed CSH shape
+may also be significant. Zhang et al. [96] recently revealed that the effect of the shape
+of cement particles (elliptical or spherical) on the chloride diffusion behavior in cement
+paste is limited. However, it is feasible that the shape of cement particles might have
+an effect on the shape of the probabilistic distribution of diffusion displacement, even
+in the system studied by Zhang et al. If we adopt the Jennings’s CM-II model for the
+CSH morphology in modeling, the CSH should possess the shape of an ellipsoid, rather
+than perfectly spherical. This discussion is worth for further investigations. Hence, it
+can be inferred that some of the elements of the microstructure have an impact on the
+shape of the probabilistic distribution of diffusional displacement, despite their limited
+influence on the diffusion coefficient. Since the tail of the diffusion distribution has
+a significant influence on the reliability (durability) assessment of reinforced concrete
+structures, it should be quite important to consider not only the diffusion coefficient,
+but also the shape of the displacement distribution in any theoretical, numerical, or
+empirical approaches.
+Liu, Liu, and Zhang [40] conducted a study investigating the dynamics of CO2, O2,
+and H2 in dry cement paste using the lattice Boltzmann method. In their research, a
+heterogeneous structure of cement paste was phenomenologically constructed, within
+which gas diffused. They determined the diffusion coefficient as a function of porosity,
+however, the probability density of displacement G(x; t) or the non-Gaussian parame-
+16
+
+2.5
+[m / s]
+2
+2.0
+1.5
+1.0
+0.5
+[nm]
+30
+40
+50
+60
+=0.10
+=0.20
+=0.30
+=0.40
+=0.50
+Figure 7: Diffusion coefficient D against the colloidal CSH size l with various porosity φ.
+ter α were not examined. The theoretical model presented here is akin to their system,
+thus their system would exhibit similar non-Gaussianity of G(x; t) and a non-negligible
+non-Gaussian parameter. In other words, their simulation methodology could verify
+our theoretical results.
+The current study addresses O2 diffusion as a case in point, however, since oxygen
+is not highly soluble in water, the estimates can be readily extrapolated even if the ce-
+ment paste in a humid environment (i.e., non-negligible amount of free or physically
+adsorbed water in capillary pores). The correction could be accomplished by incorpo-
+rating the relationship between relative humidity and water adsorption layer thickness
+[97], and incorporating it into the estimation of the diffusion coefficient of Knudsen
+diffusion. However, to extend the 2SFD model to CO2 diffusion (another crucial ag-
+gressive gas species), a slight alteration of the model may be necessary. Due to its high
+solubility in water, CO2 must be addressed as a diffusion phenomenon in conjunction
+with local solubility equilibrium, or it may be immobilized through an in-situ carbon-
+ation reaction that occurs in the pore solution and/or inside the CSH gel. To take this
+into account, it is essential to additionally incorporate as a third state a stochastic pro-
+cess that transits on a time scale so long in comparison to the observation time that
+the diffusion coefficient is virtually zero, but the trapped time can be considered effec-
+tively infinite. A comparable methodology should be necessary when addressing the
+diffusion problem of chloride ions as it also necessitates consideration of the effects
+of chloride binding. Even when it is expanded to a three-state (even for the extension
+for a multi-state model), as long as the eigenvalues and eigenvectors of the matrix Q
+in Eq. (13) (case of the 2SFD model in Eq. (16))) are obtained, all other calculations
+can always be performed. The simplicity of its mathematical structure is also another
+benefit of this model. The application to the diffusion phenomenon of chemical species
+(CO2, Cl –), which is more critical and reactive for the durability of concrete structures,
+17
+
+will be discussed in a future publication as ongoing research in the near future.
+4. Conclusion
+In conclusion, this work presents an application of the analytic method of fluc-
+tuating diffusivity to the study of gas diffusion in cementitious materials. Note that
+the concept of fluctuating diffusivity is not in opposition to the time-dependent diffu-
+sivity approach reflecting the long-term effects of changing diffusion media, such as
+prolonged hydration reactions, pore closure due to carbonation, crackings, but rather
+the target timescale is significantly different between the two approaches The fluctu-
+ating diffusivity framework effectively analyzes the diffusion of small molecules in
+cementitious materials, where the diffusivity may fluctuate spatio-temporally due to
+the heterogeneous nature of the diffusion medium, and has potential applicability to
+various diffusion phenomena in these materials. Our theoretical results of the 2SFD
+model provide a reasonable description of the diffusion coefficient of O2 in colloidal
+CSH, as measured in previous studies, by estimating input parameters from the vari-
+ables in the target systems. Furthermore, the 2SFD model highlights the presence of
+non-Gaussian diffusion, which can be attributed to the heterogeneous microstructure of
+cement pastes. The presence of non-Gaussianity in the displacement distribution, char-
+acterized by heavier tails than those of the Gaussian distribution, is quite critical for
+the accurate evaluation of the long-term reliability probability of reinforced concrete
+structures. The deviation in the shape of the tail of the Gaussian distribution obtained
+when solving the diffusion equation using a comparable diffusion coefficient in the
+2SFD model may lead to an underestimation of the conventional method’s reliability.
+In addition, while some numerical approaches utilizing the lattice Boltzmann meth-
+ods and/or random walk methods on virtual microstructures generated by previously
+established hydration models, it is important to acknowledge that there is still ample
+scope for improvement. In this regard, the development of a more conceptual stochas-
+tic model, such as the 2SFD model rooted in statistical physics, for the examination of
+diffusion phenomena in cementitious materials from a micro-perspective, and which
+can be solved analytically owing to its straightforward theoretical framework, in addi-
+tion to the structure-based model, would be of great significance to the field of cement
+and concrete materials research. We are convinced that this work contributes novel in-
+sights into the comprehension of diffusion of small molecules in cement and concrete
+materials, and has potential for further applications in the field of cement and concrete
+research.
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+

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+Video Semantic Segmentation with Inter-Frame Feature
+Fusion and Inner-Frame Feature Refinement
+Jiafan Zhuang, Zilei Wang∗, Junjie Li
+National Engineering Laboratory for Brain-inspired Intelligence Technology and
+Application, University of Science and Technology of China, Hefei 230027, China
+Abstract
+Video semantic segmentation aims to generate accurate semantic maps for each
+video frame. To this end, many works dedicate to integrate diverse information
+from consecutive frames to enhance the features for prediction, where a feature
+alignment procedure via estimated optical flow is usually required. However, the
+optical flow would inevitably suffer from inaccuracy, and then introduce noises
+in feature fusion and further result in unsatisfactory segmentation results. In
+this paper, to tackle the misalignment issue, we propose a spatial-temporal fu-
+sion (STF) module to model dense pairwise relationships among multi-frame
+features. Different from previous methods, STF uniformly and adaptively fuses
+features at different spatial and temporal positions, and avoids error-prone opti-
+cal flow estimation. Besides, we further exploit feature refinement within a sin-
+gle frame and propose a novel memory-augmented refinement (MAR) module to
+tackle difficult predictions among semantic boundaries. Specifically, MAR can
+store the boundary features and prototypes extracted from the training samples,
+which together form the task-specific memory, and then use them to refine the
+features during inference. Essentially, MAR can move the hard features closer to
+the most likely category and thus make them more discriminative. We conduct
+extensive experiments on Cityscapes and CamVid, and the results show that
+our proposed methods significantly outperform previous methods and achieves
+∗Corresponding author.
+Email addresses: jfzhuang@mail.ustc.edu.cn (Jiafan Zhuang), zlwang@ustc.edu.cn
+(Zilei Wang), hnljj@mail.ustc.edu.cn (Junjie Li)
+Preprint submitted to Pattern Recognition
+January 11, 2023
+arXiv:2301.03832v1  [cs.CV]  10 Jan 2023
+
+the state-of-the-art performance. Code and pretrained models are available at
+https://github.com/jfzhuang/ST Memory.
+Keywords:
+Video semantic segmentation, Spatial-temporal feature fusion,
+Memory mechanism, Feature refinement.
+1. Introduction
+Semantic segmentation targets to assign each pixel in scene images a se-
+mantic class, which is one of the fundamental tasks in computer vision.
+In
+recent years, image semantic segmentation has achieved unprecedented per-
+formance benefited from the great progress of deep convolutional neural net-
+work (DCNN) [1] and construction of various datasets (e.g.Cityscapes [2] and
+CamVid [3]). However, many real-world applications have strong demands for
+accurate video semantic segmentation, e.g., robotics, autonomous driving, and
+video surveillance. Actually, video data offer richer information than static im-
+ages, e.g., diverse presentations from multiple frames and temporal consistency
+prior. Thus video can provide good potential to achieve more accurate seman-
+tic segmentation. The key is how to produce more discriminative features by
+exploiting the characteristics of videos.
+A natural way to enhance video features is to integrate the diverse informa-
+tion of consecutive frames [4, 5]. Specifically, the feature alignment is commonly
+performed via the optical flow based feature warping, which ensures that pixel-
+level features at the same spatial position represent the identical object, and
+then the temporal feature fusion is conducted for each pixel. Evidently, the
+accurate optical flow is critical for feature fusion. However, the optical flow
+estimation inevitably suffers from inaccuracy in the boundary areas due to ob-
+ject occlusion and plain texture [6, 7]. If the features are not well-aligned, the
+noises would be introduced. Consequently, the quality of fused features would
+be reduced and the segmentation performance would be deteriorated.
+Besides, after aggregating information from consecutive frames, can we fur-
+ther refine the fused feature? Different from inter-frame feature fusion in video
+2
+
+(b)
+f
+f
+f𝑡−1
+f𝑡
+f𝑡
+f𝑡+1
+(a)
+f𝑡−1
+f𝑡
+f𝑡
+f𝑡+1
+f𝑡
+(c)
+Backbone
+Classifier
+Feature
+Fusion
+Feature
+Refinement
+Inter-Frame
+Feature
+Fusion
+Inner-Frame
+Feature
+Refinement
+Figure 1: Architecture illustrations for different methods. (a) Feature fusion in video
+segmentation methods (e.g., NetWarp [4] and GRFP [5]). (b) Feature refinement in image
+segmentation methods (e.g., DenseCRF [8] and SegFix [9]). (c) Our proposed method. Best
+viewed in color.
+segmentation methods, some image-based methods adopt the post-processing
+techniques to optimize the features for prediction. For example, DenseCRF [8]
+uses a graph structure to model pairwise potentials on all pixels and iteratively
+adjusts the feature by optimizing an energy function. Essentially, it uses simi-
+lar features to mutually enhance themselves. SegFix [9] proposes to replace the
+difficult boundary features with some better ones, whose locations are predicted
+by a network and often lie around the boundary areas in practice.
+Actually, feature fusion is proposed to aggregate useful information from
+different frames while feature refinement is designed for correcting error-prone
+features, which are potentially complementary. Based on this motivation, in
+this paper, we aim to improve the accuracy of video semantic segmentation by
+simultaneously considering inter-frame feature fusion and inner-frame feature
+refinement, as shown in Figure 1. For the inter-frame fusion, we need to tackle
+the feature misalignment issue.
+To this end, we propose a spatial-temporal
+fusion (STF) module that uniformly fuses the features at different spatial and
+temporal positions and does not require explicit feature alignment via error-
+prone optical flows. Here the transformer [10] is particularly adopted due to
+the power to model long-range dependencies.
+To be specific, the encoder is
+fed with the features extracted from consecutive frames, and the decoder is
+used to generate the prediction features by retrieving the current frame from
+3
+
+the encoded features. In particular, we utilize the self-attention mechanism in
+transformer to guide the feature fusion in latent space, in which more similar
+features are supposed to be more likely to represent the same object. For an
+image pixel, hence, STF would integrate multiple similar features at different
+temporal and spatial positions, rather than only the temporal-aligned features
+in the previous works [4, 5].
+In addition, an image with the resolution of (1024, 2048) would typically
+produce the features with the resolution of (128, 256). The transformer taking
+three frames needs to process 3 × 128 × 256 = 98304 pixel-level features, which
+results in unacceptable computation and memory cost with O(N 2) complexity
+when computing affinity matrix. Inspired by a recent work [11], we propose an
+interlaced cross-self attention (ICSA) attention mechanism to divide the dense
+affinity matrix computation in transformer as the product of a long-range cross-
+attention and a short-range self-attention, which can greatly reduce the memory
+consumption.
+On the other hand, we propose inner-frame feature refinement to further ad-
+just the fused features for better prediction without devising more complicated
+network structure. In this work, we focus on refining the hard features that are
+error-prone and always lie in the boundary areas of different classes [9]. To this
+end, we propose a novel memory-augmented refinement (MAR) module that
+uses the stored features in memory to augment the hard features. Actually, this
+is motivated by an intuitive observation that humans would retrieve memory to
+enhance the judgement when facing semantically ambiguous contents. Here the
+memory represents the experience from the training samples. For each semantic
+category, we particularly store the hard features and their corresponding class
+prototypes (refer to the mean feature representing a single category), which to-
+gether form a key-value memory bank. During inference, a hard feature would
+be refined by the class prototypes, where the weights of different classes are
+computed by comparing it with the stored hard features in the memory. In this
+way, the discriminativeness of boundary features would be enhanced since MAR
+would make them move closer to the most likely category. Evidently, MAR has
+4
+
+good interpretability and can be conveniently inserted into different models as
+an independent module.
+We experimentally evaluate the proposed method on the Cityscapes and
+CamVid datasets. The results validate the effectiveness of our STF and MAR
+to improve the quality of features, and their combination can achieve the state-
+of-the-art segmentation performance.
+The contributions of this work are summarized as
+• We design a novel video semantic segmentation framework by simultane-
+ously considering inter-frame feature fusion and inner-frame feature re-
+finement, which can take advantages from both two feature enhancement
+techniques and effectively improve segmentation accuracy.
+• We propose an effective spatial-temporal fusion module based on the trans-
+former, which can uniformly aggregate the features at different spatial and
+temporal positions and avoid error-prone temporal feature alignment.
+• We propose a novel memory-augmented refinement module to particularly
+refine hard features using the experience from training samples. In par-
+ticular, the key-value memory is stored to refine the hard features closer
+to the most likely category.
+• We experimentally evaluate the effectiveness of our proposed methods,
+and the results on Cityscapes and CamVid demonstrate the superiority of
+our methods to previous state-of-the-art methods.
+The rest of this paper is organized as follows. We review the related works on
+image and video semantic segmentation, transformer and memory mechanism
+in Section 2.
+Section 3 provides the details of our approach, and Section 4
+experimentally evaluates the proposed method. Finally, we conclude the work
+in Section 5.
+5
+
+2. Related Work
+2.1. Image Semantic Segmentation
+With the development of DCNN, more semantic segmentation networks
+spring up. Specifically, the fully convolutional networks (FCNs) [1] firstly uses
+the convolutional layers to replace fully-connected layers and can achieve better
+performance. Inspired by FCN, many extensions have been proposed to ad-
+vance image semantic segmentation. The dilated layers [12] are used to replace
+the pooling layers, which can better balance the computational cost and size
+of receptive fields. To further improve segmentation accuracy, spatial pyramid
+pooling and atrous spatial pyramid pooling (ASPP) are used in PSPNet [13]
+and DeepLab [12] to capture multi-scale contextual information.
+Mitivated
+by ASPP, Peng et al. [14] proposes a stride spatial pyramid pooling (SSPP)
+to capture multiscale semantic information from the high-level feature map,
+while Lian et al. [15] proposes a cascaded hierarchical atrous pyramid pooling
+module to simultaneously extract rich local detail characteristics and impor-
+tant global contextual information. CENet [16] aggregates contextual cues via
+densely usampling the convolutional features of deep layer to the shallow decon-
+volutional layers, which can fully explore multiple scale contextual information.
+GPNet [17] densely captures and filters the multi-scale information in a gated
+and pair-wise manner with a gated pyramid module and a cross-layer attention
+module. Marin et al. [18] propose a novel architecture based on shared pyrami-
+dal representation and fusion of heterogeneous features along the upsampling
+path, which is effective for dense inference in images with large scale. Different
+from enhancing features, EFNet [19] propose to produce multiple enhanced im-
+ages and fuses them to yield one new image, which can encourage the model to
+exploit complementary information.
+Differently, our proposed methods focus on exploiting both spatial and tem-
+poral contexts to further improve the performance and can build upon any
+existing image segmentation models.
+6
+
+2.2. Video Semantic Segmentation
+Different from static images, videos embody rich temporal information that
+can be exploited to improve the semantic segmentation performance. Existing
+video semantic segmentation methods mainly fall into two categories. The first
+category aims to accelerate inference speed by reusing the features in previous
+frames. DFF [20] estimates the optical flow fields [21] from the key frame to
+other frames and then propagates the high-level features using the predicted
+optical flows.
+Accel [22] proposes a reference branch to extract high-quality
+segmentation from the key frames and an update branch to efficiently extract
+low-quality segmentation from the current frames, and the fuses them to improve
+the segmentation accuracy. DAVSS [7] designs a feature correction mechanism
+to tackle distorted features after propagation due to inaccurate optical flows.
+LERNet [23] proposes to propagate multi-level features from the key frame via a
+temporal holistic attention module. TDNet [24] distributes several sub-networks
+over sequential frames and then recomposes the extracted features for segmen-
+tation via an attention propagation module. Differently, Liu et al. [25] designs
+a new temporal knowledge distillation methods to narrow the performance gap
+between compact models and large models.
+Another category focus on improving segmentation accuracy by modeling
+cross-frame relations to integrate information from consecutive frames. V2V [26]
+utilizes a 3D CNN to perform a voxel-level prediction. STFCN [27] utilizes a
+spatial-temporal LSTM over per-frame CNN features. However, these methods
+cannot achieve high performance due to rough processing of different frames.
+HDCNN [28] proposes a transition layer structure to make the pixel-wise label
+prediction consist with adjacent pixels across space and time domains. Recently,
+some works [4, 5] propose to fuse features from multiple frames to produce the
+better features for prediction. They usually adopt the optical flow to model
+cross-frame relationships and perform temporal alignment by warping features.
+In particular, NetWarp [4] uses a set of learnable weights to fuse multiple fea-
+tures, and GRFP [5] proposes the gated recurrent units STGRU to estimate the
+uncertainty of warped features and then conducts feature fusion on the areas
+7
+
+with high reliability. Obviously, the optical flow is critical for feature align-
+ment and would affect the final accuracy. However, the optical flow estimation
+inevitably suffers from inaccuracy, especially for the occlusion areas and small
+objects (e.g., pedestrian, pole) [7]. In this work, we follow the route of sec-
+ond category and focus on improving segmentation accuracy. Different from
+previous works, we propose to simultaneously model the spatial-temporal re-
+lationship without feature alignment, which can avoid error-prone optical-flow
+estimation. Furthermore, we propose to use memory to refine the prediction
+features.
+2.3. Transformer
+Transformer is originally proposed for the sequence-to-sequence machine
+translation [10], and currently has dominated various NLP tasks. As the core
+component of transformer, the self-attention is particularly suitable for mod-
+eling long-range dependencies. Due to the success of transformer in the NLP
+field, some works attempt to explore the benefits of transformer in computer
+vision. DETR [29] first builds an object detection system based on transformer,
+which can reason about relationships between objects and global context and
+directly output the final set of predictions. Swin Transformer [30] designs a
+novel shifted windowing scheme, which can limit attention computation to local
+windows while also allow for cross-window connection. It achieves an impressive
+performance on a broad range of vision tasks. In this paper, we propose STF by
+using transformer to model the spatial-temporal relationship among pixel-wise
+features extracted from consecutive frames. To our best knowledge, this is the
+first attempt to exploit the transformer in video semantic segmentation.
+Recently, Action Transformer [31] and Actor Transformer [32] also adopt
+transformer to model spatial-temporal relationship in action detection and group
+action recognition tasks, which are closely related to our proposed STF. They
+naturally adopt transformer for modeling proposal-context and proposal-proposal
+relationship. But our proposed STF is different from these two works. STF is
+designed for modeling pixel-wise relationship, which would involve huge memory
+8
+
+and computation overhead issues. In this work, we propose interlaced cross-self
+attention (ICSA) mechanism to tackle these issues and achieve efficient global
+relationship modeling.
+2.4. External Memory
+In DCNN, external memory is generally used to enhance feature represen-
+tations by storing history data, which is especially useful for the tasks without
+enough samples, e.g., life-long learning [33] and few-shot learning [34, 35]. For
+example, MM-Net [35] proposes to store the representative features in the sup-
+port set for one-shot learning, and then use them to predict the parameters of
+feature extraction network on query images. Actually, this can make the query
+features more relevant to the support features. In recent years, the memory
+mechanism is also exploited to store long-range temporal contexts for video
+tasks during inference. In video object detection, [36] proposes to store pixel-
+level and instance-level features extracted from previous frames and then use
+them to enhance the current frame.
+LFB [37] proposes a long-term feature
+bank for action localization to store supportive information extracted over the
+entire span of a video, and then uses them to enhance the short-term features
+extracted from short video clips. Different from the previous works that store
+temporal [36, 37] or sample [35] contexts, in this paper we propose to store
+the hard features and class prototypes from the training samples to form a
+task-specific memory, and then use them to refine the boundary features during
+inference.
+3. Our Approach
+In this work, we aim to boost the accuracy of video semantic segmenta-
+tion by enhancing the features for prediction. To this end, we first propose a
+spatial-temporal fusion (STF) module to perform inter-frame feature fusion at
+different spatial and temporal positions, which can avoid error-prone optical flow
+estimation. Then we propose a memory-augmented refinement (MAR) module
+9
+
+𝐼𝑇−1
+𝐼𝑇
+𝐼𝑇+1
+𝐟𝑇−1
+𝐟𝑇
+𝐟𝑇+1
+Spatial-Temporal
+Fusion Module
+Framework
+Memory-Augmented
+Refinement Module
+Backbone
+1、相邻多帧特征经过STRM实现特征融合
+2、融合特征经过MARM实现进一步优化
+መ𝐟𝑇
+Feature 
+Memory
+𝑆𝑇
+𝑆𝑇−1
+…
+…
+Sliding Window
+…
+…
+Classifier
+𝐟𝑇
+′
+Video Feature Enhancement
+መ𝐟𝑇−1
+Figure 2: The framework of our proposed approach. First, the feature is extracted by
+an image segmentation model for each frame. Then the features of consecutive frames are fed
+into our proposed STF module to perform feature fusion. After that, the fused feature f
+′
+T
+is further refined by our proposed MAR module, resulting in �fT . Finally, the segmentation
+result is obtained by applying the classifier on �fT . Best viewed in color.
+to further refine the boundary features during inference, which is essential to
+utilize the stored experience from training samples. In the following, we first
+introduce the framework of our proposed approach, and then elaborate on two
+key modules, namely, STF and MAR.
+3.1. Framework
+Our proposed video semantic segmentation framework is illustrated in Fig-
+ure. 2. Formally, given a sequence of n video frames denoted by {I1, I2, · · · ,
+In}, our purpose is to get the accurate semantic segmentation maps for every
+video frame, denoted by {S1, S2, · · · , Sn}. Specifically, we first extract features
+from each frame image using an off-the-shelf segmentation model. Then we con-
+duct video feature enhancement for the current timestamp T with a sequence
+of three-frame features {fT −1, fT , fT +1}, resulting in �fT for final prediction.
+Finally, we apply the classifier on �fT to produce the segmentation result ST .
+Since such a procedure can be performed in a sliding-window manner, we can
+obtain the corresponding segmentation sequence.
+In this work, we dedicate to enhance video features to improve the segmen-
+tation performance. To be specific, we first feed the sequence of frame features
+10
+
+𝐟𝑇−1
+𝐟𝑇
+𝐟𝑇+1
+Concat
+Interlaced
+Cross-Self
+Attention
+Add & Norm
+FFN
+Add & Norm
+Encoder
+Interlaced
+Cross-Self
+Attention
+Add & Norm
+FFN
+Add & Norm
+Decoder
+𝐟𝑇
+Interlaced
+Cross-Self
+Attention
+Add & Norm
+𝐟𝑇
+′
+𝐟𝐸𝑛𝑐
+′
+𝐟𝐸𝑛𝑐
+Figure 3: Illustration of transformer based spatial-temporal fusion module. STF
+consists of an encoder for modeling spatial-temporal relationships and feature encoding, and
+a decoder for retrieving the feature of current frame from the encoded feature f
+′
+Enc. Best
+viewed in color.
+into our proposed STF module to capture spatial-temporal dependencies and
+complete pixel-wise feature fusion, resulting in the fused feature f
+′
+T . After that,
+our proposed MAR module further refines f
+′
+T into �fT by exploiting the stored
+feature memory to enhance the discriminativeness of the boundary features. Ev-
+idently, STF and MAR are the key components of our method that determine
+the performance of video semantic segmentation.
+3.2. Spatial-Temporal Fusion
+In this work, we propose a spatial-temporal fusion module to effectively
+integrate the features of consecutive frames. Here it is expected that the spatial-
+temporal relationship among consecutive frames is well modeled and the optical
+flow estimation is avoided. In particular, we use the transformer [10] to perform
+inter-frame fusion, which recently achieves the amazing performance in both
+NLP and CV areas. Thus our STF consists of an encoder and a decoder, as
+shown in Figure. 3.
+11
+
+Encoder. In STF, the encoder is used to capture the spatial-temporal relation-
+ships of pixel-level features. To this end, we concatenate the 2D features of
+multiple frames {fT −1, fT , fT +1} to obtain a 3D feature fEnc ∈ Rd×3×H×W ,
+where d is the dimension of pixel-level features, H and W represent the spatial
+size of frame features. That is, there are 3HW features in total for processing in
+the encoder. We first pass fEnc into our proposed interlaced cross-self attention
+(ICSA) module to model dense spatial-temporal relationships, and the features
+are adjusted by weighting on all features. Then we feed the new features into
+feed-forward network (FFN) to perform feature transformation. Similar to [10],
+we employ the residual connections for the attention module and FFN followed
+by layer normalization. Finally, we obtain the encoded features f
+′
+Enc. Com-
+pared with the previous optical flow based methods [4, 5], our proposed STF
+uniformly aggregates all features at different spatial and temporal positions, and
+no explicit feature alignment is required. Essentially, a single feature in STF
+is implicitly aligned with multiple similar features by attention other than the
+temporally-aligned ones. This is reasonable since the purpose of feature fusion
+is to mutually enhance the features belonging to the same semantic class.
+Decoder. In STF, the decoder is used to get the prediction features of the current
+frame. To this end, we use the original feature of current frame to retrieve from
+the encoded features f
+′
+Enc. To be specific, we first feed the feature of current
+frame into an ICSA module to enhance the features similar in the encoder. Then
+we pass the enhanced features together with f
+′
+Enc into another ICSA module
+for cross attention and produce the features f
+′
+T with FFN. Different from the
+previous one, here the enhanced fT serves as the query and f
+′
+Enc serves as the
+key and value. Intuitively, we retrieve the encoded features from f
+′
+Enc for each
+pixel-level feature in fT , and consequently the f
+′
+T would contain rich information
+from other spatial and temporal positions.
+Interlaced Cross-Self Attention. In the original transformer, the attention op-
+eration would involve O(N 2) complexity given an input of size N (e.g., here
+N = 3HW in our case), which is impractical to the video semantic segmen-
+12
+
+query
+value
+key
+permute
+permute
+permute
+query
+value
+key
+Positional
+Encoding
+Positional
+Encoding
+BWA
+permute
+output
+output
+query
+value
+key
+BWA
+Interlaced Cross-Self Attention (ICSA)
+Interlaced Sparse Attention (ISA)
+Positional
+Encoding
+Positional
+Encoding
+Long-range Cross-Attention
+Short-range Self-Attention
+permute
+BWA
+permute
+BWA
+Long-range Attention
+Short-range Attention
+Figure 4:
+Illustration of differences between our interlaced cross-self attention
+(ICSA) and with interlaced sparse attention (ISA) [11]. ICSA takes query, key and
+value separately for long-rang cross-attention first and then conduct short-range self-attention
+on the previous enhanced feature, which can be seamlessly integrated in the transformer
+structure, especially for cross-attention module in the decoder. Besides, ICSA implements
+necessary positional encoding and can deal with features from multiple frames directly, which
+can uniformly model spatial-temporal relationships. Best viewed in color.
+tation task since computation on pixel-level features would consume too much
+memory. To tackle this issue, a recent work ISA [11] provides a successful so-
+lution. It decomposes the whole attention calculation as the combination of
+long-range and short-range sparse attention calculations, as shown in the upper
+subplot in Figure. 4. In this way, it can retain the ability of modeling global
+relationship while effectively reduce the memory consumption. However, ISA is
+designed for self-attention mechanism like non-local [38], which is not well com-
+patible with the transformer structure. Specifically, ISA takes a single feature
+as input and performs enhancement by modeling inner relationship. Thus it
+can not be directly integrated into cross attention in the transformer decoder.
+Besides, how to insert necessary positional encoding and deal with features of
+multiple frames are not considered by ISA.
+In this work, we extend the original ISA into a more general form and propose
+13
+
+Method
+query
+key
+value
+Divide
+Group
+q
+k
+v
+Block-wise Attention Module
+MHA: Multi-Head Attention
+MHA
+MHA
+MHA
+MHA
+Figure 5: Illustration of block-wise attention (BWA). The input 3D features, i.e.query,
+key and value, are spatially divided into patches with the same shape. Then we apply multi-
+head attention (MHA) [10] operation on corresponding query, key and value patches indepen-
+dently, and combine their results back to the entire one. Best viewed in color.
+interlaced cross-self attention (ICSA), which can be seamlessly integrated into
+transformer structure, as illustrated in the Figure. 4. Generally, we reorganize
+ISA with long-range cross-attention and short-range self-attention operations.
+First, we take query, key and value separately as inputs for cross-attention. Par-
+ticularly, the query, key, and value are the same feature fEnc for the STF module
+encoder, while the key and value are f
+′
+Enc and the query is the enhanced fT for
+the STF module decoder. Here we directly takes 3D features as input to uni-
+formly model spatial-temporal relationships. For query and key, we supplement
+the features with positional encoding. Particularly, we choose the learnable po-
+sitional encoding by following [10]. In this work, we extend positional encoding
+to the 3D version and they have the same shape as the corresponding input.
+Following ISA, we divide features into k blocks with the same shape (e.g.,
+k = 4 in Figure. 4 and Figure. 5). To model long-range cross-attention, we
+harvest features with same spatial positions from different blocks in query, key
+and value via permutation operation, respectively. Then we conduct block-wise
+attention (BWA) operation for relationship modeling. As shown in Figure. 5, we
+first divide input query, key and value features into pre-defined blocks. Then,
+we apply multi-head attention (MHA) [10] on corresponding query, key and
+14
+
+Relationship Computation
+Feature Refinement
+Boundary Feature of class 0
+Prototype of class 0
+Boundary Feature of class 1
+Prototype of class 1
+Figure 6: Illustration of feature refinement. We first compute the relationships between
+the stored boundary features and the test feature to estimate the class likelihoods. Then we
+refine the test feature using the class prototypes, which essentially makes the feature move
+closer to the most likely class. Best viewed in color.
+value patches independently, and combine their results back to the entire one.
+For short-range self-attention, we first permute the feature back to the original
+positions and then regard it as query, key and value for the next attention cal-
+culation. After adding positional encoding, we conduct BWA operation again
+and obtain the final enhanced feature. With ICSA, STF can conveniently har-
+vest global spatial-temporal information for feature enhancement while keeps
+an efficient attention computation. Besides, if we take a single feature from one
+frame as query, key and value, and remove positional encoding, ICSA would
+degenerate into ISA. Evidently, ISA is a special case of ICSA.
+3.3. Memory-Augmented Refinement
+In this work, we propose a novel memory-augmented refinement module to
+further refine the fused features. Different from previous works that explore
+the relationship among the inference features [8, 39, 9], we focus on refining
+the hard features (e.g., boundary features) using the memory from the training
+samples.
+The idea is illustrated in Figure. 6, and it is actually inspired by
+an intuitive mechanism of humans to process semantically ambiguous contents.
+Specifically, given a test feature during inference, it usually lies in the boundary
+area of different classes in the feature space if it is hard to distinguish (e.g., with
+15
+
+Method
+SDA: Scaled Dot-product Attention
+FFN: Feed Forward Network
+𝐟
+Key-Value Memory
+𝐟𝑅
+FFN
+SDA
+𝑆𝑖 = 𝜃(𝑄𝑝) ∅(𝐾𝑖)𝑇, 𝑖 ∈ 1, 𝐶𝐾 , p = (x, y),d is channel num
+𝑆 = 𝑆𝑜𝑓𝑡𝑚𝑎𝑥(𝑆)
+෢
+𝑄𝑝 = ෍
+𝑖=1
+𝐶𝐾
+𝑆𝑖𝑉𝑗 ,
+𝑗 is the class index of 𝐾𝑖, 𝑗 ∈ [1, 𝐶]
+SDA的计算步骤：
+Add &
+Norm
+Add &
+Norm
+Figure 7: Illustration of memory-augmented refinement module. The input feature
+f is refined into fR using the key-value memory extracted from the training samples. Here
+f serves as the query, the key is the stored boundary features, and the value is the class
+prototypes.
+Here ’SDA’ represents scaled dot-product attention and ’FFN’ represents feed
+forward network.
+low confidence score). To enhance its discriminativeness, we first estimates its
+likelihoods to different classes by computing the similarities between the feature
+and stored boundary features of each class. Then we use the class prototypes
+to refine the feature according to the estimated likelihoods, where the class
+prototype refers to the mean feature representing a category. Through this way,
+the test feature would move closer to the most likely category.
+Our proposed MAR module is used to implement such an idea and is illus-
+trated in Figure. 7. Specifically, we build the key-value memory for each class
+that stores two kinds of data from the training samples, namely, the bound-
+ary features and class prototypes.
+The boundary features serve as the keys
+K ∈ Rd×CKL and the class prototypes serve as the values V ∈ Rd×C, where C
+denotes the number of classes and KL is a hyper-parameter to control the size of
+memory. In the MAR module, the input feature F is refined into FR using the
+key-value memory. Inspired by the transformer, we use the scaled dot-product
+attention (SDA) and FFN to construct the MAR block. To be specific, we take
+the test feature as query Q ∈ Rd, and use the key-value in memory to refine it,
+resulting in Q
+′. Formally,
+si = θ(Q)T φ(Ki),
+(1)
+si =
+esi
+�CKL
+i=1 esi ,
+(2)
+Q
+′ =
+CKL
+�
+i=1
+siVj,
+(3)
+16
+
+Training
+Set
+Memory Organization
+Sky
+Features with
+Lowest Scores
+Car 
+Sky
+Features with
+Highest Scores
+Car 
+𝐾𝐻 = 4, 𝐾𝐿 = 4, 𝐶 = 2 for Visualization
+𝐾𝐻𝐶 Features
+Prototype Generation
+𝐾𝐿𝐶 Features
+Score-Based Selection
+Backbone
+Sky
+Key
+Car 
+Sky
+Value
+Car 
+𝐶 Features
+𝐾𝐿𝐶 Features
+Key-Value Groups
+Figure 8: Illustration of the key-value memory.
+From the extracted features on the
+training set, we select KH ”good” features with the highest scores and KL ”hard” features
+with the lowest scores per class.
+Then we generate the class prototypes by averaging the
+”good” features, and organize them with ”hard” features to form the key-value memory. Here
+KH = 4, KL = 4, and C = 2 are used for visualization.
+where i ∈ [1, CKL] denotes the sample in memory and j is the class index cor-
+responding to the i-th sample. Here θ(Q) = WθQ and φ(Ki) = WφKi, and Wθ
+and Wφ are two learnable matrices. Notably, in Eq. (3), we index V by j rather
+than i, which is different from the original self-attention calculation. We employ
+the residual connections for SDA and FFN followed by layer normalization, like
+in the original transformer [10].
+Next we explain how to generate the key-value memory from the training
+samples, which is shown in Figure. 8. We first train the segmentation network
+without the MAR module. Using this model, we extract the features for all
+training samples. Note that a feature would be discarded if it is misclassified by
+the classifier. According to the ground truth, for each class, we select KL ”hard”
+features with the lowest confidence scores and KH ”good” features with the
+highest confidence scores. The former are considered to suffer from semantically
+ambiguity while the latter are to accurately represent the semantic category.
+After that, we compute the mean feature of the ”good” features for each class,
+resulting in the class prototype. Finally, we store the ”hard” features as keys
+and the corresponding class prototype as values in the memory, which essentially
+represent the task-specific experience.
+17
+
+3.4. Training Strategy
+Our proposed network consists of four main components, i.e., backbone, clas-
+sifier, STF, and MAR. Here, we adopt a multi-stage training schedule, which is a
+common strategy in advanced works, e.g., Faster RCNN and knowledge distilla-
+tion. First, the backbone and classifier together are pretrained on ImageNet and
+finetuned on a particular segmentation dataset (e.g., Cityscapes and CamVid).
+The backbone would keep fixed and the classifier would be re-initialized in the
+following training procedures. Then, we train STF together with the backbone
+and classifier, and use this model to generate the key-value memory. Finally,
+we train the MAR and classifier by fixing STF. In the test phase, we perform
+the end-to-end inference with STF and MAR.
+4. Experiment
+In this section, we experimentally evaluate our proposed method on two
+challenging datasets by following the previous works, namely, Cityscapes [2] and
+CamVid [3]. Here some state-of-the-art methods are adopted for comparison
+and the results from the literatures are listed. We follow the standard protocols
+of video semantic segmentation and report the mean Intersection over Union
+(mIoU) as the performance metric.
+4.1. Dataset
+Cityscapes [2] is a popular dataset in semantic segmentation and autonomous
+driving domain. It focuses on semantic understanding of urban street scenes.
+The training and validation subsets contain 2, 975 and 500 video clips, respec-
+tively, and each video clip contains 30 frames. The 20th frame in each clip is
+annotated by the pixel-level semantic labels with 19 categories.
+CamVid [3] also focuses on the semantic understanding of urban street
+scenes, but it contains less data than Cityscapes. It only has 701 color images
+with annotations of 11 semantic classes. CamVid is divided into the trainval
+set with 468 samples and test set with 233 samples. All samples are extracted
+18
+
+Method
+Backbone
+mIoU (%)
+PSPNet [13]
+ResNet18
+69.79
++ Liu et al. [25]
+ResNet18
+73.06 (+3.27)
++ Ours
+ResNet18
+74.58 (+4.79)
+PSPNet [13]
+ResNet50
+76.24
++ Accel [22]
+ResNet50
+70.20 (-6.04)
++ TDNet [24]
+ResNet50
+76.40 (+0.27)
++ EFC [40]
+ResNet50
+78.44 (+2.31)
++ Ours
+ResNet50
+79.22 (+2.98)
+PSPNet [13]
+ResNet101
+79.70
++ TDNet [24]
+ResNet101
+79.90 (+0.20)
++ NetWarp [4]
+ResNet101
+80.60 (+0.90)
++ GRFP [5]
+ResNet101
+80.20 (+0.50)
++ Ours
+ResNet101
+80.96 (+1.26)
+Swin-B [30]
+Swin Transformer
+81.34
++ Ours
+Swin Transformer
+81.67 (+0.33)
+Table 1: Performance comparison on Cityscapes val subset. PSPNet and Swin Transformer
+are chosen as the image segmentation models.
+from the driving videos captured at daytime and dusk, and have pixel-level
+semantic annotations. Each CamVid video contains 3, 600 to 11, 000 frames at
+a resolution of 720 × 960.
+4.2. Performance Comparison
+Here we compare our proposed method with the state-of-the-art methods on
+Cityscapes and CamVid. In particular, the image segmentation model is used
+as the baseline. The PSPNet [13] with the backbone ResNet18/50/101 has been
+widely used on Cityscapes, and Dilation8 [41] is mainly adopted on CamVid.
+Table 1 and Table 2 show the results, and we have the following observations.
+First, our proposed method achieves the state-of-the-art performance on both
+datasets and various baseline model, which demonstrate the effectiveness and
+generalization of our method. Second, our proposed method can get more gains
+19
+
+Method
+Backbone
+mIoU (%)
+Dilation8 [41]
+VGG16
+65.3
++ STFCN [27]
+VGG16
+65.9 (+0.4)
++ GRFP [5]
+VGG16
+66.1 (+0.8)
++ FSO [42]
+VGG16
+66.1 (+0.8)
++ VPN [43]
+VGG16
+66.7 (+1.4)
++ NetWarp [4]
+VGG16
+67.1 (+1.8)
++ EFC [40]
+VGG16
+67.4 (+2.1)
++ Ours
+VGG16
+67.9 (+2.6)
+PSPNet [13]
+ResNet101
+76.2
++ Accel [22]
+ResNet101
+71.5 (-4.7)
++ TDNet [24]
+ResNet101
+76.0 (-0.2)
++ Ours
+ResNet101
+76.6 (+0.4)
+Swin-B [30]
+Swin Transformer
+77.6
++ Ours
+Swin Transformer
+77.9 (+0.3)
+Table 2:
+Performance comparison on CamVid test subset.
+Dilation8, PSPNet and Swin
+Transformer are chosen as the image segmentation model.
+on light-weight baseline model. This is reasonable since improving more com-
+plicated model is generally more difficult. Third, TDNet [24], Accel [22] and
+Liu et al. [25] have nearly no improvement and even degradation on accuracy
+comparing to the baseline, since they mainly focus on improving inference speed.
+Fourth, even on the strong baseline, e.g., Swin Transformer [30], our method
+can also bring improvement.
+4.3. Ablation Study
+Effectiveness of our method. In this work, we propose two key modules, namely
+STF and MAR. To investigate their effects, we also give the results of applying
+one of them, as shown in Table 3.
+We can have the following observations.
+First, our proposed STF and MAR can bring significant performance improve-
+ment separately compared with the baseline, and the version equipped with
+both of them performs best. Second, STF can brings more gains than MAR,
+20
+
+Method
+Dataset
+Backbone
+mIoU (%)
+PSPNet [13]
+Cityscapes
+ResNet50
+76.24
++ STF
+Cityscapes
+ResNet50
+78.75 (+2.62)
++ MAR
+Cityscapes
+ResNet50
+78.37 (+2.24)
++ Both
+Cityscapes
+ResNet50
+79.22 (+2.98)
+Dilation8 [41]
+CamVid
+VGG16
+65.3
++ STF
+CamVid
+VGG16
+67.5 (+2.2)
++ MAR
+CamVid
+VGG16
+67.3 (+2.0)
++ Both
+CamVid
+VGG16
+67.9 (+2.6)
+Table 3: Ablation study on key modules of our proposed method. Performance comparison
+on Cityscapes val subset and CamVid test subset. PSPNet and Dilation8 are chosen as the
+image segmentation model, respectively.
+Method
+mIoU (%)
+DeepLabv3 [44]
+79.5
++ DenseCRF [8]
+79.7 (+0.2)
++ GUM [39]
+79.8 (+0.3)
++ SegFix [9]
+80.5 (+1.0)
++ Our MAR
+81.0 (+1.5)
+Table 4:
+Comparison of different feature refinement methods on Cityscapes val subset.
+DeepLabv3 is chosen as the baseline model.
+since STF integrates the multi-frame information while MAR only optimizes the
+features within the current frame. Third, our proposed STF outperforms other
+multi-frame fusion methods (e.g., TDNet [24] in Table 1 and GRFP [5] and
+NetWarp [4] in Table 2), which indicates the effectiveness of modeling spatial-
+temporal relationship.
+Analysis of MAR. In this paper, we propose MAR to refine the video features.
+Actually, this technique can also be used in other tasks. Here we particularly
+investigate its effect on image segmentation, and show its superiority by com-
+paring with other representative refinement methods, including DenseCRF [8],
+21
+
+Current Frame
+Baseline
+w/ STF
+Ours
+Ground Truth
+Figure 9: Visualization of some sample segmentation results from Cityscapes. It
+can be seen that STF can significantly improve the baseline results, and MAR can further
+bring gains. Here the red rectangles highlight the important regions. Best viewed in color.
+GUM [39], and SegFix [9]. DenseCRF [8] establishes pairwise potentials on all
+pairs of pixels and poses segmentation refinement problem as maximum a pos-
+teriori (MAP) inference, which is a classic post-processing method. GUM [39]
+proposes to enrich bilinear upsampling operators by introducing a learnable
+transformation for semantic maps, which can steer the sampling towards the
+correct semantic class.
+SegFix [9] proposes to replace the unreliable predic-
+tions of boundary pixels with the predictions of interior pixels, which currently
+achieves the state-of-the-art performance. Table 4 provides the comparison re-
+sults, where the DeepLabv3 is adopted as the baseline by following previous
+works. We can see that our MAR outperforms the previous methods, which
+indicates the effectiveness of refining feature by memory.
+To intuitively show the effect of MAR on feature refinement, we particularly
+choose two easily ambiguous categories, i.e., wall and building, to visualize the
+features before and after applying MAR. To be specific, we randomly sample
+100 hard features (with confidence scores lower than 0.8) per category, and then
+visualize their distribution using t-SNE. The results are shown in Figure. 10.
+Before using MAR, two kinds of features are confused together. MAR can move
+features closer to their corresponding class prototypes and make them easier to
+be separated.
+22
+
+w/o MAR
+w/ MAR
+Wall
+Building
+Wall
+Building
+Figure 10: Visualization on the change of feature distribution. It can be seen that
+features become more separable after using MAR module. Best viewed in color.
+Method
+road
+side.
+build.
+wall
+fence
+pole
+light
+sign
+vege.
+terr.
+PSPNet-50
+97.82
+83.23
+91.70
+35.86
+58.07
+63.87
+70.76
+78.93
+91.95
+62.85
++ MAR
+98.13
+85.02
+92.39
+51.34
+60.81
+63.46
+71.62
+80.49
+92.55
+65.47
+Method
+sky
+pers.
+rider
+car
+truck
+bus
+train
+motor
+bike
+mean
+PSPNet-50
+94.16
+81.86
+60.96
+94.82
+76.34
+85.83
+77.67
+64.43
+77.61
+76.24
++ MAR
+94.52
+82.42
+60.88
+95.34
+80.72
+88.72
+78.73
+68.28
+78.21
+78.37
+Table 5: Category-wise performance on Cityscapes val subset. PSPNet-50 is chosen as the
+baseline model.
+Hyper-parameter KL and KH. In our MAR, KL and KH are used to control the
+number of boundary features for memory and good features for prototype per
+class. Here we explore their influence on the segmentation accuracy. Considering
+the memory size, we particularly evaluate the KL from {10, 50, 100, 300} and
+the KH from {1, 5, 10, 50} on Cityscapes val subset with PSPNet-ResNet50 as
+the base model. We found that there is almost no performance fluctuation for
+different settings. Finally, KL = 10 and KH = 10 are adopted throughout the
+experiments.
+Analysis of segmentation results. Our proposed MAR mainly handles the am-
+biguous cases, especially for the hard classes. Table 5 lists the segmentation
+accuracy of different semantic classes on Cityscapes, where PSPNet with the
+23
+
+wall
+buildingwall
+buildingModule
+MACs (G)
+PSPNet-50 (3 Frames)
+4285.9
+PSPNet-101 (3 Frames)
+6152.9
+STF
+563.5
+MAR
+31.5
+Classifier
+0.5
+Table 6: Computational cost of different modules (GMACs). The resolution of input image
+is 1024 × 2048.
+ResNet50 backbone is particularly adopted as the baseline. We can see that
+our method can consistently boost the accuracy over all classes and the gain
+is especially significant for the hard classes, e.g., wall, terrain, and truck. In
+addition, to intuitively show the effectiveness of our proposed STF and MAR,
+we visualize three sample segmentation results from Cityscapes in Figure. 9. It
+can be seen that the original segmentation results can be progressively improved
+by STF and MAR.
+Cost of STF and MAR. Here we analyze the computational cost of different
+components in our proposed method, and the statistics are provided in Table 6.
+It can be seen that our STF and MAR involve little computational cost com-
+pared with the base model. In particular, our proposed MAR is more efficient
+to achieve good segmentation performance than devising more complicated net-
+work structures.
+5. Conclusion
+In this paper, we design a novel video semantic segmentation framework
+with inter-frame feature fusion and inner-frame feature refinement, and pro-
+pose two novel techniques to boost the accuracy. Specifically, we first propose
+a spatial-temporal fusion module based on the transformer, which can effec-
+tively aggregate multi-frame features at different spatial and temporal positions,
+and meanwhile avoid error-prone optical flow estimation. Then we propose a
+24
+
+memory-augmented refinement module that exploits the stored features from
+the training samples to augment the hard features during inference. Our ex-
+perimental results on Cityscapes and CamVid show that the proposed method
+outperforms the state-of-the-art methods.
+Acknowledgements
+This work is supported by the National Natural Science Foundation of China
+under Grant No.62176246 and No.61836008. We acknowledge the support of
+GPU cluster built by MCC Lab of Information Science and Technology Institu-
+tion, USTC.
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+29
+

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+Offline Policy Evaluation with Out-of-Sample Guarantees
+Sofia Ek
+sofia.ek@it.uu.se
+Department of Information Technology
+Uppsala University
+Dave Zachariah
+dave.zachariah@it.uu.se
+Department of Information Technology
+Uppsala University
+Abstract
+We consider the problem of evaluating the performance of a decision policy using past
+observational data. The outcome of a policy is measured in terms of a loss or disutility (or
+negative reward) and the problem is to draw valid inferences about the out-of-sample loss
+of the specified policy when the past data is observed under a, possibly unknown, policy.
+Using a sample-splitting method, we show that it is possible to draw such inferences with
+finite-sample coverage guarantees that evaluate the entire loss distribution. Importantly, the
+method takes into account model misspecifications of the past policy – including unmeasured
+confounding. The evaluation method can be used to certify the performance of a policy using
+observational data under an explicitly specified range of credible model assumptions.
+1
+Introduction
+In this work, we are interested in evaluating the performance of a decision policy, denoted π, which chooses
+an action A from a discrete action set. Each action A is taken in a context with observable covariates X and
+incurs a real-valued loss L (aka. disutility or negative reward). Such policies are considered in contextual
+bandit problems and precision medicine (Langford & Zhang, 2007; Qian & Murphy, 2011; Lattimore &
+Szepesvári, 2020; Tsiatis et al., 2019). For instance, A may be one of several treatment options for a patient
+with observable characteristics X and L measures the severity of the outcome.
+A target policy π can be evaluated using experimental data obtained from trials. Such experiments are,
+however, often costly and may lead to rather restrictive sample sizes. Moreover, in safety-critical applications
+it is often unethical to test new policies without severe restrictions. A more fundamental inferential problem,
+however, is the lack of ‘external’ validity, i.e., the limited ability to extrapolate from the trial population
+to the intended target population leads to invalid inferences (Westreich, 2019; Manski, 2019). The main
+alternative is off-policy evaluation, i.e., using observational data from a past decision process to infer the
+performance of the target policy. This requires that the past process is modelled without systematic errors
+– by assuming well-specified models and no unmeasured confounding. The credibility of these assumptions
+therefore determine the ‘internal’ validity of inferences about π from observational data (Manski, 2003).
+Inferences that lack validity are particularly serious when evaluating π in decision processes that are costly
+or safety-critical. In such cases even inferences that are asymptotically valid with increasing sample size
+may not be adequate. Moreover, when the resulting distribution of losses is skewed or is widely dispersed,
+its tails are important to evaluate. Then inferring the expected loss Eπ[L], as is commonly done, provides
+a very limited evaluation of π. For instance, the average loss in a population maybe low but the tail losses
+are unacceptable (Wang et al., 2018). In such applications, we are more concerned with providing valid
+certifications of the overall performance (see Figure 1a), rather than precise but invalid inferences of a single
+distributional parameter.
+In this paper we propose a method for evaluating a specified target policy using observational data that
+1
+arXiv:2301.08649v1  [stat.ML]  20 Jan 2023
+
+• provides finite-sample coverage guarantees for the out-of-sample loss,
+• evaluates the entire loss distribution instead of the expected value,
+• and takes model misspecification, including unmeasured confounding, into account.
+0.0
+0.5
+1.0
+1.5
+ℓα
+0%
+20%
+40%
+60%
+80%
+100%
+1 − α
+π
+π1
+π0
+(a) Evaluation of out-of-sample loss L.
+0%
+20%
+40%
+60%
+80%
+100%
+Target coverage
+0%
+20%
+40%
+60%
+80%
+100%
+Actual coverage
+π
+π1
+π0
+(b) Evaluation of coverage of limit curves.
+Figure 1: Evaluating out-of-sample losses under target policy with binary decisions A ∈ {0, 1}. Policies π0
+and π1 correspond to ‘treat none’ (A ≡ 0) and ‘treat all’ (A ≡ 1), respectively, while π denotes a policy that
+adapts to context X, see Section 5.1 for more details. (a) Each curve certifies that a new loss L falls below
+the limit ℓα with a probability of least 1 − α. The certified performance of the adaptive policy π dominates
+those of the alternative policies. (b) Evaluation of actual coverage, that is, the probability of L ≤ ℓα, across
+target coverage 1 − α.
+2
+Problem formulation
+We consider a target policy π for deciding actions A in different contexts, which are described by observed
+and unobserved covariates X and U, respectively. The policy can be either deterministic or random, and
+corresponds to a distribution pπ(A|X), which can be conditional on observed covariates X. Our aim is to
+evaluate the losses L that result from applying any given π. Each instance of contextual covariates, action
+and loss, i.e., (X, U, A, L), is drawn independently from a target distribution pπ(X, U, A, L). At our disposal
+is an observational data set
+D =
+�
+(Xi, Ai, Li)
+�n
+i=1,
+(1)
+and our goal is to use it to characterize the out-of-sample loss Ln+1. Specifically, for any miscoverage level
+α ∈ (0, 1), we seek an informative limit ℓα(D) on the loss such that
+Pπ
+�
+Ln+1 ≤ ℓα(D)
+�
+≥ 1 − α,
+(2)
+In other words, ℓα(D) evaluated across α yields a finite-sample performance certification of π as is illustrated
+in Figure 1a. Unlike a single point estimate, the limit curve characterizes the distribution of losses under π.
+The causal structure of the decision process is illustrated in Figure 2a and the target distribution admits a
+causal factorization
+pπ(X, U, A, L) = p(X, U) pπ(A|X) p(L|A, X, U),
+(3)
+where p(X, U) and p(L|A, X, U) are unknown. The central challenge in off-policy evaluation of π is that (1)
+is not sampled from (3) but from a shifted training distribution which admits a causal factorization
+p(X, U, A, L) = p(X, U) p(A|X, U) p(L|A, X, U),
+(4)
+2
+
+where p(A|X, U) characterizes a different, past policy (aka.
+behavioral policy).
+The causal structure is
+illustrated in Figure 2b. If the past policy were known, it is possible to adjust for the distribution shift from
+training to target distribution using the ratio
+pπ(X, U, A, L)
+p(X, U, A, L) ≡ pπ(A|X)
+p(A|X, U) ≥ 0.
+(5)
+This is feasible in certain problems with fully automated decision-making, such as online recommendation
+systems, where the past policy is designed using observable covariates only, i.e., p(A|X, U) ≡ p(A|X). In more
+general problems, however, we have only a nominal model of the past policy �p(A|X) (aka. propensity model),
+typically estimated from prior observable data. The nominal model may therefore diverge from p(A|X, U)
+due to various modelling errors that persist even in the large-sample scenario: model misspecification and
+unmeasured confounding via U (Peters et al., 2017; Westreich, 2019). Here we follow the marginal sensitivity
+methodology of (Tan, 2006) and characterize the model divergence with respect to the odds of taking action
+A. That is, the nominal odds diverge from the unknown odds by some bounded factor Γ ≥ 1 as follows:
+1
+Γ ≤
+p(A|X, U)
+1 − p(A|X, U)
+�
+��
+�
+unknown odds
+�
+�p(A|X)
+1 − �p(A|X)
+�
+��
+�
+nominal odds
+≤ Γ.
+(6)
+When the bound equals Γ = 1, the nominal model is perfectly specified and there is no unmeasured con-
+founding. In general, we consider all cases where the nominal odds diverge by at most a factor Γ.
+In summary, the problem we consider is to construct a limit ℓα(D) for target policy π using training data
+D and a nominal model �p(A|X). The resulting limit should satisfy the finite-sample guarantee (2) across
+all miscoverage levels α, and thereby certify the target policy performance for any specified bound Γ in
+(6). This enables a robust off-policy evaluation of target policies using observational data, since it can be
+performed across a range of credible odds bounds Γ as we will illustrate in the numerical experiments below.
+By increasing the odds bound Γ, the credibility of our assumptions on �p(A|X) increases, but the strength of
+our conclusions about Ln+1 decrease, cf. (Manski, 2003). The trade-off between credibility of assumptions
+and informativeness of inferences will be quantified as well.
+X
+A
+L
+U
+(a) Causal structure that yields target distribution (3).
+X
+A
+L
+U
+(b) Causal structure that yields training distribution (4).
+Figure 2: Directed acyclic diagrams (DAGs) representing the causal structure of decision process under (a)
+target policy and (b) past policy. In (b), both contextual covariates (X, U) may affect actions A as well as
+the outcome loss L and thus U gives rise to unmeasured confounding.
+3
+Background
+We situate the problem considered in this paper and our proposed method within the context of off-policy
+evaluation.
+Expected loss: In most off-policy evaluation literature, the target quantity is the unknown expected loss
+Eπ[L] of policy π. A standard estimator of the mean, that dates back to Horvitz & Thompson (1952), is
+based on inverse propensity weighting:
+VIPW(D) = 1
+n
+n
+�
+i=1
+�w(Xi, Ai) Li,
+(7)
+3
+
+where �w(X, A) = pπ(A|X)
+�p(A|X) is a model of (5), see for instance (Rosenbaum & Rubin, 1983; Beygelzimer et al.,
+2009; Qian & Murphy, 2011; Zhang et al., 2012; Zhao et al., 2012; Kallus, 2018). We note that the estimator
+is unbiased when Γ = 1. An alternative standard estimator is based on regression modeling:
+VRM(D) = 1
+n
+n
+�
+i=1
+�
+a∈A
+pπ(a|Xi) �ℓ(a, Xi),
+(8)
+where �ℓ(A, X) is a model of E[L|A, X].
+The approaches in (7) and (8) have complementary strenghts and weaknesses based on the challenges of
+modelling the past policy or the conditional mean of losses, respectively.
+Even when models are well-
+specified, the accuracy of the estimators depend highly on the overlap of covariates X across decisions A
+in the training data Oberst et al. (2020); D’Amour et al. (2021). When the overlap is weak, the variance
+of VIPW(D) can become excessively large, even when it is unbiased. This can be mitigated by clipping the
+weights (Rubin, 2001; Kang & Schafer, 2007; Schafer & Kang, 2008; Strehl et al., 2010).
+When the models �w(X, A) or �ℓ(A, X) are systematically in error, however, their corresponding estimators
+are biased and may invalidate the evaluation of π. The ‘doubly robust’ estimator
+VDR(D) = 1
+n
+n
+�
+i=1
+�w(Xi, Ai)
+�
+Li − �ℓ(Ai, Xi)
+�
++
+�
+a∈A
+pπ(a|Xi) �ℓ(a, Xi),
+is one way to protect against one of the models being misspecified and reduces the estimator variance provided
+�ℓ(A, X) is sufficiently accurate (Bang & Robins, 2005; Dudík et al., 2011; Rotnitzky et al., 2012).
+Distribution of losses: When loss distribution is highly skewed and/or the tails are wide, the expected loss
+can be inadequate to evaluate policies, especially in high-stakes problems. There are alternative parameters
+of the loss distribution, decribed by the cumulative distribution function F(ℓ) = Pπ{Ln+1 ≤ ℓ} (cdf), that
+one can consider in such problems, such as the Conditional Value at Risk or a quantile (Wang et al., 2018;
+Chandak et al., 2021; Huang et al., 2021).
+Off-policy evaluation of π with respect to some alternative parameter can be achieved using cdf-estimators
+that are analogous to the mean estimators above, see (Huang et al., 2021). In analogy to (7), the inverse
+propensity weighted cdf-estimator
+�FIPW(ℓ; D) = 1
+n
+n
+�
+i=1
+�w(Xi, Ai) 1(Li ≤ ℓ),
+(9)
+is point-wise unbiased when Γ = 1. Similar to (8), the estimator
+�FRM(ℓ; D) = 1
+n
+n
+�
+i=1
+�
+a∈A
+pπ(a|Xi) �c(ℓ; a, Xi),
+requires a model �c(ℓ; a, x) of the conditional distribution P{L ≤ ℓ|A, X}. To mitigate against model mis-
+specification that threaten the validity of the evaluation of π, one can use the ‘doubly robust’ estimator
+�FDR(ℓ; D) = 1
+n
+n
+�
+i=1
+�w(Xi, Ai)
+�
+1(Li ≤ ℓ) − �c(ℓ; Ai, Xi)
+�
++
+�
+a∈A
+pπ(a|Xi) �c(ℓ; a, Xi),
+which protects against one of the models being in misspecified. While this estimator is consistent, it is not
+guaranteed yield a valid cdf.
+In this paper, we are interested in limiting the out-of-sample loss Ln+1 and the quantile is the smallest ℓα
+such that Pπ{Ln+1 ≤ ℓα} ≥ 1 − α. It can be estimated as
+ℓα(D) = inf
+�
+ℓ : �F(ℓ; D) ≥ 1 − α
+�
+,
+4
+
+using a cdf-estimator above.
+Distribution-free inference: Derivations of finite-sample valid, nonparametric limits on random variables
+date back to the works of Wilks (1941); Wald (1943); Scheffe & Tukey (1945). More recently, the related
+methodology of conformal prediction has focused on developing covariate-specific prediction regions (Vovk
+et al., 2005; Shafer & Vovk, 2008; Vovk, 2012). See Lei & Wasserman (2014); Lei et al. (2018); Romano
+et al. (2019) for further developments. Tibshirani et al. (2019) adapt the methodology to be valid also under
+known covariate shifts. This result was subsequently used to provide context-specific prediction intervals for
+any given policy π that are statistically valid under the assumption that the past policy p(A|X, U) is known
+Osama et al. (2020); Taufiq et al. (2022).
+The marginal sensitivity methodology developed in Tan (2006) enables us to specify a more credible range
+of assumptions using (6). This methodology was used for robust policy learning in Kallus & Zhou (2021)
+and sensitivity analysis of treatment effects in Jin et al. (2021) under unobserved confounding. This paper
+considers the overall performance of π, similar to Huang et al. (2021).
+However, we focus on ensuring
+inferences on the out-of-sample losses that are valid even with finite training data and under systematic
+modelling errors – including unobserved confounding – using a sample-splitting technique that leverages
+results derived in Jin et al. (2021).
+4
+Method
+We show that one can limit the out-of-sample losses under π using a sample-splitting technique and by
+bounding the unknown ratio in (5).
+For any specified odds bound Γ ≥ 1 in (6), we have that the ratio in (5) is bounded as:
+W ≤ pπ(X, U, A, L)
+p(X, U, A, L) ≤ W,
+(10)
+where the bounds equal
+W = pπ(A|X) ·
+�
+1 + Γ−1�
+�p(A|X
+�−1 − 1)
+�
+and
+W = pπ(A|X) ·
+�
+1 + Γ
+�
+�p(A|X)−1 − 1
+��
+.
+(11)
+That is, the bounds are functions of X and A drawn from the training distribution (4).
+We randomly split the training data (1) into two separate sets,
+D = D0 ∪ D1,
+with samples sizes n0 and n − n0, respectively. The first set D0 is used to construct a set of upper bounds
+�
+W i
+�n0
+i=1 via (11). The remaining set D1 is used to form the function
+�F(ℓ; D1, w) =
+�n
+i=n0+1 W i1(Li ≤ ℓ)
+�n
+i=n0+1 W i1(Li ≤ ℓ) + �n
+i=n0+1 W i1(Li > ℓ) + w,
+(12)
+as a proxy for the unknown cdf of the out-of-sample loss Ln+1. As the following result shows, (12) enables
+the construction of a valid limit ℓα.
+Theorem 4.1. Define the quantile function of (12) as
+�F −1(·; D1, w) = inf
+�
+ℓ : �F(ℓ; D1, w) ≥ ·
+�
+.
+For any miscoverage probability α ∈ (0, 1), construct the limit
+ℓα(D) =
+min
+β:0<β<α
+�F −1
+�1 − α
+1 − β ; D1, wβ(D0)
+�
+,
+(13)
+5
+
+where
+wβ(D0) =
+�
+W [⌈(n0+1)(1−β)⌉],
+⌈(n0 + 1)(1 − β)⌉ ≤ n0,
+∞,
+otherwise,
+(14)
+and W [k] denotes the kth order statistic of (W i)n0
+i=1.
+Then ℓα(D) limits the out-of-sample loss Ln+1 with a probability of at least 1 − α as specified in (2).
+4.1
+Implementation
+We note that (12) is piecewise constant and can readily be represented using a vector with n − n0 elements.
+The limit curve can be evaluated across a discrete grid of miscoverage levels α and the computation is
+summarized in Algorithm 1. Also, note that wβ as a function of β changes in discrete steps in (14), therefore
+the relevant values of β form a discrete set.
+Algorithm 1 Limit curve of policy π
+Input: Training data D, model �p(A|X), bound Γ ≥ 1 and sample split size n0.
+1: Randomly split D into D0 and D1.
+2: for α ∈ {0, . . . , 1} do
+3:
+for β ∈ {0, . . . , α} do
+4:
+Compute wβ using (14).
+5:
+Compute ℓα,β = inf
+�
+ℓ : �F(ℓ; D1, wβ) ≥ 1−α
+1−β
+�
+using (12).
+6:
+end for
+7:
+Set ℓα to the smallest ℓα,β above.
+8:
+Store (α, ℓα).
+9: end for
+Output: {(α, ℓα)}
+4.2
+Derivation of result
+Proof. The first part of the proof builds on techniques used to derive weighted conformal prediction intervals
+in Tibshirani et al. (2019).
+Let us consider a sequence of n − n0 samples drawn from (4) followed by a new sample drawn from (3), i.e.,
+D+ =
+�
+(Xn0+1, Un0+1, An0+1, Ln0+1), . . . , (Xn, Un, An, Ln), (Xn+1, Un+1, An+1, Ln+1)
+�
+.
+The joint distribution of this sequence can be expressed using:
+n
+�
+i=n+
+p(xi, ui, ai, ℓi) · p(xn+1, un+1, an+1, ℓn+1)wn+1 = p(D+)wn+1 = p(S+)wn+1,
+where n+ = n0 + 1 for notational simplicity, S+ is an unordered set of elements from D+, and the weight
+wi = pπ(xi, ui, ai, ℓi)
+p(xi, ui, ai, ℓi) ,
+is the (unobservable) ratio (5) that quantifies the distribution shift from training to target distribution. We
+shall use the expression for the joint distribution to derive the distribution function for the new loss Ln+1.
+Suppose we are given unordered set S+ alone, then the particular sequence D+ is unknown. Let Ei denote
+the event that the sample (Xn+1, Un+1, An+1, Ln+1) equals the ith sample (xi, ui, ai, ℓi) in the unknown
+sequence D+. We consider all possible sequences D+ obtained by permutations σ of elements in the set S+.
+The joint probability the event Ei and S+ is then
+P{Ei, S+} =
+�
+σ:σ(n+1)=n+i
+p(S+)wi = p(S+)win!.
+6
+
+The conditional probability of event Ei can now be expressed as
+pi = P{Ei|S+} =
+P{Ei, S+}
+�n+1
+j=n+ P{Ej, S+}
+=
+wi
+�n+1
+j=n+ wj
+,
+where the first equality follows from the law of total probability. The probability that the loss Ln+1 of the
+new sample equals ℓi, when conditioning on the unordered set S+, is equal to
+P{Ln+1 = ℓi|S+} = P{Ei|S+} = pi.
+Thus conditional on S+, the new loss Ln+1 follows the cdf:
+P{Ln+1 ≤ ℓ|S+} =
+n+1
+�
+i=n+
+pi1(ℓi ≤ ℓ) =
+�n+1
+i=n+ wi1(Li ≤ ℓ)
+�n+1
+i=n+ wi
+.
+(15)
+Before marginalizing out S+ from (15), we consider a limit ℓ that is a function of the observable elements in
+S+. For this part, we will build on the proof technique in (Jin et al., 2021, thm. 2.2).
+Specifically, using (12) we define the following limit:
+ℓα(D1, W n+1) = inf
+�
+ℓ : �F(ℓ; D1, W n+1) ≥ 1 − α
+1 − β
+�
+,
+(16)
+for any 0 < β < α, where W n+1 ≥ Wn+1 is given in (11). Now insert the limit ℓα(D1, W n+1) into (15) and
+use the law of total expectation to marginalize out S+:
+P{Ln+1 ≤ ℓα(D1, W n+1)} = E
+�
+Pπ{Ln+1 ≤ ℓα(D1, W n+1)|S+}
+�
+= E
+��n+1
+i=n+ Wi1(Li ≤ ℓα(D1, W n+1))
+�n+1
+i=n+ Wi
+�
+.
+We now proceed to lower bound this probability. Note that by construction:
+E
+�
+�F(ℓα; D1, W n+1)
+�
+= E
+�
+�
+i∈D1 W i1(Li ≤ ℓα)
+�
+i∈D1 W i1(Li ≤ ℓα) + �
+i∈D1 W i1(Li > ℓα) + W n+1
+�
+≥ (1 − α)
+(1 − β).
+Using this expression, we have that
+P{Ln+1 ≤ ℓα(D1, W n+1)} − (1 − α)
+(1 − β)
+≥ E
+��n+1
+i=n+ Wi1(Li ≤ ℓα)
+�n+1
+i=n+ Wi
+�
+− E
+�
+�n
+i=n+ W i1(Li ≤ ℓα)
+�n
+i=n+ W i1(Li ≤ ℓα) + �n
+i=n+ W i1(Li > ℓα) + W n+1
+�
+= E
+�
+�
+(∗)
+��n+1
+i=n+ Wi
+� ��n
+i=n+ W i1(Li ≤ ℓα) + �n
+i=n+ W i1(Li > ℓα) + W n+1
+�
+�
+� ,
+7
+
+where
+(∗) =
+� n+1
+�
+i=n+
+Wi1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+W i1(Li ≤ ℓα) +
+n
+�
+i=n+
+W i1(Li > ℓα) + W n+1
+�
+−
+�
+n
+�
+i=n+
+W i1(Li ≤ ℓ��)
+�� n+1
+�
+i=n+
+Wi
+�
+≥
+�
+n
+�
+i=n+
+Wi1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+W i1(Li ≤ ℓα) +
+n
+�
+i=n+
+W i1(Li > ℓα) + W n+1
+�
+−
+�
+n
+�
+i=n+
+W i1(Li ≤ ℓα)
+�� n+1
+�
+i=n+
+Wi
+�
+≥
+�
+n
+�
+i=n+
+Wi1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+W i1(Li > ℓα) + W n+1
+�
+−
+�
+n
+�
+i=n+
+W i1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+Wi1(Li > ℓα) + Wn+1
+�
+≥
+�
+n
+�
+i=n+
+Wi1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+Wi1(Li > ℓα) + Wn+1
+�
+−
+�
+n
+�
+i=n+
+Wi1(Li ≤ ℓα)
+��
+n
+�
+i=n+
+Wi1(Li > ℓα) + Wn+1
+�
+= 0,
+using the bounds (10) on the fourth line. Therefore we obtain a valid limit:
+P{Ln+1 ≤ ℓα(D1, W n+1)} ≥ (1 − α)
+(1 − β).
+(17)
+However, W n+1 depends on a new sample drawn from the training distribution and thus the limit is unusable.
+In lieu of W n+1, we use wβ(D0) in (14) to define the modified limit
+ℓα(D) = inf
+�
+ℓ : �F(ℓ; D1, wβ(D0)) ≥ 1 − α
+1 − β
+�
+.
+(18)
+Comparing it with (16), we see that
+ℓα(D) ≥ ℓα(D1, W n+1),
+(19)
+whenever W n+1 ≤ wβ(D0). By the construction in (14), the probability of this event is lower bounded by
+P{W n+1 ≤ wβ(D0)} ≥ 1 − β,
+(20)
+see (Vovk et al., 2005; Lei et al., 2018).
+We use this property to lower bound the probability of Ln+1 ≤ ℓα(D). First note that
+P{Ln+1 ≤ ℓα(D)} = P{Ln+1 ≤ ℓα(D) | W n+1 ≤ wβ(D0)} P{W n+1 ≤ wβ(D0)}
++ P{Ln+1 ≤ ℓα(D) | W n+1 > wβ(D0)} P{ W n+1 > wβ(D0)}
+≥ P{Ln+1 ≤ ℓα(D) | W n+1 ≤ wβ(D0)} P{W n+1 ≤ wβ(D0)} + 0.
+The first factor can be lower bounded using (19), so that
+P{Ln+1 ≤ ℓα(D)} ≥ P{Ln+1 ≤ ℓα(D1, W n+1) | W n+1 ≤ wβ(D0)} P{W n+1 ≤ wβ(D0)}
+= P{Ln+1 ≤ ℓα(D1, W n+1)} P{W n+1 ≤ wβ(D0)}
+≥ (1 − α)
+(1 − β) P{W n+1 ≤ wβ(D0)}
+≥ 1 − α.
+(21)
+The second line follows from using sample splitting, which ensures that Ln+1 ≤ ℓα(D1, W n+1) and W n+1 ≤
+wβ(D0) are independent events. The third and fourth lines follow from (17) and (20), respectively. Since
+(21) holds for any 0 < β < α, we choose β in (18) that yields the tightest limit, cf. (13).
+8
+
+5
+Numerical experiments
+In the experiments below, we evaluate policies using the limit curves (α, ℓα). Note that the extremum loss
+ℓmax in a given problem provides a valid but uninformative limit across all α. We therefore quantify the
+informativeness of a valid limit curve as follows:
+Informativeness = 1 − α∗, where α∗ = sup{α : ℓα < ℓmax}.
+That is, the lowest coverage probability at which we can informatively certify the performance of π. We can
+therefore quantify increasing the credibility of our model assumption by Γ affects the informativeness of the
+limit curve. We also consider the coverage probability of the curves:
+Miscoverage gap = Pπ{Ln+1 > ℓα(D)} − α.
+(22)
+When this gap is positive, the limit is conservative and when the gap is negative the limit is invalid,
+respectively, at level α.
+A natural benchmark for the proposed limit (13) in this problem setting is the estimated quantile
+ℓα(D) = inf
+�
+ℓ : �FIPW(ℓ; D) ≥ 1 − α
+�
+,
+(23)
+using the inverse propensity weighted cdf-estimator (9).
+In all examples below, the limit (13) is computed using sample splits of equal size, i.e., n0 = n/2.
+5.1
+Synthetic data
+In the first example, we consider synthetic data in order to evaluate the coverage of the derived limit curves.
+We use a simulation setting similar to Jin et al. (2021). The miscoverage gap (22) is estimated by Monte
+Carlo simulation using n′ = 1000 independent samples over N = 1000 independent runs, i.e., in total 106
+samples.
+We consider a population of individuals with two-dimensional covariates distributed uniformly as
+X =
+�X1
+X2
+�
+∼ U(0, 1)2.
+The actions are binary A ∈ {0, 1} corresponding to ‘not treat’ and ‘treat’, respectively. We want to evaluate
+a deterministic target policy, described by
+pπ(A = 0|X) = 1(X1X2 ≥ τ),
+(24)
+for different τ ∈ [0, 1]. That is, all individuals whose covariate product X1X2 falls below τ are treated. Note
+that τ = 0 corresponds a ‘treat none’ policy (A ≡ 0 for all X) and τ = 1 corresponds to a ‘treat all’ policy
+(A ≡ 1 for all X). What can we say about the resulting losses under this policy using observational data
+with sample sizes n ∈ {250, 500, 1000}.
+Case: Known past policy (Γ = 1) In the first scenario, we assume that the past policy is known exactly
+and there is therefore no unmeasured confounding.
+For the training data, the past policy determined actions as a Bernoulli process, where
+p(A = 0|X) ≡ �p(A = 0|X) = f
+�
+c(X1X2 + 1)
+�
+,
+c ∈
+�1
+2, 2
+�
+,
+(25)
+and f(·) is the sigmoid function. The conditional loss distribution is given by
+L|A = 0, X ∼ N(1 − X1X2, 0.1)
+and
+L|A = 1, X ∼ N(X1X2, 0.1).
+9
+
+We consider three configurations c of past policies (25), which yield inverse propensity weights in three
+ranges:
+1
+p1(A|X) < 3.72 (c = 1/2),
+1
+p2(A|X) < 8.39 (c = 1), and
+1
+p3(A|X) < 55.6 (c = 2). Thus we anticipate
+p3(A|X) to be the most challenging case.
+Here we evaluate three target policies τ = {0, 0.5, 1} in (24) and present their resulting limit curves using
+data from different past policies (25) in Figure 3. The differing dashed lines shows the corresponding past
+policy.
+The limit curves for a given target policy are very similar across training distributions and are
+informative above the 90% level. The main difference is in the inferred tail losses and is notable for when
+τ = 1 under the more challenging past policy p3(A|X).
+We now turn to evaluating miscoverage gap (22). Figure 4 presents gaps for target policy τ = 0.5 in (24).
+The solid lines illustrate the proposed method and the dashed lines show the benchmark (23). We see that
+the proposed method is slightly conservative, but remains valid for all α. By contrast, the benchmark is not
+valid in the tail of the distribution, but is less conservative for higher α in this well-specified case.
+−0.5
+0.0
+0.5
+1.0
+1.5
+0%
+20%
+40%
+60%
+80%
+100%
+1 − α
+p1(A|X)
+−0.5
+0.0
+0.5
+1.0
+1.5
+ℓα
+0%
+20%
+40%
+60%
+80%
+100%
+p2(A|X)
+−0.5
+0.0
+0.5
+1.0
+1.5
+0%
+20%
+40%
+60%
+80%
+100%
+p3(A|X)
+pπ(A|X)
+τ = 0.0
+τ = 0.5
+τ = 1.0
+pπ = p
+Figure 3: Limit curves (ℓα, 1 − α) when the past policy is known (Γ = 1) for three different potential target
+policies (i.e. τ = {0.0, 0.5, 1.0} in (24)). Dashed curve denotes the past policy. n = 1000.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+Miscoverage gap
+n = 250
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Target α
+−0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+n = 500
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+n = 1000
+Past policy
+p1(A|X)
+p2(A|X)
+p3(A|X)
+Type
+Proposed
+Benchmark
+Figure 4: Miscoverage gaps (22) across α, when the past policy is known (Γ = 1). Dashed curve denotes the
+benchmark (23).
+Case: Estimated past policy (Γ > 1) In the second scenario, we assume that we only have access to an
+estimate �p(A|X) and that there is unmeasured confounding in the training distribution. To render visually
+distinct curves from the previous case, we consider here a rather extreme case of confounding following Jin
+et al. (2021).
+In this case we have an unobserved variable drawn as
+U|X ∼ N(0, 0.1(X1 + X2)),
+and the loss L|A, X, U is generated as
+L =
+�
+1 − X1X2 + U,
+A = 0,
+X1X2 + U,
+A = 1.
+10
+
+We define the past policy in a manner that enables us to control the divergence from the nominal model
+�p(A|X) in (25):
+p(A = 0|X, U) = 1(U ≤ t(X))
+�
+1 + Γ−1
+0
+�
+�p(A = 0|X
+�−1 − 1)
+�
++ 1(U > t(X))
+�
+1 + Γ0
+�
+�p(A = 0|X)−1 − 1
+��
+,
+(26)
+where the threshold function t(X) is designed empirically to ensure that the resulting median loss of the
+past policy for A = 1 is maximized. Our design of the past policy can be seen as a worst case among all
+unknown past policies that diverge by a factor Γ0 in (6). We fix Γ0 = 2 here, but treat it as unknown.
+For clarity, we consider a ‘treat all’ target policy (τ = 1). Its limit curves, under different assumed odds
+bounds Γ = {1, 2, 3}, are presented in Figure 5. Note that under unmeasured confounding, the resulting
+curves differ notably across the training distributions unlike in Figure 3. We see that under the first and
+second distributions, the informativeness of all curves stays around the 90% level. However, in the most
+extreme third case, the informativeness drops to barely above the 60% level when we increase the credibility
+of our model assumption to an odds bound of Γ = 3. This example illustrates an inherent trade-off between
+credibility and informativeness.
+Figure 6 validates our guarantees using data drawn from p1(A|X). When Γ ≥ Γ0 = 2, the limit curves are
+valid and as Γ increases to 3, the limits become quite conservative. Note that the conservativeness persists
+even as the sample size n is increased fourfold. For Γ = 1, there is no guarantee of coverage and in this worst
+case scenario the limit curve is invalid. The benchmark does not take unmeasured confounding into account
+and is consequently invalid throughout.
+−0.5
+0.0
+0.5
+1.0
+0%
+20%
+40%
+60%
+80%
+100%
+1 − α
+p1(A|X)
+−0.5
+0.0
+0.5
+1.0
+ℓα
+0%
+20%
+40%
+60%
+80%
+100%
+p2(A|X)
+−0.5
+0.0
+0.5
+1.0
+0%
+20%
+40%
+60%
+80%
+100%
+p3(A|X)
+Gamma Γ
+1
+2
+3
+Figure 5: Limit curves (ℓα, 1 − α) for ‘treat all’ target policy using odds bounds Γ = {1, 2, 3}, when the past
+policy is unknown and subject to unmeasured confounding (Γ0 = 2 in (26)). n = 1000.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.2
+−0.1
+0.0
+0.1
+0.2
+Miscoverage gap
+n = 250
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Target α
+−0.2
+−0.1
+0.0
+0.1
+0.2
+n = 500
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−0.2
+−0.1
+0.0
+0.1
+0.2
+n = 1000
+Gamma Γ
+1
+2
+3
+Type
+Proposed
+Benchmark
+Figure 6: Miscoverage gaps (22) across α, when the past policy is unknown and subject to unmeasured
+confounding. Dashed curve denotes the benchmark (23) which does not take confounding into account.
+11
+
+6
+Conclusion
+We have considered the problem of off-policy evaluation, i.e., drawing valid inferences of a target policy
+using past observational data obtained under a different decision process with a, possibly unknown, policy.
+Using the marginal sensitivity model, we derive a sample-splitting method that provides limit curves with
+finite-sample coverage guarantees and, importantly, takes into account model misspecifications and unmea-
+sured confounding. The validity, informativeness, and conservativeness of the resulting limit curves were
+demonstrated in the numerical experiments.
+Using this method in any specific problem, we can specify range of credible model assumptions and assess the
+corresponding degrees of informativeness of the limits, which are guaranteed to be valid up to any specified
+odds ratio bound.
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+

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+arXiv:2301.01112v1  [quant-ph]  3 Jan 2023
+Time-Optimal Transport of a Harmonic Oscillator: Analytic Solution
+Gerhard C. Hegerfeldt1
+1Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen,
+Friedrich-Hund-Platz 1, D-37077 G¨ottingen, Germany
+Motivated by the experimental transport of a trap with a quantum mechanical system modeled
+as a harmonic oscillator (h.o.) the corresponding classical problem is investigated. Protocols for
+the fastest possible transport of a classical h.o. in a wagon over a distance d are derived where
+both initially and finally the wagon is at rest and the h.o. is in its equilibrium position and also at
+rest. The acceleration of the wagon is assumed to be bounded. For fixed oscillator frequency Ω it is
+shown that there are in general three switches in the acceleration and for special values of Ω only
+one switch. In the latter case the optimal transport time is Tabs, that of a wagon without oscillator.
+The optimal transport time and the switch times are determined. It is shown that in some cases it
+is advantageous to go backwards for a while. In addition a time-dependent Ω(t), bounded by Ω±,
+is allowed. In this case the behavior depends sensitively on Ω± and is spelled out in detail. In
+particular, depending on Ω±, Tabs may be obtained in continuously many ways.
+PACS numbers:
+I.
+INTRODUCTION
+Adiabatic processes may serve to transform an initial
+state of a system to a proscribed final state. Such pro-
+cesses, however, are very slow and, in principle, infinitely
+slow. Protocols for speeding up the time development
+have been introduced in the past, with numerous appli-
+cations in quantum optics [1–22] and to classical systems,
+e.g.
+cranes [23].
+These methods include ‘shortcuts to
+adiabadicity’ (STA) [1–10], ‘counterdiabatic’ approaches
+[11–13] and the ‘fast-forward’ approach [15–18]. In gen-
+eral the above mentioned protocols yield a speed-up, but
+not necessarily the fastest possible time development.
+Other methods are combinations with control theory [24–
+26], cf. e.g. [19, 27, 28]. While a time development as
+fast as possible is often desired, other considerations like
+robustness and further conditions may prolong the re-
+sulting time duration.
+A particular example is the efficient transport of ul-
+tra cold atoms and ions by moving the confining trap.
+An atom or ion in a harmonic trap can be treated to
+good approximation as a quantum harmonic oscillator.
+For harmonic traps efficient protocols have been inves-
+tigated with STA and the invariant-based inverse engi-
+neering method to obtain transitionless evolutions under
+imposed constraints, faster than by an adiabatic process
+[6, 28]. It is therefore natural to ask how fast the trans-
+port of a quantum harmonic oscillator can be made. This
+depends of course on the particular question one is inter-
+ested in, for example a time-optimal transport a a har-
+monic oscillator under additional conditions.
+Insight for the quantum case may be obtained by ask-
+ing the same question for a classical harmonic oscillator.
+Therefore in this paper the time-optimal transport of a
+classical harmonic oscillator will be investigated.
+Consider a classical one-dimensional harmonic oscilla-
+tor (h.o.) without friction in the center of a long wagon,
+such as depicted in Fig. 1 where a small mass m is at-
+tached to a spring on the wagon. When the wagon is
+accelerated the h.o. will start to perform oscillations. In
+this case the frequency Ω of the h.o.
+depends on the
+spring constant and on m.
+The problem to be investigated is the following:
+(i) Initially the wagon is at rest and the h.o. is in its
+equilibrium position, also at rest.
+(ii) Then the wagon undergoes an acceleration a(t),
+where a(t) can vary between ±amax, until it has traveled
+a prescribed distance d.
+(iii) Upon arrival at the end point the system should
+again be in its initial state, i.e.
+the wagon should be
+at rest, and the h.o. should again be in its equilibrium
+position and at rest.
+The questions to be answered here are: Is this achiev-
+able, and if so what is the shortest time possible? Can
+this time be further lowered by allowing the h.o.
+fre-
+quency Ω to be time dependent, i.e. Ω(t)? Both ques-
+tions will be answered in the affirmative.
+a(t)
+everything
+at rest
+everything
+at rest
+a(t)
+FIG. 1: Oscillating mass m attached to a spring in an
+accelerated wagon
+The plan of the paper is as follows. First, in Section II,
+a fixed oscillator frequency will be considered, examples
+will be given and a complete solution of the problem and
+an explicit protocol for fixed Ω will be formulated. In
+Section III detailed proofs are provided. In Section IV the
+case of a time-dependent oscillator frequency is treated
+where Ω(t) satisfies Ω− ≤ Ω(t) ≤ Ω+, with arbitrary Ω±.
+The results and protocols will be seen to depend critically
+on the particular choice of Ω±. Finally, in Section V the
+results are summarized and discussed.
+
+2
+II.
+OPTIMAL PROTOCOL FOR FIXED
+OSCILLATOR FREQUENCY
+We consider a classical one-dimensional harmonic os-
+cillator on a long wagon. The position of the h.o. (i.e.
+mass point) relative to the wagon center will be denoted
+by xh and the position of the wagon center in the external
+rest frame by xw. When the wagon is accelerated with
+acceleration a(t), the mass point additionally experiences
+the corresponding inertial force −ma in the rest frame of
+the wagon so that one has
+¨xh = −Ω2xh − a
+(1)
+¨xw = a .
+It is assumed that a(t) can vary between ±amax.
+Example 1. With no h.o. present, to move a wagon a
+distance d in shortest time, with initial and final veloc-
+ity equal to zero, it is optimal to accelerate with amax
+for half the distance and then decelerate with −amax [25]
+(cf. solid line in Fig.2). The corresponding time Tabs(d),
+T 2
+abs = 4 d/amax, can at most be achieved, but not un-
+dercut, if a h.o. in the wagon is to be initially and finally
+at rest in its equilibrium position.
+Example 2. For special ’resonant values’ of Ω this time
+can indeed be achieved, e.g. for
+Ω = n Ωres(d) ,
+n = 1, 2, · · ·
+Ωres(d) =
+�
+4π2amax/d = 4π/Tabs .
+(2)
+To see this consider n = 1. Initially, the wagon and h.o.
+are at rest. Upon accelerating the wagon by amax the h.o.
+experiences, in the wagon frame, the additional inertial
+force −ma and starts to move to the left. During the
+time Tabs/2 it has just performed a single oscillation, has
+returned to its initial position in the wagon and is at rest
+relative to the wagon. In this instant, the acceleration
+of the wagon is reversed, the h.o. starts moving to the
+right and at a further time duration of Tabs/2 is back
+at rest at the initial position, with the wagon at rest
+and having traveled the distance d. For n > 1 one has
+correspondingly more oscillations.
+For fixed Ω a protocol to obtain the unique optimal
+transport time is constructed as follows.
+(i) For given d determine the unique optimal time tf by
+the equation
+d = 1
+4amax t2
+f [1 −
+8
+(Ω tf)2
+�
+arccos(cos2(Ω tf/4))
+�2] . (3)
+(ii) With wagon and oscillator at rest at t = 0, accelerate
+with amax until time 1
+2tf − t1 where t1, 0 ≤ Ωt1 ≤ π/2, is
+given by
+t1 = 1
+Ω arccos(cos2(Ω tf/4)) .
+(4)
+(iii) Decelerate with −amax until time 1
+2tf.
+(iv) Accelerate with amax until time 1
+2tf + t1.
+FIG. 2: Typical wagon velocities for the acceleration
+alternating between±1. Solid curve: No oscillator
+present and Example 2 with resonant Ω. Dashed and
+dotted curves: General Ω. For the dotted curve the
+wagon velocity becomes partially negative, i.e. the
+wagon moves backwards for some time.
+(v) Finally decelerate with −amax until time tf.
+Typical wagon velocities are depicted in Fig. 2. At the
+end the wagon is obviously at rest. The oscillator may
+perform several oscillations. That finally it is also again
+at rest and in its equilibrium position will be shown at
+the end of this section.
+In the next section it will be
+shown that tf is indeed the unique optimal time. The
+above protocol has a certain symmetry; there may, or
+may not, be other, nonsymmetric, protocols which lead
+to the same unique optimal time.
+Note that t1 = 0 if Ω tf = 4nπ, n = 1, 2, · · · , which
+recovers Example 2 with tf = Tabs. If t1 > 1
+4tf the wagon
+velocity temporarily becomes negative (dotted curve in
+Fig. 2), i.e. then it is advantageous to go backwards for
+a while. From Eqs. (3, 4) this is seen to happen if
+Ω2 < 1
+4 Ωres(d)2,
+(5)
+i.e. for small oscillator frequency. However, it can easily
+be shown that the backward motion will not go back as
+far as the original starting position of the wagon.
+If one plots d as a function of tf in Eq.(3) then tf
+as a function of d is given by reflecting it at the diag-
+onal. In dimensionless scaled variables, the solid curve
+in Fig.
+3 displays Ω tf as a function of d/dΩ where
+dΩ = 4π2amax Ω−2 is the distance for which Ω is reso-
+nant, i.e. Ωres(dΩ) = Ω. The dashed curve is the corre-
+sponding Tabs(d). Note that at d/dΩ = n2, n = 1, 2, · · ·
+the two transport times coincide, which is again Example
+2.
+For fixed d, one can also obtain tf as a function of Ω
+from Eq. (3). In dimensionless scaled variables the result
+is plotted in Fig. 4. It is seen that tf diverges for Ω → 0.
+This can be made more explicit by expanding Eq. (3) in
+terms of Ω tf. A short calculation gives, in dimensionless
+
+3
+1
+2
+3
+4
+5
+10
+15
+20
+25
+FIG. 3: Solid curve: Optimal transport time tf as a
+function of distance d in units of dΩ = 4π2amax Ω−2, for
+fixed Ω. Dashed curve: Tabs(d) (without oscillator). For
+d/dΩ = 1, 22, · · · the times coincide.
+scaled variables,
+tf/Tabs(d) ≈ {6/π2}1/4 (Ω/Ωabs(d))−1/2.
+(6)
+Replacing 6 by 5.3 in Eq.(6) one obtains an excel-
+lent approximation for tf/Tabs(d) in the range 0.05 ≤
+Ω/Ωabs(d) ≤ 0.7.
+0.5
+1.0
+1.5
+2.0
+2.5
+1.02
+1.04
+1.06
+1.08
+1.10
+1.12
+FIG. 4: Fixed d: Optimal transport time tf in units of
+Tabs(d) as a function of Ω in units of Ωabs(d).
+Protocol
+evaluation.
+For
+the
+oscillator
+time-
+development Eq.
+(1) has to be evaluated with a =
+±amax. This is conveniently done in the complex plane.
+With
+z = xh + i Ω−1 ˙xh ± amax/Ω2
+(7)
+one finds ˙z = −iΩ z and thus z(t) = exp[−iΩ(t−t0) z(t0).
+Hence
+xh(t) + i Ω−1 ˙xh(t) = exp[−iΩ(t − t0)]
+(8)
+· (xh(t0) + i Ω−1 ˙xh(t0) ± amax/Ω2) ∓ amax/Ω2.
+In the complex plane the right-hand side corresponds to
+a clock-wise rotation of xh(t0) + i Ω−1 ˙xh(t0) by the an-
+gle Ω(t − t0) around the point −amax/Ω2 and amax/Ω2,
+respectively.
+In the protocol one starts with xh(0)
+=
+0 and
+˙xh(0) = 0 and rotates clock-wise around −amax/Ω2, then
+around amax/Ω2, then again around −amax/Ω2 and fi-
+nally around amax/Ω2.
+Analytically this gives for the
+-1
+1
+FIG. 5: Time-development of xh in complex
+phase-space for Ω = 1, amax = 1, d = 2.82 π2,
+tf = 3.41 π, and t1 = .205 π. Starting at the origin, i.e.
+equilibrium position and at rest, there is first a rotation
+around -1, then around 1, then around -1 and finally
+again around 1, back to the origin.
+first two rotations
+ζ1 ≡ xh(tf/2 − t1) + i Ω−1 ˙xh(tf/2 − t1)
+= exp[−iΩ(tf/2 − t1)] amax/Ω2 − amax/Ω2
+ζ2 ≡ xh(tf/2) + i Ω−1 ˙xh(tf/2)
+= exp[−iΩt1](ζ1 − amax/Ω2) + amax/Ω2.
+(9)
+xh(tf/2) is the real part of ζ2 and one finds
+xh(tf/2) = 2 a max/Ω2 (cos2(Ω tf/4)) − cos(Ω t1))
+= 0
+(10)
+by Eq. (4), i.e. ζ2 lies on the imaginary axis. The cor-
+responding trajectories in the complex plane correspond
+
+LALTL3L24
+to the two curves in the left half-plane in Fig. 5. By
+the symmetry of the protocol the next two steps give the
+two curves in the right half-plane where the last one ends
+again at the origin. This follows of course also analyti-
+cally. Hence after the final step the oscillator is again at
+rest in its equilibrium position. Thus the protocol satis-
+fies the initial and final conditions.
+III.
+PROOF OF OPTIMALITY FOR FIXED Ω
+First the equivalent converse problem will be consid-
+ered: Finding the longest distance d for a given time
+duration tf under the conditions (i) - (iii) in Section I
+and a corresponding protocol.
+Symmetry. Consider some given tf and d. In the fol-
+lowing it is convenient to let time run from − 1
+2tf to 1
+2tf.
+Let xh and xw satisfy Eqs. (1) for some a(t) and the
+boundary conditions at ± 1
+2tf. Then 1
+2(xh(t) − xh(−t))
+and
+1
+2(xw(t) − xw(−t) + d) satisfy Eqs.
+(1) with a(t)
+replaced by 1
+2(a(t) − a(−t)) and the same boundary con-
+ditions. Hence without loss of generality one can assume
+that xh and a are anti-symmetric while ˙xh and ˙xw are
+symmetric under time reversal.
+Scaled variables. We go over to dimensionless scaled
+variables. We choose some fixed length unit d0 and put
+Ω2
+0
+= amax/d0
+ω
+= Ω/Ω0
+τ
+= Ω0t
+u(τ)
+= a(t)/amax
+ξ1(τ)
+= xh(t)/d0
+ξ2(τ)
+= d
+dτ ξ1(τ)
+ξ3(τ) = xw(t)/d0
+ξ4(τ) = d
+dτ ξ3(τ)
+(11)
+so that u(τ) can vary between −1 and 1.
+Then one
+obtains
+¨ξ1 ≡ d2
+dτ 2 ξ1 = −ω2ξ1 − u(τ)
+(12)
+¨ξ3 = u(τ) .
+For fixed Ω and a suitable d0 one can assume Ω0 = Ω
+and then ω = 1.
+Pontryagin Maximum (or Minimum) Principle (PMP)
+[24–26]. This is a far-reaching generalization of the cal-
+culus of variations and regarded as a milestone in control
+theory. A simple example is a car moving in shortest time
+from standstill at A to standstill at B, under the only
+condition that the time-dependent acceleration resp. de-
+celeration (the ’control’) is bounded, but not necessarily
+continuous.
+The PMP serves to determine necessary conditions for
+an optimal control function u∗(t) (or possibly several con-
+trol functions) which minimizes a given cost function J
+of the form J =
+� T
+0 L(u(τ), ...)dτ, where L is a func-
+tion of the control u(τ) and some state functions ξi and
+their derivatives. For the present distance-optimal con-
+trol problem, one can take L = ξ4 since J =
+� T
+0
+˙ξ3dτ is
+the (scaled) distance. To minimize it, the PMP considers
+a control Hamiltonian Hc,
+Hc = −L+p1 ˙ξ1 + p2 ˙ξ2 + p3 ˙ξ3 + p4 ˙ξ4,
+(13)
+where one inserts ˙ξi from Eqs. (11-12) and where the
+adjoint states pi are Lagrange multipliers which can not
+all be identically zero.
+Then, for an extremal control
+u(τ) = u ∗ (t), Hamilton’s equations
+˙pi = −∂Hc/∂ξi,
+˙ξi = ∂Hc/∂pi
+(14)
+hold.
+For almost all −τf/2 ≤ τ ≤ τf/2, the function
+Hc(pi(t), ξi(t), u(t)) attains its maximum at u(t) = u∗(t),
+and Hc = const. For simplicity we omit the asterisk on
+u∗. Inserting for ˙ξi, Hc becomes
+Hc = −ξ4 + p1ξ2 + p2(−ω2ξ1 − u) + p3ξ4 + p4u .
+(15)
+From the term (p4−p2) u it follows that for a maximum
+one has to choose u(τ) = 1 if p4 − p2 > 0 and -1 if
+p4 − p2 < 0. When p4 − p2 = 0, or more precisely, when
+p4 − p2 changes sign, there is a switch from ±1 to ∓1 in
+u. Hamilton’s equations become
+˙p1 = ω2p2,
+˙p2 = −p1
+˙p3 = 0,
+˙p4 = −p3 + 1
+(16)
+The solutions are
+p2(τ) = A cos τ + B sin ωτ,
+p1 = − ˙p2
+p3 = c3,
+p4 = (−c3 + 1) τ + c4
+(17)
+where A, B, c3, and c4 are constants. If p4 − p2 ≡ 0
+in some extended interval, then p4 = p2 ≡ 0, by linear
+independence. Therefore it is not possible to have u ≡ 0
+and ξ4 ≡ const in some extended interval so that there
+are only isolated switches. Hence, by anti-symmetry of
+u, there is a switch at τ = 0, i.e. −p2(0) + p4(0) = 0,
+and thus A = c4. By the boundary conditions on ξi at
+±τf/2 only the terms containing u remain in Hc which
+by antisemitic of u lead to two equations and to
+A (cos(ωτf/2) − 1) = 0 .
+(18)
+Thus either A = 0 or ωτf = 4πn. In the latter case the
+situation is analogous to Example 2, i.e. the h.o. can per-
+form 2n complete oscillations and the optimal distance
+is the same as without oscillator. We can therefore as-
+sume A = c4 = 0.
+For ωτf ̸= 4πn there are at least
+two switches of u and therefore B ̸= 0 since otherwise
+−c3 + 1 = 0, c3 = 1, and ξ4 ≡ const. The explicit values
+of B and c3 are not needed, they can in principle be cal-
+culated at the end; it suffices to discuss the cases B < 0
+and B > 0.
+Note: From the remark after Eq. (15) it follows that
+u(τ) = 1 when the line p4(τ) lies above the sine curve
+p2(τ) and u(τ) = −1 when it lies below.
+
+5
+Case B < 0. (i) Single switch for τ < 0, at −τ1, say.
+Then the line p4(τ), denoted by L1 in Fig. 6, intersects
+with the -sine curve p2(τ) once.
+The analog of Eqs.
+(9) for ξ + iω−1 ˙ξ in the scaled
+variables, now with initial time -τf/2 and final time 0
+yields
+ξ1(0) = cos(ωτf/2) − 2 cos(ωτ1) + 1.
+(19)
+From the anti-symmetry of ξ1 one has ξ1(0) = 0, and
+from this one obtains
+cos ωτ1 = cos2(ωτf/4)
+(20)
+with −π/2ω < −τ1 < 0. Thus line L1 in Fig. 6 is typical
+in this case, while line L2 is not possible.
+-3 π
+-2 π
+-π
+FIG. 6: Case B < 0. With ω = 1. L1 and L2 denote
+possible lines for p4(τ). Their intersections with p2(τ)
+(-sine curve) are possible switching points. In regions
+where p4(τ) is above p2(τ) one has acceleration,
+otherwise deceleration. Only L1 with a single switch is
+optimal.
+(ii) If there are two or more switches for τ < 0, e.g. if
+p4(τ) is given by line L2 in Fig. 6, then the last decelera-
+tion period before τ = 0 is longer than π/2ω. Hence the
+total acceleration time is less than in (i) and the distance
+traveled by the wagon during τf is less than that in (i).
+Hence for B < 0 there is only a single switch for τ < 0.
+Case B > 0. From Fig. 7 this is case B < 0 reflected at
+the τ axis, with u = ±1 interchanged and thus positive
+wagon distances for B < 0 now become negative. But
+there might also be negative distances for B < 0, corre-
+sponding to positive distances for B > 0, and therefore a
+more detailed discussion is required. Here we use ω = 1.
+(i) Single switch for τ < 0: As for B < 0 there is only a
+single solution for fixed τf, and this is the corresponding
+optimal backward motion, with p4(τ) typically given by
+L3 in Fig. 7.
+(ii) Exactly two switches for τ < 0. Typical for this
+would be lines L4 and L5 in Fig.
+7, with switches at
+−τ2 < −τ1 < 0, say.
+a) Case τ2 − τ1 > π/2.
+From Fig. 7 one easily finds ˙ξ3(0) = τf/2 − 2(τ2 − τ1) <
+τf/2 − π while, from case B < 0, ˙ξ3opt ≥ τf/2 − π since
+here the switching point lies to the right of −π/2. Hence
+in case B < 0 the distance is larger.
+b) Case τ2 − τ1 < π/2.
+This will be shown to be incompatible with the bound-
+ary conditions on the h.o.. One has ξ1(0) = 0, by anti-
+symmetry, while ˙ξ1(0) ≡ λ is unknown. Reversing the
+time development from τ = 0 to τ = −τ2 one obtains
+ξ1(−τ1) + i ˙ξ1(−τ1) = exp[−iτ1]{iλ + 1} − 1
+ξ1(−τ2) + i−1 ˙ξ1(−τ2) =
+exp[i(−τ1 + τ2)]{ξ1(−τ1) + i ˙ξ1(−τ1) − 1} + 1
+= exp[i(−τ1 + τ2){exp[iτ1](iλ + 1) − 2} + 1
+(21)
+Since this must lie on the circle around −1 passing
+through 0, upon adding 1 the rhs becomes a number of
+modulus 1:
+1 = | exp[i(−τ1 + τ2)]{exp[iτ1](iλ + 1) − 2} + 2|
+= |iλ + 1 − 2 exp[−iτ1] + 2 exp[−iτ2]|
+(22)
+Hence the modulus of the real part,
+|1 − 2 cos τ1 + 2 cosτ2|,
+(23)
+must be less than, or equal to, 1. However, from Fig.
+7, one has −3π/2 < −τ1 < −π and so cos τ1 < 0.
+For −2π < −τ2 < −3π/2 one has cos τ2 > 0 while for
+−3π/2 < −τ2 < −π one has −2 cosτ1 + 2 cosτ2 > 0.
+Hence the bracket in Eq.
+(23) is larger than 1, a
+contradiction. Thus this case can not occur.
+(iii) Three or more switches for τ < 0: A typical line is
+L5 in Fig. 7. From Fig.7 it is evident that the area under
+the curve (i.e. distance) decreases.
+FIG. 7: Case B > 0. With ω = 1. L3, L4 and L5 denote
+possible lines for p4(τ). Their intersections with p2(τ)
+(sine curve) are possible switching points. Dashed: ˙ξ3
+with 2 intersection points −τ1 and −τ2. Dotdashed:
+˙ξ3opt from case B < 0. For τ2 − τ1 > π/2 one has
+˙ξopt > ˙ξ. L3 is typical for the optimal backwards
+motion.
+
+L5.L3L4L2L16
+As a consequence, case B > 0 is not possible and case
+B < 0 (i) gives the unique optimal distance for given τf
+and fixed ω in scaled variables. This distance is easily
+calculated to be τ 2
+f /4 − 2τ 2
+1, with τ1, 0 ≤ τ1 ≤ π/2, given
+by Eq. (20). In the original variables one has
+d = 1
+4amaxt2
+f − 2amaxt2
+1.
+(24)
+Going back to the original problem one obtains the
+protocol of Section II.
+IV.
+PROTOCOLS FOR TIME-DEPENDENT
+OSCILLATOR FREQUENCY
+In this case one allows in addition to a(t) also Ω(t) to
+be time-dependent and seeks a minimal transport time
+tf for a distance d under the condition that the wagon
+is initially and finally at rest and the oscillator is at rest
+in its equilibrium position. This situation is more com-
+plicated. If there are no bounds on Ω then for Ω → ∞
+one obtains the absolute minimal time as without oscilla-
+tor. Therefore, in addition to |a(t)| ≤ amax one imposes
+bounds
+0 ≤ Ω− ≤ Ω(t) ≤ Ω+ < ∞.
+(25)
+If a ’resonant value’ from Eq.
+(2) lies in this interval
+then, from Example 2, one chooses this value for Ω and
+then obtains the absolute minimal time.
+Distance optimization.
+Again we first consider the
+equivalent problem of finding a protocol that maximizes
+the distance d for given time tf and let time run from
+− 1
+2tf to 1
+2tf. We will seek solutions that satisfy the same
+symmetry properties as in Section III, i.e.
+we assume
+that Ω(t) is symmetric.
+The same scaled variables as in Eq.
+(11) are used.
+Introducing
+u1(τ) ≡ ω2(τ)
+(26)
+as a second control variable, Eq. (12) reads
+¨ξ1 ≡ d2
+dτ 2 ξ1 = −u1(τ)ξ1 − u(τ)
+(27)
+¨ξ3 = u(τ) .
+The condition on Ω(t) becomes ω2
+− ≤ u1(τ) ≤ ω2
++. The
+control Hamiltonian for the PMP now reads
+Hc = −ξ4 + p1ξ2 + p2(−u1ξ1 − u) + p3ξ4 + p4u .
+(28)
+As before it follows that for a maximum one has to choose
+u(τ) = 1 if p4 > p2 and -1 if p4 < p2. When p4 − p2 = 0,
+or more precisely, when p4 − p2 changes sign, there is
+a switch from ±1 to ∓1 in u.
+Similarly, u1 = ω2
++ if
+p2ξ1 < 0, and u1 = ω2
+− if p2ξ1 > 0. A switch occurs
+when p2ξ1 changes sign.
+Depending on whether u1 = ω2
++ or u1 = ω2
+−, Hamil-
+ton’s equations in the respective τ intervals become
+˙p1 = ω2
+± p2,
+˙p2 = −p1
+˙p3 = 0,
+˙p4 = −p3 + 1.
+(29)
+Between switches of u1 the solutions are of the form
+p2(τ) = A± cos ω±τ + B± sin ω±τ = C± sin(ω±τ − ϕ±)
+(30)
+p1 = − ˙p2,
+p3 = c3,
+p4 = (−c3 + 1) τ + c4
+where c3, c4, C± are constants, and A±, B±, ϕ± are con-
+stants which may dependent on the respective interval.
+If p2(τ) ≡ 0 in some interval then it is zero everywhere
+because it cannot be joined continuously to the a nonzero
+p2 from Eq. (30).
+Since ω(τ) is symmetric there must be intervals of
+equal length with ω(τ) = ω+ directly to the left and right
+of τ = 0 (or ω− intervals, but this will not be optimal as
+shown later). Hence one must have ϕ+ = 0 in this inter-
+val since then there are switches in ω(τ) at τ = ±π/ω+
+because p2ξ1 vanishes there. It also vanishes at τ = 0
+but does not change sign because of anti-symmetry of ξ1
+and p2 so that ω has no switch at τ = 0 although u does.
+Thus p2 is of the form
+p2(τ) = B+ sin(ω+τ)
+(31)
+in the interval −π/ω+ ≤ τ ≤ π/ω+.
+To the left of τ = −π/ω+ there is an interval with ω−,
+then again an ω+ interval and so on, and similarly to the
+right of τ = π/ω+. Since p2(τ) is differentiable different
+parts of p2 have to be joined accordingly. This yields an
+anti-symmetric p2 as typically displayed in Fig. 8.
+FIG. 8: Solid: p2(τ) with symmetric ω± sequence.
+Dashed: p4(τ).
+The procedure for the determination of τ1 uses the
+time-development of ξ1 and depends on the interval in
+which 1
+2τf lies. This will be exemplified for 1
+2τf ≤ π/ω++
+π/ω−.When 1
+2τf ≤ π/ω+ the situation is the same as in
+Section III and τ1 is given by Eq. (20), with ω replaced
+by ω+.
+
+7
+When π/ω+ <
+1
+2τf ≤ π/ω+ + π/ω− we calculate
+ξ1(τf/2) and ˙ξ1(τf/2) from ξ1(0) and ˙ξ10).
+By anti-
+symmetry one has ξ1(0) = 0 and we put ˙ξ1(0) = λ, the
+exact value of which will not be needed. Using Eq. (8)
+one obtains
+η1 ≡ ξ1(τ1) +
+i
+ω+
+˙ξ1(τ1)
+= exp[−iω+(τ1 − 0)]( i
+ω+
+λ + 1
+ω2
++
+) − 1
+ω2
++
+η2 ≡ ξ1(π/ω+) +
+i
+ω+
+˙ξ1(π/ω+)
+= exp[−iω+( π
+ω+
+− τ1)]{ℜη1 +
+i
+ω+ω+ℑη1 − 1
+ω2
++
+} + 1
+ω2
++
+˜η3 ≡ ξ1(τf/2) +
+i
+ω−
+˙ξ1(τf/2)
+= exp[−iω−(τf/2 − π
+ω+
+)]{ℜη2 +
+i
+ω−
+ω+ℑη2 − 1
+ω2
+−
+} + 1
+ω2
+−
+(32)
+By the boundary conditions at 1
+2τf one has ˜η3 = 0, and
+thus
+0 = ℜη2 +
+i
+ω−
+ω+ℑη2 − 1
+ω2
+−
++ exp[iω−(τf/2 − π
+ω+
+)] 1
+ω2
+−
+.
+(33)
+Taking the real part of this one obtains after a short
+calculation
+cos[ω+τ1] = ω2
++
+2ω2
+−
+{1 + cos(ω−τf/2 + ω+ − ω−
+ω+
+π)}. (34)
+The l.h.s. cannot exceed 1, while the r.h.s. becomes 1
+for τf = τopt where
+τopt/2 = π
+ω+
++ π
+ω−
+− 2
+ω−
+arccos[ω−
+ω+
+],
+(35)
+which lies between π/ω+ and π/ω++π/ω−. Then τ1 = 0
+and the distance becomes the absolute optimum for this
+particular τf = τopt.
+Example 3. Let ω− = ω+/2. Then Eq. (35) yields
+τopt/2 = 5
+3π/ω+ and the distance d/d0 becomes 1
+4τ 2
+opt.
+If one considered only ω+ and the corresponding τopt,
+one would have ω+τ1 = arccos[3/4] ̸= 0 and the distance
+would be less.
+How to proceed when the r.h.s. of Eq. (34) is larger
+than 1? To answer this question we recall that p2 has
+also the trivial solution p2(τ) ≡ 0. Then there are no
+restrictions on the choice of ω(τ). If one decreases ω+
+on the r.h.s of Eq. (34) to ω− the r.h.s. becomes less
+or equal to 1. Hence there must be an intermediate ω,
+denoted by ˜ω+, such that the r.h.s becomes 1. Hence if
+one uses [ω−, ˜ω+] instead of [ω−, ω+] one gets a solution
+for τ1, namely τ1 = 0, so that the sequence ω− and ˜ω+
+gives the largest distance for the given τf. This means
+going over to a sub-interval [ω−, ˜ω+] of [ω, ω+] optimizes
+the distance in this case. There are many sub-intervals
+with the same property, as seen further below.
+In the case π/ω+ + π/ω− < τf/2 ≤ π/ω+ + π/ω− +
+π/ω+, i.e. if one starts with ω+, switches to ω−, and to
+ω+ before τ = 0, i.e. a sequence +−+|+−+ in Fig. 8, then
+η1 and η2 in Eq. (32) remain unchanged while in η3 one
+replaces τf/2 by π/ω+ + π/ω− and there is an additional
+η4,
+η3 = −ℜη2 + 2/ω2
+− −
+i
+ω−
+ω−ℑη2
+η4 ≡ ξ1(τf/2) +
+i
+ω+
+˙ξ1(τf/2)
+= exp[−iω+(τf/2 − π/ω+ − π/ω−)]
+{ℜη3 +
+i
+ω+
+ω−ℑη3 + 1
+ω2
++
+} − 1
+ω2
++
+.
+(36)
+The condition η4 = 0 now gives
+cos ω+τ1 = ω2
++
+ω2
+−
+− 1 + 1
+2{1 + cos(ω+τf/2 − ω+ − ω−
+ω−
+π)}.
+(37)
+For complete ω± intervals the exponentials in Eqs. (32)
+and (36) equal -1 and using this the results are easily
+generalized. In particular, for the ω± sequence − + − + | +
+− + − one obtains
+cos(ω+τ1) = ω2
++
+ω2
+−
+− 1 + ω2
++
+2ω2
+−
+{1 + cos(ω−τf/2 − 2π ω−
+ω+
+)}.
+(38)
+Time optimization. These results will now be applied
+to the original problem in which a distance, now denoted
+by d0, is fixed and the shortest transport time for given
+Ω± is sought. If this d0 is taken for the definition of the
+scaled variables, d0 becomes ξ3(τf/2) = 1. The absolutely
+shortest possible time, τabs, and corresponding ωres is
+then, by Example 2, given by
+τabs = 2
+ωres = 2π.
+(39)
+From Fig. 2 the distance traveled in time τf is 1
+4τ 2
+f − 2τ 2
+1
+and if τf is to be optimal it must satisfy
+1 = 1
+4τ 2
+f − 2τ 2
+1
+(40)
+where τf = τf(ω−, ω+). For given ω± one obtains τ1 from
+Eqs. (20, 34, 37) and generalizations thereof, depending
+on in which interval the as yet unknown τf/2 lies.
+If
+ωres or an integer multiple n thereof lies in [ω−, ω+] one
+chooses ω(τ) ≡ nωres and obtains the absolute optimal
+τabs. Different case of increasing complexity will now be
+discussed.
+Case: ω− = 0, 0 < ω+ < 2π and the distance 1. If
+the spring constant is 0 then in the lab frame the mass
+point m travels free of force and in the the wagon frame
+under the inertial force. It can happen that it is optimal
+
+8
+to start with ω−.
+Then m initially remains at rest in
+the lab frame until a switch to ω+ occurs. If the time
+development starts with ω+ there can be no switch to
+ω− because the associated time interval π/ω− is infinite.
+Hence in this case the results of Section II and III apply.
+From Fig. 4 it is seen that τf decreases with increasing
+ω+ < 2π. Since τf/2 ≤ π/ω+ one has, for optimality,
+τf = 2π/ω+ and τ1 = 0, by Eqs. (3,4). From Eq. (40)
+one then obtains τ 2
+f = 4 so that in this case one must
+have ω+ = π/
+√
+2 ≡ ˜ω+. Thus if ω+ > ˜ω+ one starts
+with ω− = 0 and then there is a switch to ω+ at some
+later time.
+In this case Eq.
+(34) holds for τ1 and it
+becomes 0 for τf = τopt given by Eq. (35). Taking the
+limit ω− → 0 one finds τopt = (2π + 4)/ω+. This must
+equal τabs = 2 which gives ω+ = π + 2 ≡ ωabs. From this
+value of ω+ on one obtains the absolute time minimum.
+The optimal time as a function of ω+ is displayed in Fig.
+9.
+Protocol. This depends on ω+ and is as in Section II
+when ω+ ≤ ˜ω+. When ˜ω+ < ω+ ≤ ωabs one determines
+τf and τ1 from Eqs. (37) and (40), starts with ω− = 0 for
+the time duration −π/˜ω+ + τf/2 and with u = 1, then
+switches to ω+ and continues for the time −τ1 + π/˜ω+,
+then switches to u = −1 for the time τ1 and continues by
+symmetry, resp. anti-symmetry. When ωabs = 2 + π <
+ω+ ≤ ωres one chooses the protocol for ω+ = ωabs.
+0.4
+0.6
+0.8
+1.0
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+FIG. 9: Shortest transport time tf for fixed distance d0,
+Ω− = 0 and 0 ≤ Ω+/Ωres(d0) ≤ 1. Dotted: tf for fixed
+Ω+ without switch in Ω. Solid: Ω+/Ωres(d0) >
+√
+2/4;
+initially Ω(t) ≡ 0 and then a switch to Ω+. For
+1/2 + 1/π ≤ Ω+/Ωres(d0) ≤ 1 one has Tabs(d). The
+switch in Ω can thus lead to a shorter transport time
+than for Ω+ alone.
+Case: 0 < ω− < ω+ < ωres = 2π.
+As in the pre-
+ceding case, only ω+ is relevant if ω+ ≤ ˜ω+ = π/
+√
+2.
+Then τf(ω−, ω+)/2 ≤ π/ω+ and is independent of ω−.
+This is the upper close meshed region in Fig. 10. For
+ω+ > ˜ω+ there are on the l.h.s. of Fig. 8 two or more
+alternating ω±’s for the time development. If there are
+two, one starts with ω−, and the initial time −τf/2 sat-
+isfies π/ω+ ≤ τf/2 ≤ π/ω+ + π/ω−. In this case Eqs.
+(40) and (34) apply. If the l.h.s. of Eq. (34) is less or
+equal to 1 then one can determine τ1 and τf(ω−, ω+), dis-
+played by the coarse meshed region in Fig. 10. Putting
+cos[ω+τ1] = 1 one obtains with τf = τabs = 2π from
+Eq. (34) the boundary curve at the bottom of the coarse
+meshed surface which borders the region denoted by Tabs.
+In this region there is no solution for τ1. As before, here
+the solution p2(τ) ≡ 0 can be used and then there are
+no restrictions on ω(τ).
+If one starts from the point
+{ω−, ω+} and first decreases ω+ until one hits the bound-
+ary curve and then similarly increases ω− one obtains the
+end points of an arc on the boundary curve. Every point
+{ˆω−, ˆω+} on this arc satisfies {ω− ≤ ˆω− ≤ ˆω+ ≤ ω+} and
+yields τabs. Thus there is again an improvement over the
+single ω+ case.
+If there were a third, preceding, interval, i.e.
+with
+ω+, then τf(ω−, ω+)/2 > π/ω+ + π/ω− and τf would
+thus be larger than that with only two periods. Hence
+a third period does not occur. By a similar calculation,
+interchanging ω+ and ω− leads to a larger transport time.
+Protocol: When ω+ ≤ ˜ω+ = π/
+√
+2 one proceeds with
+ω+ as in Section II. When ω+ > ˜ω+ one determines
+τf(ω−, ω+) and τ1 from Eqs.
+(34) and (40), provided
+a solution for τ1 exists. Then one has an ω± sequence
+of the form − + | + − and thus one starts with u = 1 and
+ω− from time −τf/2 to time −π/ω+ where one switches
+to ω+. Then one continues until time −τ1, where one
+switches to u = −1 and continues to τ = 0 where there
+is a switch back to u = 1. For τ > 0 one continues by
+symmetry, resp. anti-symmetry. When there is no solu-
+tion for τ1, i.e when the point {ω−, ω+} lies in the region
+denoted by Tabs in Fig. 10, then one can choose a proto-
+col for any point on the above arc. This will yield τabs
+and in this case the protocol is not unique.
+Case: ωres = 2π ≤ ω− < ω+ < 2 ωres. Arguing as
+before, one has + − +| + −+ and − + − + | + − + − as possible
+ω± sequences.
+To the first sequence Eq.
+(37) applies
+and to the second Eq. (38). One now solves Eq. (40)
+together with Eq. (37) for τf under the condition thatτf/2
+lies in the last ω+ interval. In Fig. 11 this gives the left
+surface outside of which there is no solution for τ1. In a
+similar way one obtains the right surface for the second
+sequence.
+On the boundary curve at the bottom one
+has τabs and the curve is obtained from cos(ω+τ1) = 1.
+The two ω± sequences are separated by the dashed curve
+under the surface.
+This curve is obtained by putting
+τf/2 = 2π/ω+ + π/ω− in Eqs. (37, 40). Its end point on
+the boundary curve is given by { 1
+2 + 1
+2
+√
+2, 1 + 1
+2
+√
+2} ωres
+and on the diagonal by 1
+4
+√
+34 ωres.
+In the region denoted by Tabs there is no solution for
+τ1.
+Again one can choose any point {ˆω−, ˆω+} on the
+arc constructed as before to obtain τabs. Reversing the
+sequence to − + −| − +− leads to larger transport times.
+Protocol: If for a given {ω−, ω+} one has ω− ≤ ( 1
+2 +
+
+9
+FIG. 10: Shortest transport time tf for fixed distance d0
+and 0 ≤ Ω−/Ωres(d0) ≤ Ω+/Ωres(d0) ≤ 1. For
+Ω+/Ωres(d0) ≤ π/
+√
+2 there is only Ω+ and no switch
+(close meshed region). For {Ω−, Ω+} in the region
+denoted by Tabs at the r.h.s. one has the shortest time
+Tabs. The intersection of the surface with the front
+plane is the curve of Fig. 9 and that with the diagonal
+plane is the left part of the curve of Fig. 4 until 1.
+1
+2
+√
+2) ωres or if a solution for τ1 in Eq. (37) exists, one has
+a sequence +−+|+−+, from Fig. 11. If a solution exists the
+protocol is analogous to the previous case above. If not,
+one picks a point {ˆω−, ˆω+} on the arc on the boundary
+curve, as before, and uses the protocol for this point with
+τf = τabs. Otherwise, one has a sequence
+− + − + | + − + −
+and the procedure is analogous.
+V.
+SUMMARY AND DISCUSSION
+Protocols for the fastest possible transport of a classi-
+cal harmonic oscillator (h.o.) over a distance d have been
+derived where both initially and finally everything is at
+rest, i.e. the position of the h.o. is at rest and the h.o.
+is in its equilibrium position and also at rest. The accel-
+eration a(t) is assumed to satisfy −amax ≤ a(t) ≤ amax.
+First, with fixed h.o.
+frequency Ω, for the shortest
+transport time the optimal acceleration alternates be-
+tween ±amax. It was shown that one starts with amax
+and that there are three switches or, for special values
+Ω = nΩres(d) = 2πn
+�
+amax/d, n = 1, 2, · · ·, only one
+switch. The switch times were determined.
+The dependence of the shortest transport time, de-
+noted by tf, on d, Ω and amax was found, cf. Figs. 3 and 4.
+The optimal time tf is proportional to 1/√amax, diverges
+FIG. 11: Shortest transport time tf for fixed distance d0
+and 1 ≤ Ω−/Ωres(d0) ≤ Ω+/Ωres(d0) ≤ 2. The left side
+of the surface belongs to an Ω± sequence + − +| + −+,
+the right side to − + − + | + − + −, separated by the
+dashed line in the bottom plane. For {Ω−, Ω+} in the
+region denoted by Tabs one obtains the shortest time
+Tabs by going over to a point on the boundary
+corresponding to a sub-interval of [Ω−, Ω+].
+for Ω → 0 and, not surprisingly, for Ω → ∞ converges
+to Tabs(d) = 2
+�
+d/amax, the optimal time for a wagon
+without h.o.. The function tf(d) approaches Tabs(d) for
+large d. Surprisingly, sometimes it is advantageous to go
+backwards for a while, but not as far back as the initial
+position.
+Second, in addition to a(t) a time-dependent Ω(t) sat-
+isfying Ω− ≤ Ω(t) ≤ Ω+ was considered. In this case
+the behavior of tf depends sensitively on Ω±. If n Ωres(d)
+lies in the interval [Ω−, Ω+] for some n then choosing
+n Ωres(d) will give the minimal time Tabs(d).
+If Ω+ ≤
+1
+2
+√
+2Ωres then Ω(t) ≡ Ω+, there is no switch
+in Ω, and Ω− does not enter. Otherwise there are two
+alternatives if Ω+ < Ωres:
+(i) One starts with Ω−, switches to Ω+ and then back to
+Ω−.
+(ii) Or there are ˜Ω±, depending on Ω±, with Ω− ≤ ˜Ω− ≤
+˜Ω+ ≤ Ω+ and one starts with ˜Ω−, switches to ˜Ω+ and
+then back to ˜Ω−. In this case one obtains the minimal
+time Tabs(d).
+In the Ω− − Ω+ plane this happens for
+{Ω−, Ω+} in a region, cf. Fig. 10.
+If n Ωres < Ω− ≤ Ω+ < (n + 1)Ωres the situation is
+similarly involved and depicted for n = 1 in Fig. 11 .
+The Pontryagin Maximum Principle was employed,
+first for constant Ω with a(t) as a control variable, and
+then with a(t) and Ω(t) as control variables. Symmetry
+
+2.0
+2- /Sres(do)
+1.5
+1.0
+1.03
+1.02
+t /Tabs(do)
+1.01
+1.00
+1.0
+1.5
+(op)s/+
+2.01.0
+2- /Sres(do)
+0.5
+0.0
+1.4
+t /Tabs(do)
+1.2
+1.0
+0.0
+a.n
+0.5
+(p)sa/+
+1.010
+properties played an important role which were proved
+for constant Ω and assumed in an analogous form for
+time-dependent Ω.
+One may also want to impose restrictions on the veloc-
+ities ˙xw and ˙xh or on the relative displacement xh of the
+h.o.. Within the PMP this may be formulated by means
+of Lagrangian multipliers. In [28] the relative displace-
+ment was assumed to be bounded and taken as the only
+control. However, in this case there are δ(t)-like forces at
+the time of a switch acting on the h.o., and no oscillations
+occur.
+The above results for constant Ω have immediate appli-
+cations to cranes for small-angle oscillations of the pay-
+load where the the rope length l is constant. For time
+dependent l(t) modifications are needed since l(t) is not
+related to the frequency Ω(t) in the same way as the
+spring constant.
+The harmonic oscillator considered here is an idealized
+system. However, it may serve as a benchmark for more
+realistic models, e.g. if the switches are short but smooth
+rather than instantaneous.
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+Towards Computer-Vision Based Vineyard
+Navigation for Quadruped Robots
+Lee Milburn
+Dynamic Legged Systems Lab
+Istituto Italiano di Tecnologia
+Genova, Italy
+lee.milburn@iit.it
+Juan Gamba
+Dynamic Legged Systems Lab
+Istituto Italiano di Tecnologia
+Genova, Italy
+juan.gamba@iit.it
+Claudio Semini
+Dynamic Legged Systems Lab
+Istituto Italiano di Tecnologia
+Genova, Italy
+claudio.semini@iit.it
+Abstract—There is a dramatic shortage of skilled labor for
+modern vineyards. The Vinum project is developing a mobile
+robotic solution to autonomously navigate through vineyards for
+winter grapevine pruning. This necessitates an autonomous navi-
+gation stack for the robot pruning a vineyard. The Vinum project
+is using the quadruped robot HyQReal. This paper introduces
+an architecture for a quadruped robot to autonomously move
+through a vineyard by identifying and approaching grapevines
+for pruning. The higher level control is a state machine switch-
+ing between searching for destination positions, autonomously
+navigating towards those locations, and stopping for the robot
+to complete a task. The destination points are determined by
+identifying grapevine trunks using instance segmentation from
+a Mask Region-Based Convolutional Neural Network (Mask-
+RCNN). These detections are sent through a filter to avoid
+redundancy and remove noisy detections. The combination of
+these features is the basis for the proposed architecture.
+Index Terms—Agricultural Robotics, Computer-Vision, Vine-
+yard Navigation, Quadruped Control
+I. INTRODUCTION
+Fig. 1: HyQReal in Vineyard.
+There is a major shortage of labor in vineyards across the
+world. Vineyards rely on seasonal labor, which in a lot of cases
+includes international workforces. Seasonal labor shortages
+began with the COVID-19 pandemic and have continued
+since1. Vineyards have looked towards robotic automation of
+seasonal work to account for the labor shortage.
+The Vinum project is built on the HyQReal quadruped
+robot that is being developed to autonomously do the winter
+pruning of grapevines, see Fig. 1 [8]. To accomplish this,
+1https://www.winemag.com/2021/12/07/wine-industry-labor-supply/
+the Vinum robot has to autonomously navigate vineyards,
+arriving at each grapevine that needs winter pruning. This
+extended abstract introduces a navigation architecture based
+on computer vision for quadruped robots. Previous vineyard
+navigation has described moving down each row, using a
+laser sensor, until there are no more grapevines in a row
+[7]. Other navigation stacks have been developed which also
+move down rows but they use laser scanners for perception
+[2]. Our proposed navigation stack initializes itself with a
+search of a vineyard row and will choose whether to start
+from right or left. It uses computer vision to detect the
+grapevines and a filter to average the detections and eliminate
+noise. In other papers, grapevine trunks were identified using
+instance segmentation [5]. We implemented a similar sensor
+navigation control using a RGB-D for grapevine trunk image
+segmentation. So, detections of the grapevine trunks are made
+using a Mask-RCNN trained off a created dataset with 100
+images. The combination of the higher level control with the
+grapevine detections makes the basis for the Vinum navigation
+stack.
+The contribution of this extended abstract is a navigation
+for precise placement of quadruped robots moving through
+vineyard rows. It will allow for precise robot placement within
+the vineyard that is ideal for a robotic workspace. This allows
+the robot to perform selective, plant-by-plant task automation
+within the vineyard. A series of experiments were preformed
+with the Aliengo robot and our approach achieved a mean of
+3.36cm and standard deviation of 2.19cm of distance from the
+desired position, which is sufficient for an automated task.
+II. STATE OF THE ART
+As of today, different robots and vehicles have been de-
+veloped that can move autonomously throughout vineyards.
+These robots either move continuously throughout the row
+and/or are not quadrupeds. The EU Project BACCHUS robot
+is a wheeled vehicle that is under development to harvest
+grapes and take care of vineyards. The BACCHUS robot uses
+semantic segmentation of vineyard trunks for its localization
+[5]. Our proposed navigation architecture takes the same
+segmentation approach but it is used to identify positions for
+the robot to walk to instead. The EU Project CANOPIES is
+aimed at developing a human-robot collaborative paradigm for
+arXiv:2301.00887v1  [cs.RO]  2 Jan 2023
+
+harvesting and pruning in vineyards1. It is a wheeled robot
+that works over the vineyard row. A similar autonomous over
+the vineyard row robot is the ViTiBOT Bakus which is used
+to improve vineyard help by removing herbicides and using
+precision spraying. This solution does not include stopping at
+each grapevine. YANMAR’s autonomous over-the-row robot,
+YV01, does a similar task that autonomously sprays vineyard
+rows, without stopping at a specific grapevine2. A proposed
+wheeled robot for precision agriculture is the Agri.q02 which
+is meant to work in unstructured environments in collaboration
+with a UAV [6]. A navigation stack was created for the
+wheeled Ackerman Vehicles in percision farming, path plan-
+ning from pose to pose [3]. There was autonomous navigation
+outlined in the Echord++ GRAPE experiment which maps a
+vineyard that uses a wheeled robot and moves to locations on
+the map to perform tasks [1]. These autonomous robots are
+all wheeled and most do not have to stop at precise locations
+in the vineyard. The proposed navigation architecture of this
+paper is quadruped navigation based on previous techniques
+used for localization to find precise positions for automated
+tasks to take place such as winter pruning and harvesting
+grapes.
+III. NAVIGATION ARCHITECTURE
+The navigation architecture is a combination of higher-level
+control and object detection. The higher level control will
+make decisions on its movement path through a vineyard row
+based on the grapevine trunks detected. The object detection
+was done by training a Mask-RCNN from Detectron2 [9].
+A. Higher Level Control
+Fig. 2: Navigation Flow.
+The higher level control is a state machine for the robot
+to move throughout a vineyard row, as illustrated in Fig. 2.
+It begins with an initial search to find the starting lines for
+both sides of the row. The user can set initially if they want
+the robot to move to the left or right of the row. The initial
+detections get sent through a filter which will find the rolling
+averages of each detection. From the filtered detection points,
+the control will find the lines on which the vineyard rows
+begin.
+The robot has to approach parallel to the grapevines for it to
+be able to prune properly. To find the correct destination point,
+initially, the robot determines the orientation of the approach
+by calculating the vector of the vineyards in a row. This is
+1www.canopies-project.eu
+2https://www.yanmar.com/eu/campaign/2021/10/vineyard/
+derived from a list of points found in the initial search. It
+updates the vector for possible deviances of grapevines as the
+robot moves along the row. The robot then approaches the
+grapevines in parallel at a desired distance that depends on
+the robot size and the workspace of the arm.
+After the robot has reached the determined location in the
+vineyard, it removes that grapevine from the list of vines to
+approach. Next, the control will choose the closest grapevine
+to the robot as its next target. It will continue this method until
+there are no more grapevines to identify in a row.
+B. Grapevine Identification
+Instance segmentation using a Mask-RCNN is used to detect
+the grapevine trunks in a vineyard. The training of the neural
+network was done in Detectron2 using 100 hand annotated
+images of potted grapevines. The corresponding depth of the
+detections is found by using the aligned depth image and
+from there the grapevine locations are found in relation to
+the quadruped.
+Fig. 3: Result of the image segmentation to detect grapevine
+trunks. (4 examples).
+IV. EXPERIMENTS
+A. Higher Level Control
+1) Goals: The goals of these experiments are to determine
+the precision of moving the robot’s center of mass to desired
+positions. They are aimed to align the geometric center of
+the robot with the grapevine trunk, this way an arm mounted
+on the front of the robot is in the center of the grapevine’s
+main cordon, and thus optimizes the workspace of the arm for
+single-plant operations, such as pruning.
+Fig. 4: Experiment setup.
+2) Setup: The higher level control was tested in a lab using
+Unitree’s Aliengo robot. Aliengo was used for simplification
+
+RowEnd
+No
+Yes
+Segment
+Filter segmented
+If next
+Derive pose to
+Move to next
+Grapevine Trunks
+detections
+Grapevineexists
+approach
+Grapevine
+Update detected grapevine trunkslit
+lit
+ISTITUTO ITALIANO
+ISTITUTO ITALIANO
+DI TECNOLOGIA
+DI TECNOLOGIA
+STIC
+DLS
+DYNAMIC LEGGED SYSTEMS
+DYNAMIC LEGGED SYSTEMSFig. 5: Measurement of Aliengo’s arrival at a position.
+of experiments since it is 21kg and 61cm in length. Aliengo
+is equipped with Intel’s Realsense D435 RGB-D camera. Red
+balls for segmenting were used to test in lab instead of the
+grapevine trunks. The red balls are spaced out at about 80cm
+from each other, the approximate distance that grapevines are
+from each other. The setup of the experiment can be seen in
+Fig. 4. How the precision of the robot approaching a position
+was measured is shown in Fig. 5.
+3) Tests: The robot does an initial search of the area using
+its RGB-D camera to segment the red balls. After it finds the
+row of red balls, it approaches the first position in the row.
+After the robot’s arrival at the initial position, it pauses for
+an automated task and update its detections. It repeats this
+process until the row is finished and then stops.
+Ten trials were conducted with five balls. To measure the
+error between the destination point and the red ball, a laser
+pointer was used to show the point that Aliengo’s center of
+mass reached.
+4) Results: The error of reaching the destination point is
+a mean of 3.36cm and standard deviation of 2.19cm. The
+accompanying video shows complete trials.
+B. Grapevine Identification
+1) Goals: The goal of this is to test how well the Mask-
+RCNN was trained for working in vineyards.
+2) Setup: The training of the neural network was done in
+Detectron2 using the framework set up in the paper [4].
+3) Tests: The results were tested on a previously recorded
+video of a potted vineyard at University Cattolica of Piacenza
+during winter.
+4) Results: Outputs from the model are shown in Fig. 3.
+Currently the model needs to be trained on more data for
+robustness and for functionality in other vineyards as well.
+V. CONCLUSION
+This paper presented a method of computer-vision based
+navigation in vineyards for quadruped robots. This method
+will allow for precise placement to preform selective task
+automation.
+The control architecture works accurately with the exper-
+iments in the lab, and the trunk detections from the image
+segmentation can accurately identify grapevine trunks. The
+quadruped can reach a desired destination position with a mean
+error of 3.36cm error.
+The next steps for this architecture is combining the
+grapevine trunk semantic segmentation with the higher level
+control to test in the field. The dataset created for this project
+has to be expanded to train a more robust Mask-RCNN as
+well.
+VI. ACKNOWLEDGMENTS
+Thanks to the contributions of Miguel Fernandes for helping
+train the dataset and Lorenzo Amatucci for the configuration
+of the robot’s controllers for experiments.
+REFERENCES
+[1]
+P. Astolfi, A. Gabrielli, L. Bascetta, and M. Matteucci.
+“Vineyard Autonomous Navigation in the Echord++ GRAPE
+Experiment (FP7-601116). http://echord.eu/grape/”. In: IFAC-
+PapersOnLine 51.11 (2018). 16th IFAC Symposium on In-
+formation Control Problems in Manufacturing INCOM 2018,
+pp. 704–709. ISSN: 2405-8963. DOI: https://doi.org/10.1016/
+j.ifacol.2018.08.401. URL: https://www.sciencedirect.com/
+science/article/pii/S2405896318315271.
+[2]
+M. Bergerman, S. M. Maeta, J. Zhang, G. M. Freitas, B. Ham-
+ner, S. Singh, and G. Kantor. “Robot Farmers: Autonomous Or-
+chard Vehicles Help Tree Fruit Production”. In: IEEE Robotics
+& Automation Magazine 22.1 (2015), pp. 54–63. DOI: 10.1109/
+MRA.2014.2369292.
+[3]
+R. F. Carpio, C. Potena, J. Maiolini, G. Ulivi, N. B. Rossell´o,
+E. Garone, and A. Gasparri. “A Navigation Architecture for
+Ackermann Vehicles in Precision Farming”. In: IEEE Robotics
+and Automation Letters 5.2 (2020), pp. 1103–1110. DOI: 10.
+1109/LRA.2020.2967306.
+[4]
+M. Fernandes, A. Scaldaferri, P. Guadagna, G. Fiameni, T. Teng,
+M. Gatti, S. Poni, C. Semini, D. G. Caldwell, and F. Chen.
+“Towards Precise Pruning Points Detection using Semantic-
+Instance-Aware Plant Models for Grapevine Winter Pruning
+Automation”. In: CoRR abs/2109.07247 (2021). arXiv: 2109.
+07247. URL: https://arxiv.org/abs/2109.07247.
+[5]
+A. Papadimitriou, I. Kleitsiotis, I. Kostavelis, I. Mariolis, D.
+Giakoumis, S. Likothanassis, and D. Tzovaras. “Loop Closure
+Detection and SLAM in Vineyards with Deep Semantic Cues”.
+In: 2022 International Conference on Robotics and Automation
+(ICRA). 2022, pp. 2251–2258. DOI: 10.1109/ICRA46639.2022.
+9812419.
+[6]
+G. Quaglia, C. Visconte, L. S. Scimmi, M. Melchiorre, P.
+Cavallone, and S. Pastorelli. “Design of a UGV Powered
+by Solar Energy for Precision Agriculture”. In: Robotics 9.1
+(2020). ISSN: 2218-6581. DOI: 10.3390/robotics9010013. URL:
+https://www.mdpi.com/2218-6581/9/1/13.
+[7]
+G. Riggio, C. Fantuzzi, and C. Secchi. “A Low-Cost Navigation
+Strategy for Yield Estimation in Vineyards”. In: May 2018,
+pp. 2200–2205. DOI: 10.1109/ICRA.2018.8462839.
+[8]
+C. Semini, V. Barasuol, M. Focchi, C. Boelens, M. Emara,
+S. Casella, O. Villarreal, R. Orsolino, G. Fink, S. Fahmi,
+et al. “Brief introduction to the quadruped robot HyQReal”.
+In: Istituto di Robotica e Macchine Intelligenti (I-RIM) (2019).
+[9]
+Y. Wu, A. Kirillov, F. Massa, W.-Y. Lo, and R. Girshick.
+“Detectron2”. In: (2019).
+
+w.burster.it
+7
+61
+81
+14

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+page_content='Towards Computer-Vision Based Vineyard  Navigation for Quadruped Robots  Lee Milburn  Dynamic Legged Systems Lab  Istituto Italiano di Tecnologia  Genova, Italy  lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='milburn@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='it  Juan Gamba  Dynamic Legged Systems Lab  Istituto Italiano di Tecnologia  Genova, Italy  juan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='gamba@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='it  Claudio Semini  Dynamic Legged Systems Lab  Istituto Italiano di Tecnologia  Genova, Italy  claudio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='semini@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='it  Abstract—There is a dramatic shortage of skilled labor for  modern vineyards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The Vinum project is developing a mobile  robotic solution to autonomously navigate through vineyards for  winter grapevine pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This necessitates an autonomous navi-  gation stack for the robot pruning a vineyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The Vinum project  is using the quadruped robot HyQReal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This paper introduces  an architecture for a quadruped robot to autonomously move  through a vineyard by identifying and approaching grapevines  for pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The higher level control is a state machine switch-  ing between searching for destination positions, autonomously  navigating towards those locations, and stopping for the robot  to complete a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The destination points are determined by  identifying grapevine trunks using instance segmentation from  a Mask Region-Based Convolutional Neural Network (Mask-  RCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' These detections are sent through a filter to avoid  redundancy and remove noisy detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The combination of  these features is the basis for the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  Index Terms—Agricultural Robotics, Computer-Vision, Vine-  yard Navigation, Quadruped Control  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' INTRODUCTION  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 1: HyQReal in Vineyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  There is a major shortage of labor in vineyards across the  world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Vineyards rely on seasonal labor, which in a lot of cases  includes international workforces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Seasonal labor shortages  began with the COVID-19 pandemic and have continued  since1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Vineyards have looked towards robotic automation of  seasonal work to account for the labor shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The Vinum project is built on the HyQReal quadruped  robot that is being developed to autonomously do the winter  pruning of grapevines, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 1 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' To accomplish this,  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='winemag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='com/2021/12/07/wine-industry-labor-supply/  the Vinum robot has to autonomously navigate vineyards,  arriving at each grapevine that needs winter pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This  extended abstract introduces a navigation architecture based  on computer vision for quadruped robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Previous vineyard  navigation has described moving down each row, using a  laser sensor, until there are no more grapevines in a row  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Other navigation stacks have been developed which also  move down rows but they use laser scanners for perception  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Our proposed navigation stack initializes itself with a  search of a vineyard row and will choose whether to start  from right or left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It uses computer vision to detect the  grapevines and a filter to average the detections and eliminate  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' In other papers, grapevine trunks were identified using  instance segmentation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' We implemented a similar sensor  navigation control using a RGB-D for grapevine trunk image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' So, detections of the grapevine trunks are made  using a Mask-RCNN trained off a created dataset with 100  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The combination of the higher level control with the  grapevine detections makes the basis for the Vinum navigation  stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The contribution of this extended abstract is a navigation  for precise placement of quadruped robots moving through  vineyard rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It will allow for precise robot placement within  the vineyard that is ideal for a robotic workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This allows  the robot to perform selective, plant-by-plant task automation  within the vineyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' A series of experiments were preformed  with the Aliengo robot and our approach achieved a mean of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='36cm and standard deviation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='19cm of distance from the  desired position, which is sufficient for an automated task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' STATE OF THE ART  As of today, different robots and vehicles have been de-  veloped that can move autonomously throughout vineyards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  These robots either move continuously throughout the row  and/or are not quadrupeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The EU Project BACCHUS robot  is a wheeled vehicle that is under development to harvest  grapes and take care of vineyards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The BACCHUS robot uses  semantic segmentation of vineyard trunks for its localization  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Our proposed navigation architecture takes the same  segmentation approach but it is used to identify positions for  the robot to walk to instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The EU Project CANOPIES is  aimed at developing a human-robot collaborative paradigm for  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='00887v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='RO]  2 Jan 2023  harvesting and pruning in vineyards1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It is a wheeled robot  that works over the vineyard row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' A similar autonomous over  the vineyard row robot is the ViTiBOT Bakus which is used  to improve vineyard help by removing herbicides and using  precision spraying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This solution does not include stopping at  each grapevine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' YANMAR’s autonomous over-the-row robot,  YV01, does a similar task that autonomously sprays vineyard  rows, without stopping at a specific grapevine2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' A proposed  wheeled robot for precision agriculture is the Agri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='q02 which  is meant to work in unstructured environments in collaboration  with a UAV [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' A navigation stack was created for the  wheeled Ackerman Vehicles in percision farming, path plan-  ning from pose to pose [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' There was autonomous navigation  outlined in the Echord++ GRAPE experiment which maps a  vineyard that uses a wheeled robot and moves to locations on  the map to perform tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' These autonomous robots are  all wheeled and most do not have to stop at precise locations  in the vineyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The proposed navigation architecture of this  paper is quadruped navigation based on previous techniques  used for localization to find precise positions for automated  tasks to take place such as winter pruning and harvesting  grapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' NAVIGATION ARCHITECTURE  The navigation architecture is a combination of higher-level  control and object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The higher level control will  make decisions on its movement path through a vineyard row  based on the grapevine trunks detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The object detection  was done by training a Mask-RCNN from Detectron2 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Higher Level Control  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 2: Navigation Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The higher level control is a state machine for the robot  to move throughout a vineyard row, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  It begins with an initial search to find the starting lines for  both sides of the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The user can set initially if they want  the robot to move to the left or right of the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The initial  detections get sent through a filter which will find the rolling  averages of each detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' From the filtered detection points,  the control will find the lines on which the vineyard rows  begin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The robot has to approach parallel to the grapevines for it to  be able to prune properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' To find the correct destination point,  initially, the robot determines the orientation of the approach  by calculating the vector of the vineyards in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This is  1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='canopies-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='eu  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='yanmar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='com/eu/campaign/2021/10/vineyard/  derived from a list of points found in the initial search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It  updates the vector for possible deviances of grapevines as the  robot moves along the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The robot then approaches the  grapevines in parallel at a desired distance that depends on  the robot size and the workspace of the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  After the robot has reached the determined location in the  vineyard, it removes that grapevine from the list of vines to  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Next, the control will choose the closest grapevine  to the robot as its next target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It will continue this method until  there are no more grapevines to identify in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Grapevine Identification  Instance segmentation using a Mask-RCNN is used to detect  the grapevine trunks in a vineyard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The training of the neural  network was done in Detectron2 using 100 hand annotated  images of potted grapevines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The corresponding depth of the  detections is found by using the aligned depth image and  from there the grapevine locations are found in relation to  the quadruped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 3: Result of the image segmentation to detect grapevine  trunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' (4 examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Higher Level Control  1) Goals: The goals of these experiments are to determine  the precision of moving the robot’s center of mass to desired  positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' They are aimed to align the geometric center of  the robot with the grapevine trunk, this way an arm mounted  on the front of the robot is in the center of the grapevine’s  main cordon, and thus optimizes the workspace of the arm for  single-plant operations, such as pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 4: Experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  2) Setup: The higher level control was tested in a lab using  Unitree’s Aliengo robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Aliengo was used for simplification  RowEnd  No  Yes  Segment  Filter segmented  If next  Derive pose to  Move to next  Grapevine Trunks  detections  Grapevineexists  approach  Grapevine  Update detected grapevine trunkslit  lit  ISTITUTO ITALIANO  ISTITUTO ITALIANO  DI TECNOLOGIA  DI TECNOLOGIA  STIC  DLS  DYNAMIC LEGGED SYSTEMS  DYNAMIC LEGGED SYSTEMSFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 5: Measurement of Aliengo’s arrival at a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  of experiments since it is 21kg and 61cm in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Aliengo  is equipped with Intel’s Realsense D435 RGB-D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Red  balls for segmenting were used to test in lab instead of the  grapevine trunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The red balls are spaced out at about 80cm  from each other, the approximate distance that grapevines are  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The setup of the experiment can be seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' How the precision of the robot approaching a position  was measured is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  3) Tests: The robot does an initial search of the area using  its RGB-D camera to segment the red balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' After it finds the  row of red balls, it approaches the first position in the row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  After the robot’s arrival at the initial position, it pauses for  an automated task and update its detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' It repeats this  process until the row is finished and then stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  Ten trials were conducted with five balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' To measure the  error between the destination point and the red ball, a laser  pointer was used to show the point that Aliengo’s center of  mass reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  4) Results: The error of reaching the destination point is  a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='36cm and standard deviation of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='19cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The  accompanying video shows complete trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Grapevine Identification  1) Goals: The goal of this is to test how well the Mask-  RCNN was trained for working in vineyards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  2) Setup: The training of the neural network was done in  Detectron2 using the framework set up in the paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  3) Tests: The results were tested on a previously recorded  video of a potted vineyard at University Cattolica of Piacenza  during winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  4) Results: Outputs from the model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  Currently the model needs to be trained on more data for  robustness and for functionality in other vineyards as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' CONCLUSION  This paper presented a method of computer-vision based  navigation in vineyards for quadruped robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' This method  will allow for precise placement to preform selective task  automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The control architecture works accurately with the exper-  iments in the lab, and the trunk detections from the image  segmentation can accurately identify grapevine trunks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The  quadruped can reach a desired destination position with a mean  error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='36cm error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  The next steps for this architecture is combining the  grapevine trunk semantic segmentation with the higher level  control to test in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' The dataset created for this project  has to be expanded to train a more robust Mask-RCNN as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' ACKNOWLEDGMENTS  Thanks to the contributions of Miguel Fernandes for helping  train the dataset and Lorenzo Amatucci for the configuration  of the robot’s controllers for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  REFERENCES  [1]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Astolfi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Gabrielli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Bascetta, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Matteucci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  “Vineyard Autonomous Navigation in the Echord++ GRAPE  Experiment (FP7-601116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' http://echord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='eu/grape/”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' In: IFAC-  PapersOnLine 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='11 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 16th IFAC Symposium on In-  formation Control Problems in Manufacturing INCOM 2018,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' 704–709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' ISSN: 2405-8963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='1016/  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='ifacol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='com/  science/article/pii/S2405896318315271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content='  [2]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Bergerman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Maeta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+page_content='  In: Istituto di Robotica e Macchine Intelligenti (I-RIM) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+page_content='  “Detectron2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQf-PoQ/content/2301.00887v1.pdf'}
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+Diseases Based on the Assessment 
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+Difficulties 
+ 
+Zoltan Galaz1, Jiri Mekyska1, Jan Mucha1, Vojtech Zvoncak1, Zdenek Smekal1, 
+Marcos Faundez-Zanuy2, Lubos Brabenec3, Ivona Moravkova3,4,5, and Irena 
+Rektorova3,4 
+1 Department of Telecommunications, Faculty of Electrical Engineering and 
+Communication, Brno University of Technology, Brno, Czech Republic 
+xgalaz00@gmail.com 
+2 Escola Superior Politecnica, Tecnocampus, Mataro, Barcelona, Spain 
+3 Applied Neuroscience Research Group, Central European Institute 
+of Technology – CEITEC, Masaryk University, Brno, Czech Republic 
+4 First Department of Neurology, Faculty of Medicine and St. Anne’s University 
+Hospital, Masaryk University, Brno, Czech Republic 
+5 Faculty of Medicine, Masaryk University, Brno, Czech Republic 
+ 
+Abstract. To this date, studies focusing on the prodromal diagnosis of 
+Lewy body diseases (LBDs) based on quantitative analysis of graphomo- 
+tor and handwriting difficulties are missing. In this work, we enrolled 18 
+subjects diagnosed with possible or probable mild cognitive impairment 
+with Lewy bodies (MCI-LB), 7 subjects having more than 50% prob- 
+ability of developing Parkinson’s disease (PD), 21 subjects with both 
+possible/probable MCI-LB and probability of PD > 50%, and 37  age- 
+and gender-matched healthy controls (HC). Each participant performed 
+three tasks: Archimedean spiral drawing (to quantify graphomotor diffi- 
+culties), sentence writing task (to quantify handwriting difficulties), and 
+pentagon copying test (to quantify cognitive decline). Next, we parame- 
+terized the acquired data by various temporal, kinematic, dynamic, spa- 
+tial, and task-specific features. And finally, we trained classification mod- 
+els for each task separately as well as a model for their combination to 
+estimate the predictive power of the features for the identification of 
+LBDs. Using this approach we were able to identify prodromal LBDs 
+with 74% accuracy and showed the promising potential of computerized 
+objective and non-invasive diagnosis of LBDs based on the assessment 
+of graphomotor and handwriting difficulties. 
+ 
+ 
+This work was supported by grant no. NU20-04-00294 (Diagnostics  of  Lewy  body 
+diseases in prodromal stage based on multimodal data analysis) of the Czech Ministry 
+of Health and by Spanish grant of the Ministerio de Ciencia e Innovacio´n no. PID2020- 
+113242RB-I00. 
+
+ 
+ 
+· 
+· 
+· 
+· 
+· 
+ 
+Keywords:   Lewy  body  diseases    Online  handwriting    Graphomotor 
+difficulties     Handwriting  difficulties    Machine  learning     Prodromal 
+diagnosis 
+ 
+1 Introduction 
+ 
+Lewy body diseases (LBDs) is a term describing a group of neurodegenerative 
+disorders characterized by a pathophysiological process of α-synuclein accumu- 
+lation in specific brain regions leading to the formation of Lewy bodies and 
+Lewy neurites resulting in cell death. LBDs consists of two major clinical enti- 
+ties: Parkinson’s disease (PD) and dementia with Lewy bodies (DLB) [29,38]. 
+Although the phenotypes and temporal evolution of motor and cognitive symp- 
+toms of these two diseases vary, they share many clinical and pathophysiolog- 
+ical features and are therefore referred to as LBDs spectrum. Together with 
+Alzheimer’s disease (AD), LBDs comprise the major part of all cases of neu- 
+rodegenerative disorders. 
+It is known that LBDs do not start suddenly. At the time the clinical symp- 
+toms occur, the neurodegenerative process has reached a severe degree in which 
+most of the targeted neurons have already been damaged. Before the clinical 
+diagnosis based on the presence of typical clinical symptoms becomes possible, 
+there is a long period of the underlying neurodegenerative process with subtle or 
+nonspecific symptoms [18,29] such as sleep disturbances, mood changes, smell 
+loss, constipation, etc. This period of LBDs is called the prodromal stage. 
+One of the early markers of PD is PD dysgraphia (micrographia and other 
+alterations in handwriting, e.g. kinematic and dynamic) [21,32,33]. Similarly, 
+some manifestations of dysgraphia have been observed in the prodromal DLB 
+as well [23]. Although modern approaches to the analysis of graphomotor and 
+handwriting difficulties (utilising digitising tablets) were proved to work well 
+during e.g. diagnosis of the clinical stage of PD [9,11,35], assessment of cogni- 
+tion in PD patients [4], or discrimination of AD and mild cognitive impairment 
+(MCI) [15], to the best of our knowledge, no studies employed this technology 
+(with high potential) in the prodromal diagnosis of LBDs in a larger scale. 
+Identification of the early stages of LBDs is crucial for the development 
+of disease-modifying treatment since the neurodegeneration may be possibly 
+stopped or treated before the pathological cascades start. Therefore, the goal 
+of this study is to explore whether the computerised assessment of graphomo- 
+tor and handwriting difficulties could support the prodromal diagnosis of LBDs, 
+more specifically, we aim to: 
+1. identify which task significantly discriminates LBD patients and age- and 
+gender-matched healthy controls (HC), 
+2. identify what conventional online handwriting features have good discrimina- 
+tion power. 
+
+ 
+ 
+± 
+± 
+± 
+ 
+2 Materials and Methods 
+ 
+2.1 
+Dataset 
+We enrolled 39 subjects (19 females, 20 males, age = 69.53 
+6.61) diagnosed 
+with possible or probable MCI (based on the scores of the MoCA – Montreal 
+Cognitive Assessment [25] and based on the CCB – Complex Cognitive Battery, 
+see the explanation below) who were simultaneously diagnosed with possible or 
+probable MCI-LB (i.e. mild cognitive impairment with Lewy bodies) based on 
+the criteria published by McKeith et al. [22]. In this group, 21 subjects also 
+had more than 50% probability of developing PD (calculated following the MDS 
+criteria published in [18]). In addition, we enrolled 7 subjects (2 females, 5 males, 
+age = 66.41    4.32) without possible/probable MCI-LB, but still with more than 
+50% probability of developing PD. Finally, we enrolled 37 HC (26 females, 11 
+males, age = 67.60 
+5.61). In the experiments, we stratified the subjects into 
+two groups, HC vs. LBD (i.e. people with a high risk of developing PD or DLB). 
+CCB was used to evaluate four cognitive domains: 1) memory (The Brief 
+Visuospatial memory test–revised [2], Philadelphia Verbal Learning Test [3]); 
+2) attention (Wechsler Adult Intelligence Scale-III: Letter-Number Sequencing, 
+Digit Symbol Substitution [37]); 3) executive functions (Semantic and phonemic 
+verbal fluency [30], Picture arrangement test [37]); and 4) visuospatial functions 
+(Judgment of Line Orientation [36]). The cognitive domain z-scores were com- 
+puted as the average z-scores of the tests included in the particular domain. 
+The participants were asked to perform a set of three tasks: 
+1. Archimedean spiral (spiral) – we consider this task as a graphomotor one, i.e. 
+it is a building block of some letter shapes; in addition, it is a golden standard 
+in PD dysgraphia diagnosis [35] 
+2. sentence  “Tramvaj   dnes  uˇz  nepojede”  (translation:   “A  tram  will  not  go 
+today.”) writing (sentence) – this handwriting task was used e.g. in the 
+PaHaW database [11] 
+3. pentagon copying test (pentagons) – it is a task frequently used for quantifi- 
+cation of cognitive decline [4] 
+All participants were right-handed and had Czech as their native language. 
+They all signed an informed consent form that was approved by the local ethics 
+committee. 
+ 
+2.2 
+Feature Extraction 
+The participants were asked to perform the tasks (using the Wacom Ink pen) 
+on an A4 paper that was laid down and fixed to a digitizing tablet Wacom 
+Intuos 4 M (sampling frequency fs = 130 Hz). Before the acquisition, they had 
+some time to get familiar with the hardware. The recorded time series (x and 
+y position; timestamp; a binary variable, being 0 for in-air movement and 1 for 
+on-surface movement, respectively; pressure exert on the tablet’s surface during 
+
+ 
+ 
+ 
+writing; pen tilt; azimuth) were consequently parameterised utilising the follow- 
+ing set of features (we selected the set based on available reviews and based on 
+our experience [9,11,35]): 
+1. temporal – duration of writing, ratio of the on-surface/in-air duration, dura- 
+tion of strokes, and ratio of the on-surface/in-air stroke duration 
+2. kinematic – velocity, and acceleration 
+3. dynamic – pressure, tilt, and azimuth 
+4. spatial – width, height, and length of the whole product, as well as its partic- 
+ular strokes, i.e. stroke width, height, and length 
+5. spiral-specific – degree of spiral drawing severity [31], mean drawing speed of 
+spiral [31], second-order smoothness of spiral [31], spiral precision index [5], 
+spiral tightness [31], variability of spiral width [31], and first-order zero- 
+crossing rate of spiral [31] 
+6. other – number of interruptions (pen elevations), number of pen stops [27], 
+tempo (number of strokes normalised by duration), number of on-surface 
+intra-stroke intersections, relative number of on-surface intra-stroke intersec- 
+tions, number of on-surface inter-stroke intersections, and relative number of 
+on-surface inter-stroke intersections, Shannon entropy [4], number of changes 
+in the velocity profile, relative number of changes in the velocity profile 
+Most of the features were extracted using the recently released Python library 
+handwriting-features (v 1.0.1) [14], the rest of them were coded in Matlab. Some 
+features (mainly spatial, temporal and kinematic) were extracted from both on- 
+surface and in-air movements. In addition, kinematic features were also analysed 
+in horizontal and vertical projection. Features represented by vectors were con- 
+sequently transformed to a scalar value using median, non-parametric coefficient 
+of variation (nCV; interquartile range of feature divided by its median), slope 
+and 95th percentile (95p). 
+ 
+2.3 
+Statistical Analysis and Machine Learning 
+To compare the distribution of features between the HC and LBD subjects, we 
+conducted Mann-Whitney U-test with the significance level of 0.05. Moreover, 
+to assess the strength of a relationship between the features and the subject’s 
+clinical status (HC/LBD), we computed Spearman’s correlation coefficient (ρ) 
+with the significance level of 0.05. Finally, during this exploratory step, we calcu- 
+lated Spearman’s correlation with the domains of CCB and the overall score of 
+MDS–Unified Parkinson’s Disease Rating Scale (MDS–UPDRS), part III (motor 
+part) [16]. 
+To identify the presence of graphomotor or handwriting difficulties, we built 
+binary classification models using an ensemble extreme gradient boosting algo- 
+rithm known as XGBoost [6] (with 100 estimators). This algorithm was chosen 
+due to its robustness to outliers, ability to find complex interactions among fea- 
+tures as well as the possibility of ranking their importance. To build models with 
+an optimal set of hyperparameters, we conducted 1000 iteration of randomized 
+
+ 
+ 
+× 
+× 
+× 
+2 
+TP + FN TN + FP  
+ 
+search strategy via stratified 5-fold cross-validation with 10 repetitions aiming 
+to optimize balanced accuracy score (BACC; described in more detail along with 
+other evaluation scores below). The following set of hyperparameters were opti- 
+mized: the learning rate [0.001, 0.01, 0.1, 0.2, 0.3], γ [0, 0.05, 0.10, 0.15, 0.20, 
+0.25, 0.5], the maximum tree depth [6, 8, 10, 12, 15], the fraction of observations 
+to be randomly sampled for each tree (subsample ratio) [0.5, 0.6, 0.7, 0.8, 0.9, 
+1.0], the subsample ratio for the columns at each level [0.4, 0.5, 0.6, 0.7, 0.8, 
+0.9, 1.0], the subsample ratio for the columns when constructing each tree [0.4, 
+0.5, 0.6, 0.7, 0.8, 0.9, 1.0], the minimum sum of the weights of all observations 
+required in a child node [0.5, 1.0, 3.0, 5.0, 7.0, 10.0], and the balance between 
+positive and negative weights [1, 2, 3, 4]. 
+The classification test performance was determined using the following clas- 
+sification metrics: Matthew’s correlation coefficient (MCC), balanced accuracy 
+(BACC), sensitivity (SEN) also known as recall (REC), specificity (SPE), pre- 
+cision (PRE) and F1 score (F1). These metrics are defined as follows: 
+TP × TN + FP × FN  
+ 
+MCC = 
+√N 
+, 
+(1) 
+BACC = 1 
+TP 
+TN 
+, 
+(2) 
+ 
+SPE = 
+TN  
+TN + FP  
+PRE = 
+TP  
+TP + FP  
+REC = 
+TP  
+, 
+(3) 
+ 
+, 
+(4) 
+ , 
+(5) 
+ 
+TP + FN  
+F1 = 2 PRE × REC 
+PRE + REC 
+ 
+(6) 
+where N  = (TP + FP )    (TP + FN )      (TN + FP )     (TN + FN ), TP (true 
+positive) and FP (false positive) represent the number of correctly identified 
+LBD subjects and the number of subjects incorrectly identified as having LBDs, 
+respectively. Similarly, TN (true negative) and FN (false negative) represent 
+the number of correctly identified HC and the number of subjects with LBDs 
+incorrectly identified as being healthy. 
+To further optimize the trained classification models, we fine-tuned the mod- 
+els’ decision thresholds via the receiver operating characteristics (ROC) curve. 
+Using the fine-tuned decision thresholds, we evaluated the classification perfor- 
+mance of the models using the leave-one-out cross-validation. The ROC curves 
+were plotted using the probabilities of the predicted labels obtained via the 
+cross-validation procedure that was employed during the final evaluation of the 
+fine-tuned models. 
+And finally, to evaluate the statistical significance of the prediction perfor- 
+mance obtained by the built classification models, a non-parametric statisti- 
+cal method named permutation test was employed [7,28]. For this purpose, we 
+applied 1 000 permutations with the significance level of 0.05. To estimate the 
+
+ 
+ 
+− 
+ 
+performance of the models on the permuted data, we used the same classification 
+setup as employed during the training phase [26]. 
+ 
+3 Results 
+ 
+The results of the exploratory data analysis are summarized in Table 1 (sorted 
+based on the p-value for the Mann-Whitney U-test). The following features were 
+found as the most distinguishing ones in terms of the differentiation between HC 
+and subjects with LBD (the top 4 features are listed; *, **, and *** denote the p- 
+values for both the Mann-Whitney U-test and Spearman’s correlation coefficient 
+being bellow the significance level of 0.05, 0.01, and 0.001, respectively; if both p- 
+values are bellow a different significance level, the weaker statistical significance 
+is selected): a) spiral – nCV of acceleration (on-surface) ρ =    0.2438∗, variability 
+of spiral width ρ = 0.2439∗, median of azimuth ρ = 0.2378∗, and spiral precision 
+index ρ = 0.2367∗; b) sentence – number of pen stops ρ = 0.3460∗∗, slope of 
+duration of stroke (in-air) ρ = 0.2823∗∗, median of vertical velocity (on-surface) 
+ρ = −0.2438∗, and median of vertical acceleration (on-surface) ρ = 0.2317∗; and 
+c) pentagons – width of writing (on-surface) ρ = −0.3045∗∗, median of length 
+of stroke (on-surface) ρ = −0.2894∗∗, nCV of length of stroke (on-surface) ρ = 
+0.2489∗, and median of duration of stroke (on-surface) ρ = −0.2327∗. 
+ 
+Table 1. Results of the exploratory analysis. 
+ 
+Feature 
+p(U) 
+ρ 
+p(ρ) 
+Spiral 
+nCV of acceleration (s) 
+Variability of spiral width 
+0.0138 
+0.0138 
+−0.2438 
+0.2439 
+0.0263 
+0.0263 
+Median of azimuth 
+0.0158 0.2378 
+0.0304 
+Spiral precision index 
+0.0162 0.2367 
+0.0312 
+nCV of duration of stroke (s) 
+0.0438 −0.1892 0.0867 
+Sentence 
+Number of pen stops 
+0.0009 0.3460 
+0.0014 
+Slope of duration of stroke (a) 
+0.0054 0.2823 
+0.0097 
+Median of vertical velocity (s) 
+Median of vertical acceleration (s) 
+0.0138 
+0.0182 
+−0.2438 
+0.2317 
+0.0263 
+0.0351 
+Rel. total number of intra-stroke intersections 0.0232 −0.2206 0.0451 
+Pentagons 
+Width of writing (s) 
+Median of length of stroke (s) 
+nCV of length of stroke (s) 
+Median of duration of stroke (s) 
+Median of horizontal acceleration (s) 
+0.0030 
+0.0045 
+0.0123 
+0.0178 
+0.0182 
+−0.3045 
+−0.2894 
+0.2489 
+−0.2327 
+0.2317 
+0.0051 
+0.0080 
+0.0233 
+0.0343 
+0.0351 
+p(U) – p-value of Mann-Whitney U-test; ρ – Spearman’s correlation coeffi- 
+cient; p(ρ)– p-value of ρ; (s) – on-surface movement; (a) – in-air movement. 
+
+ 
+ 
+∗ 
+∗∗ 
+ 
+Next, Table 2 presents the results of the correlation analysis (*, and ** denote 
+the p-values for Spearman’s correlation coefficient being below the significance 
+level of 0.05 and 0.01, respectively) between the features summarized in Table 1 
+and the following clinical information: a) MDS–UPDRS, and b) CCB domains. 
+ 
+Table 2. Results of the correlation analysis. 
+ 
+Feature 
+ρ (UPDRS) ρ (V) 
+ρ (A) 
+ρ (E) 
+Spiral 
+nCV of acceleration (s) 
+Variability of spiral width 
+Median of azimuth 
+Spiral precision index 
+nCV of duration of stroke (s) 
+−0.3411∗ 
+0.1653 
+0.0442 
+0.0606 
+−0.1089 
+−0.0013 
+−0.3973∗∗ 
+−0.3656∗ 
+−0.0942 
+−0.1344 
+0.1130 
+−0.2981∗ 
+−0.1029 
+−0.3987∗∗ 
+−0.1618 
+0.1899 
+−0.1666 
+−0.0490 
+−0.2126 
+−0.0469 
+Sentence 
+Num. of pen stops 
+Slope of duration of stroke (a) 
+Median of vertical velocity (s) 
+−0.1018 
+0.2620 
+0.0314 
+−0.1181 
+−0.1928 
+0.1106 
+0.1012 
+−0.0513 
+0.0025 
+−0.1956 
+−0.1025 
+0.1794 
+Median of vertical acceleration (s) 
+Rel. total num. of intra-stroke intersections 
+−0.2641 
+0.0477 
+−0.0301 
+0.1647 
+0.3246∗ 
+0.1143 
+0.0193 
+0.0962 
+Pentagons 
+Width of writing (s) 
+Median of length of stroke (s) 
+nCV of length of stroke (s) 
+Median of duration of stroke (s) 
+Median of horizontal acceleration (s) 
+−0.3448∗ 
+−0.1545 
+0.3065∗ 
+−0.0348 
+0.3215∗ 
+0.2947∗ 
+0.1607 
+−0.2435 
+0.0080 
+−0.0226 
+0.1351 
+0.0501 
+−0.1126 
+−0.0085 
+−0.1632 
+0.1362 
+0.1511 
+−0.1155 
+−0.0269 
+−0.2060 
+ρ – Spearman’s correlation coefficient (∗ denotes p-value < 0.05 and ∗∗ denotes p-value 
+< 0.01);  UPDRS – MDS–Unified  Parkinson’s  Disease  Rating  Scale,  part  III  (motor 
+part) [16]; V – visuospatial domain of CCB; A – attention domain of CCB; E – executive 
+functions domain of CCB; (s) – on-surface movement; (a) – in-air movement. 
+ 
+To visualize the difference in the distribution of the top 4 features summarized 
+above for HC and subjects with LBD, the box-violin plots are presented in 
+Figs. 1, 2 and 3. The Fig. 1 shows the distribution of the features for the spiral 
+drawing, the Fig. 2 shows the distribution of the features for the sentence writing, 
+and the Fig. 3 is dedicated to the distribution of the features for the pentagon 
+copying test. 
+The results of the classification analysis are summarized in Table 3. We 
+trained 4 models in total: 3 models dedicated to each task separately and 
+a model combining all of the tasks. The following results were achieved (where 
+and 
+denote p-value of the permutation test bellow < 0.05 and < 0.01, 
+respectively): a) spiral – BACC = 0.6848∗∗, SEN = 0.8696, SPE = 0.5000; b) 
+sentence – BACC = 0.7283∗∗, SEN = 0.9783, SPE = 0.4783 c) pentagons – 
+BACC = 0.6848∗∗, SEN = 0.9348, SPE = 0.4348; and d) all tasks combined – 
+
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 1. Distribution of the top 4 most discriminating features (spiral drawing). 
+ 
+ 
+Fig. 2. Distribution of the top 4 most discriminating features (sentence writing). 
+ 
+ 
+BACC = 0.7391∗∗, SEN = 0.8043, SPE = 0.6739. The ROC curves of the trained 
+models are shown in Fig. 4. 
+ 
+4 Discussion 
+As mentioned in the methodology, the Archimedean spiral is considered as a 
+gold standard, especially in the assessment of graphomotor difficulties in PD 
+patients [5,8,31], nevertheless, it has been utilised during the quantitative anal- 
+ysis of Huntington’s disease, essential tremor, or brachial dystonia as well [13]. 
+Concerning the spiral features with the highest discrimination power (as identi- 
+fied by the Mann-Whitney U-test), we observed that the LBD group was asso- 
+ciated with a lower range in on-surface acceleration, which we suppose is caused 
+
+1.0
+0.5
+0.0
+0.5
+1.0
+HC
+LBD
+3500
+meden cf amtn
+3000
+2500
+000
+1500
+1000
+500
+.0.5
+*
+0.4
+0.3
+0.2
+0.1
+0.0
+HC
+JBJD
+40
+35
+*
+30
+25
+20
+15
+10
+5
+H50
+***
+40
+30
+20
+10
+10
+-20
+HC
+T
+80
+mscliaun qf weehel welotb fo-surtsco
+40
+10
+0
+-10
+HY0.10
+**
+0.05
+0.05
+0.10
+0.15
+HO
+L13
+100
+megbisn of wgrbicu sxalrabioa
+-50
+-100
+-150
+-200
+250 
+ 
+− 
+− 
+− 
+− 
+− 
+ 
+ 
+ 
+ 
+Fig. 3. Distribution of the top 4 most discriminating features (pentagons copying test). 
+ 
+Table 3. Results of the classification analysis. 
+ 
+Task 
+MCC BACC SEN 
+SPE 
+PRE 
+F1 
+threshold p 
+Spiral 
+0.3977 0.6848 0.8696 0.5000 0.6349 0.7339 0.26 
+∗∗ 
+Sentence 
+0.5271 0.7283 0.9783 0.4783 0.6522 0.7826 0.36 
+∗∗ 
+Pentagons 
+0.4267 0.6848 0.9348 0.4348 0.6232 0.7478 0.13 
+∗∗ 
+All tasks combined 0.4824 0.7391 0.8043 0.6739 0.7115 0.7551 0.48 
+∗∗ 
+MCC – Matthew’s correlation coefficient; BACC – balanced accuracy; SEN – 
+sensitivity; SPE – specificity; PRE – precision; F1 – F1 score; p – p-values computed by 
+the permutation test (1 000 permutations, ∗ denotes p-value < 0.05 and ∗∗ denotes p-
+value < 0.01); threshold – fine-tuned decision threshold. 
+ 
+ 
+by rigidity. This assumption is supported by the fact that the measure signifi- 
+cantly correlates (ρ = 0.3, p < 0.05) with the overall score of MDS–UPDRS III. 
+Next, the LBD group was not able to keep small variability of loop-to-loop spi- 
+ral width index, which is in line with findings reported in [31]. We also observed 
+a significant correlation between this feature and the visuospatial (ρ =  0.4, 
+p < 0.01) and the attention (ρ = 0.3, p < 0.05) domain of CCB. On the other 
+hand, the LBD group had generally higher values of the spiral precision index 
+than the HC one, which is against our initial assumptions (also the correlation 
+with the attention domain of CCB is surprisingly negative; ρ = 0.4, p < 0.01). 
+Finally, the last significant correlation with the clinical status was identified in 
+the median of azimuth, which was higher in the LBD group (in addition we 
+observed a negative correlation with the visuospatial domain of CCB; ρ = 0.4, 
+p < 0.05). 
+Regarding the classification analysis, based on the spiral features, we were 
+able to discriminate the LBD and HC groups with 68% balanced accuracy (area 
+under the curve (AUC) = 71%), which is the worst result when compared to other 
+
+90
+80
+*?
+02
+80
+50
+40
+30
+20
+10
+LBD
+9.5
+mocf emortm
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+-3.200
+150
+100
+50
+50
+TLSE
+12
+10
+683
+2
+HE 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4. Receiver operating characteristic curves for the trained models. 
+ 
+ 
+tasks and which supports our previous findings that even though the spiral is 
+considered as a gold standard the sentence copy task accents the manifestations 
+of dysgraphia much better [11]. 
+Regarding the sentence, the most discriminative feature extracted from this 
+task is the number of pen stops (i.e. a pen is in contact with the paper and 
+does not vary its position for at least 30 ms [8]), which was higher in the LBD 
+group. This parameter has been mainly employed in the diagnosis of develop- 
+mental dysgraphia in children population [27], however, in one study, Danna et 
+al. observed that this measure (but extracted from the spiral) was significantly 
+different between PD patients in the OFF state and HC [8]. Initially, we assumed 
+that the feature could be theoretically linked with cognitive deficits, but we did 
+not observe any significant correlation with the visuospatial, attention, or execu- 
+tive functions domain of CCB. The second most significant feature was the slope 
+
+0.4
+0.@Roc (pentagoms)
+1.0
++++++++++
+hreshold:0.13
+0.8
+0.6
+0.4
+AL
+...
+ROC curve (area = 0.73)
+Random guess
+0.00.4
+0.@1.0
+Roctantaskscompined
+0.a
+Threshold : 0.48
+0.6
+FD
+0.4
+ROC curve (area =0.76)
+Random guess
+0.00.6Roe spirall
+1.0
+Threshold:0.26
+0.8
+Ruai
+0.6
+tie
+0.4
+ru
+ROC curve (area = 0.71)
+Random guess
+0.00.61.0
+Roc tsenbence
+++++++++++++++++++++++++++++++++++
+Threshold:0.36
+0.8
+Ruai
+0.6
+ositive
+0.4
+ru
+ROC curve (area = 0.80)
+Random guess
+0.0 
+ 
+− 
+− 
+ 
+of the duration of in-air strokes. The positive correlation coefficient suggests that 
+the LBD subjects were associated with progressing fatigue [1,12,17]. Next, in 
+the LBD group, we observed lower on-surface vertical velocity (this is in line 
+with e.g. [21,35]), but increased on-surface vertical acceleration. This could be 
+probably explained by the slow and less smooth handwriting. In terms of pro- 
+jection, the reason why these deficits dominate in the vertical movement could 
+be explained by the fact that the finger system (which is mainly involved in the 
+vertical movement) is more affected by muscular fatigue than the wrist system 
+(which controls horizontal movement) [20]. The vertical movement requires coor- 
+dinated movement and finer flexions/extensions of more joints (interphalangeal 
+and metacarpophalangeal), thus it is more complex than ulnar abductions of the 
+wrist [10,34] and could more accent the rigidity and bradykinesia. In addition, 
+this manifestation could be associated with the progressive/consistent vertical 
+micrographia, i.e., progressive/consistent reduction in letter amplitude [33]. 
+In terms of classification, by modelling features extracted from the sentence, 
+we were able to differentiate both groups with 73% balanced accuracy (AUC 
+= 80%). In comparison with the state of the art in supportive LBD or PD 
+diagnosis [9,19,35], it is not a competitive result, but on the other hand, we 
+would like to highlight that we deal with results evaluating diagnosis of LBDs 
+in the prodromal state that has not been targeted by other research teams yet. 
+Concerning the last (cognitive) task, all the top 5 discriminative features were 
+extracted from the on-surface movement. In our recent article [4] we proved that 
+in-air entropy-based parameters could be used to identify early cognitive deficits 
+in PD without major cognitive impairment and that they correlate with the 
+level of attention. In the current study, these in-air measures were not signifi- 
+cant, but on the other hand, their on-surface variants (i.e. median of Shannon 
+entropy calculated from the global/vertical movement) had the p-values of the 
+Mann-Whitney U-test < 0.05, moreover, they significantly correlated with the 
+visuospatial domain of CCB (e.g. ρ = 
+0.3, p < 0.05). The top 5 parameters 
+consist of the width of the product, which was smaller in the LBD group. It 
+slightly correlates with the lower median of the length of strokes (ρ = 0.3) and 
+lower median of the duration of strokes (ρ = 0.2) and probably means that the 
+subjects in the LBD group made the overlapped pentagons smaller. In addition, 
+since the non-parametric coefficient of variation of the length of strokes was 
+higher, we assume that the LBD subjects were not able to keep a stable length 
+of strokes (nevertheless, based on the scoring published in [24], this is assumed 
+as a very small deviation). Regarding the width, we also observed a negative 
+correlation (ρ =   0.3, p < 0.05) with the overall score of MDS–UPDRS III. 
+The classification based on the pentagon copying test provided 68% balanced 
+accuracy (AUC = 0.73%), which is slightly better than in the case of the spiral, 
+but not as high as in the case of the sentence. 
+And finally, a machine learning model based on the whole set of features 
+(tasks) enabled us to improve the accuracy to 74% (AUC = 76%). This shows 
+that the combination of the graphomotor, handwriting and cognitive deficits can 
+be used to achieve reasonable performance in the prodromal diagnosis of LBDs. 
+
+ 
+ 
+ 
+5 Conclusion 
+ 
+This study has several limitations. Our dataset has a small sample size and the 
+HC and LBD groups are imbalanced, therefore to get better results in terms 
+of their generalisation, a bigger database must be analysed. Next, due to the 
+small sample size, we fused subjects with a high risk of developing PD or MCI- 
+LB into one LBD group. Nevertheless, subjects with MCI-LB in its prodromal 
+stage are associated mainly with cognitive (executive or visuospatial) decline, 
+while subjects with prodromal PD experience mainly motor deficits. In other 
+words, we suppose that further stratification of these participants into two groups 
+could increase the classification accuracy (we hypothesise that MCI-LB would 
+be more pronounced in the pentagon copying task and PD in the handwriting 
+one). Finally, although we tried a correction of multiple comparisons during the 
+statistical analysis, almost no significant features appeared after this adjustment. 
+To sum up, concerning the limitations mentioned above, the study should be 
+considered as a pilot one. 
+In conclusion, despite the limitations, to the best of our knowledge, it is 
+the first work exploring the impact of computerised analysis of a graphomotor, 
+cognitive, and handwriting task on the prodromal diagnosis of these neurodegen- 
+erative disorders. It bridges the knowledge gap in the field of LBDs, and provides 
+baseline results for future studies focusing on the prodromal diagnosis of LBDs 
+via a computerized and objective analysis of graphomotor and handwriting dif- 
+ficulties. 
+ 
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+

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+AMD: Adaptive Masked Distillation for Object
+Detection
+Guang Yanga, Yin Tangb, Jun Lia�,Jianhua Xua,Xili Wanb
+a School of Computer and Electronic Information, Nanjing Normal University, Nanjing, China
+b School of Computer Science and Technology, Nanjing Tech University, Nanjing, China
+{Jun Li}lijuncst@njnu.edu.cn
+Abstract—As a general model compression paradigm, feature-
+based knowledge distillation allows the student model to learn
+expressive features from the teacher counterpart. In this paper,
+we mainly focus on designing an effective feature-distillation
+framework and propose a spatial-channel adaptive masked distil-
+lation (AMD) network for object detection. More specifically, in
+order to accurately reconstruct important feature regions, we first
+perform attention-guided feature masking on the feature map of
+the student network, such that we can identify the important
+features via spatially adaptive feature masking instead of random
+masking in the previous methods. In addition, we employ a simple
+and efficient module to allow the student network channel to
+be adaptive, improving its model capability in object perception
+and detection. In contrast to the previous methods, more crucial
+object-aware features can be reconstructed and learned from
+the proposed network, which is conducive to accurate object
+detection. The empirical experiments demonstrate the superiority
+of our method: with the help of our proposed distillation method,
+the student networks report 41.3%, 42.4%, and 42.7% mAP
+scores when RetinaNet, Cascade Mask-RCNN and RepPoints are
+respectively used as the teacher framework for object detection,
+which outperforms the previous state-of-the-art distillation meth-
+ods including FGD and MGD.
+Index Terms—Feature-based Knowledge Distillation, Object
+Detection, Adaptive Masked Distillation, Object-Aware Features
+I. INTRODUCTION
+Recent years have witnessed successful and pervasive ap-
+plications of Deep Convolutional Neural Networks (CNNs) in
+various computer vision tasks. However, deep CNNs usually
+cost a huge amount of computational resources in pursuit
+of higher performance, which adversely affects their deploy-
+ment in practical applications and leads to severe parameter
+redundancy. It is therefore necessary to transfer the dark
+knowledge learned in the complex networks (teacher) to
+another lightweight network (student). This is also termed as
+knowledge distillation [1] which allows the student model to
+generate expressive features learned from the teacher model.
+Thus, it is more preferable to deploy the student model
+with compact network architecture sacrificing minimal loss of
+performance.
+The earliest distillation algorithms function mainly at the
+output head. The representative examples include logit-based
+distillation for classification and head-based distillation for
+detection [2]. Recently, a more common distillation strategy
+emerges as feature-based distillation mechanism. Since only
+0.3
+0.4
+0.8
+0.4
+3.6
+2.3
+1.2
+1.1
+0.8
+0.5
+0.3
+0.1
+1.8
+1.0
+1.3
+0.7
+1.2
+1.0
+0.4
+1.8
+0.8
+0.5
+1.8
+1.2
+0.8
+0.7
+0.3
+0.2
+0.3
+1.1
+0.4
+0.7
+1.3
+1.2
+0.5
+Fig. 1: Different regions are quantified with varying attention
+scores in the feature map of teacher model. The regions
+with higher scores encode the region importance and should
+outweigh the low-score regions in the feature masking.
+the head or projector after the generated feature varies within
+different networks, the feature-based distillation approaches
+can potentially be employed in a variety of tasks. There-
+fore, it has become a prominent line of research for both
+model compression and performance improvement due to
+its simplicity and efficacy. In object detection, in particular,
+a variety of feature-based distillation approaches have been
+developed. The earlier research, such as FitNet [3], performs
+distillation at the global level. FGFI [4] operates by distilling
+the features of high IoU between ground truth and anchors.
+FGD [5] was developed to separate distillation of foreground
+and background. Recent research suggests it is preferable for
+the student model to reconstruct and learn expressive features
+from the teacher model in the first place instead of following
+the teacher for generating competitive representations. For
+instance, MGD [6] was proposed to randomly mask pixels in
+the feature map of student network, leading to reconstructed
+features of the teacher model via a simple block.
+Although MGD further improves the feature distillation
+by reconstructing the features of masked areas, the masked
+regions are generated in a random manner. This random
+operation fails to identify the region-specific importance, and
+is likely to cause the student model to generate features of
+the teacher in unimportant regions. As illustrated in Fig. 1,
+arXiv:2301.13538v1  [cs.CV]  31 Jan 2023
+
+Spatial Attention
+Mask
+Feature
+Mask
+Backbone
+(Neck)
+SE block
+3×3
+3×3
+Distillation
+Teacher
+Spatial Attention
+R
+1× H× W
+R
+C×H× W
+R
+C×1× 1
+R
+C×H× W
+R
+C×H× W
+Avg pooling
+RELU
+FC
+FC
+Sigmoid
+Backbone
+(Neck)
+SE block
+ReLU
+Student
+Teacher
+Generation 
+Block
+.
+Fig. 2: The proposed AMD distillation framework. It first learns the adaptive Region-of-Interest (RoI) via attention-guided
+feature masking, generating the spatial mask clue from the teacher model imposed on the student feature. Furthermore, we
+apply the simple and efficient SE layer to the feature of the teacher model, leading to the channel adaptive clues. The auxiliary
+clues are then fused with the output from the generation block via a Hadamard product, such that the generated feature from
+the student model is channel adaptive.
+the importance of different regions in the feature map of
+a teacher model can be quantified using the region-specific
+attention scores. Only the regions with higher scores play
+critical role in feature masking while the low-score regions
+should be downplayed.
+To alleviate the above-mentioned drawback, we propose an
+adaptive masked distillation (AMD) framework which enjoys
+object-aware spatial and channel adaptivity. On the one hand,
+we perform attention-guided spatial masking instead of ran-
+dom masking on the feature map of the student network. More
+specifically, we first learn a spatial attention map from the
+feature map of the teacher model, producing a region-specific
+mask. Then, the feature of the student network is adaptively
+masked by using this attention map. Benefiting from this
+selective feature masking, it allows subsequent generation
+block to focus on those adaptively masked important areas,
+leading to robust and expressive representations. On the other
+hand, to further explore the object-awareness capability, we
+leverage a simple and effective SE layer [7] for modeling the
+channel attention of the resulting feature of the teacher model.
+The learned clue and the output from the generation block
+of students will be fused via a Hadamard product, achieving
+desirable object-aware channel adaptivity.
+To summarize, the contributions of this paper are threefold.
+• First, we develop a spatially adaptive feature masking
+mechanism for the student model, such that the region-
+specific importance can be encoded in the features recon-
+structed and learned from the teacher network.
+• Second, we further explore the channel adaptivity by
+introducing a simple and efficient SE module to improve
+the object-aware capability of the student model.
+• Third, we evaluate our proposed feature distillation net-
+work AMD using various detection frameworks includ-
+ing one-stage detector RetinaNet [8], two-stage detec-
+tor Faster-RCNN [9], and anchor free model RepPoint
+[10]. Extensive experimental results demonstrate that
+our method can help to learn features with sufficient
+descriptive capability and achieve significant performance
+gains over the previous state-of-the-art methods.
+The remainder of this paper is structured as follows. After
+reviewing the related work in Section II, we elaborate on our
+method in Section III. Next, we conduct extensive experi-
+mental evaluations in Section IV before the paper is finally
+concluded in Section V.
+II. RELATED WORK
+In this section, we comprehensively review the recent ad-
+vance in object detection and knowledge distillation, both of
+which are closely related to our method.
+A. Object Detection
+As one fundamental vision task, object detection aims to
+determine the category and location of the objects in an
+image. Over recent years, the success of CNNs has enormously
+advanced the research in object detection. In general, the
+detectors based on deep CNNs can be classified into three
+categories including anchor-based detectors [9, 11], anchor-
+free detectors [12] and end-to-end detectors [13]. In particular,
+anchor-based detection models are divided into two-stage
+[9, 14–16] and one-stage detectors [11, 17, 18]. The former
+detection method, represented by R-CNN like [9, 19] algo-
+rithms, has a higher detection accuracy, whereas its inference
+speed is usually unsatisfactory due to expensive computational
+costs incurred by region proposal network (RPN). As a result,
+
+it is impractical for some real-time scenarios. In contrast, one-
+stage detectors directly perform classification and regression
+on the anchors without generating proposals beforehand. Thus,
+they run faster with guaranteed detection performance.
+While recent deep networks achieve high detection accu-
+racy, they usually rely on complex backbone structure and
+significant computational resources [13, 20–22]. In this sense,
+designing lightweight and efficient backbone networks has
+emerged as a major line of research in object detection. In
+particular, knowledge distillation, which can transfer sufficient
+descriptive power from a large network to a small network, is
+beneficial for designing lightweight backbone with maintained
+performance close to the large network.
+B. Knowledge Distillation
+Recently, knowledge distillation has received increasing
+attention in model compression, since it is capable of retaining
+compact model structure with promoted performance. Hinton
+et al. [1] first came up with the concept of knowledge
+distillation by introducing the soft label of the teacher network
+as part of the loss of the student network, allowing the student
+network to learn probability distribution fitting of the teacher
+model for classification task. Moreover, Romero et al. [3]
+demonstrated that semantic information in the intermediate
+layer can also be learned as dark knowledge by student
+networks. Thus, knowledge distillation can therefore be widely
+applied to a wide range of downstream tasks. Chen et al. [2]
+distilled the neck feature, classification head, and regression
+head by setting up three loss functions, respectively. Tang
+et al. [23] carefully designed the distillation weights and
+distillation loss functions such that they are automatically
+adjusted between samples for the single-stage object detector.
+Li et al. [24] used region proposals of the larger network to
+help the smaller network learn higher semantic information.
+Zheng et al. [25] transferred the knowledge distillation of the
+classification head to the location head of object detection,
+leading to a new distillation mechanism termed Localization
+Distillation (LD). LD makes logit mimicking become a better
+alternative to feature imitation, and reveals the knowledge
+of object category and object location should be handled
+separately. Dai et al. [26] developed GID framework which
+selects distillation areas based on differences between the
+student and teacher networks. Yang et al. proposed FGD [5]
+which separates the foreground and background, enabling the
+student model to learn from the teacher network areas of
+interest and global knowledge via local and global distillation
+respectively. Besides, MGD [6] imposes random masking on
+the feature map of the student model, and then generates
+the feature map reconstructing from the teacher network.
+However, the uncertainty of random masking may introduce
+additional noise, producing biased feature map with compro-
+mised representation capability.
+III. THE PROPOSED APPROACH
+Recently, a massive amount of distillation methods are
+carefully designed for various model architectures and tasks.
+Typically, the feature maps used for distillation usually have
+high-level semantics and spatial information about adjacent
+pixels. Therefore, learning these features from the teacher
+model can significantly improve the performance of the stu-
+dent model. Mathematically, basic feature distillation can be
+formulated as:
+Lfea =
+1
+CHW
+C
+�
+k=1
+H
+�
+i=1
+W
+�
+j=1
+�
+F T
+k,i,j − f
+�
+F S
+k,i,j
+��2
+(1)
+where C, H, and W denote the channel, height, and width of
+the feature map, respectively. F T and F S denote the feature
+generated from the teacher model and its counterpart from the
+student model. f represents the adaptation layer that aligns the
+shape of F S and F T .
+Recent research suggests learning and reconstructing the
+features of the teacher model is a desirable alternative to
+feature imitation [6]. More specifically, expressive features
+can be generated from the masked regions on the feature
+map of the student network. However, previous state-of-the-
+art method mainly performs random feature masking without
+identifying the importance of different regions on the feature
+map. In this paper, we attempt to make the student model
+generate features corresponding to the important areas on the
+feature map of the teacher network. Towards this end, we
+propose a spatial-channel adaptive masked distillation strategy
+termed AMD. In contrast to the random masking strategy in
+the previous method, we perform feature masking via region-
+aware attention for identifying the important areas in the
+feature map of the teacher network. In order to improve the
+object-aware capability, we further introduce a simple and
+efficient SE module such that the resulting features are channel
+adaptive. The framework of our proposed method is illustrated
+in Fig. 2.
+A. Spatially adaptive feature masking
+Using random pixels to recover the complete feature map,
+MGD allows the masked features of the student model to
+generate features of the teacher model. Thus, it is beneficial for
+the student network to obtain a better representation. However,
+the region-specific importance is discarded due to the random
+masking in MGD. To alleviate this drawback, we carefully
+design the region-aware feature masking with the help of
+spatial attention. To begin with, we calculate the absolute mean
+value of the teacher network along the channel dimension:
+GS(F) = 1
+C
+C
+�
+k=1
+��F T
+k
+��
+(2)
+where C denotes the channel number of the feature. F T is the
+feature of the teacher. GS(F) is the spatial representation map.
+Then, the spatial attention mask resulting from the teacher
+model can be formulated as:
+AS(F) = H · W · softmax
+�
+GS(F)/T
+�
+(3)
+where T is a hyper-parameter introduced in [1] to change the
+probability distribution such that the shape of the resulting AS
+
+FGD
+mAP:40.7
+MGD
+mAP:41.0
+Ours
+mAP:41.3
+Fig. 3: Visualisation of the feature maps obtained by different distillation methods. Teacher detector is RetinaNet-ResNeXt101
+while student detector is RetinaNet-ResNet50.
+is 1×H ×W. The attention score for each location represents
+the level of interest in the teacher network. Furthermore, the
+mask value is set to 0 when the attention score is greater than
+λ and the rest are set to 1. This can be expressed as:
+Mi,j =
+� 0,
+if AS
+i,j > λ
+1,
+Otherwise
+(4)
+where AS
+i,j is the spatial attention score at the point with
+coordinates (i, j) on the feature map of the teacher network.
+λ is a hyper-parameter to control the number of pixels in the
+mask. Next, we cover the feature map of the student model
+with the mask M, which can be formulated as follows:
+F S
+mask = F S · M
+(5)
+In a nutshell, with the help of this attention-guided feature
+masking, we can mask out the student feature map according
+to the important regions of interest on the teacher counterpart,
+and the resulting feature will contain more important semantic
+information.
+B. Channel adaptive clues generation
+Different from single-object recognization tasks such as
+image classification, object detection is a dense prediction task
+focusing on detecting multiple objects. Except for the effective
+receptive field (ERF), the capability of capturing the object
+information in different scales can also bring a significant
+performance fluctuation for a detector, which is not considered
+in the previous work [5, 6, 13]. Therefore, we utilize a simple
+and lightweight SE layer [7] to learn the channel adaptive clue
+from the teacher feature. The resulting channel adaptive clue
+will be applied to enhance the student’s feature, and further
+improve the object-awareness capability:
+F T
+clue = σ
+�
+WL1
+�
+WL2
+�
+F T
+avg; θ1
+�
+; θ2
+��
+,
+GS(F S
+mask) = WC1
+�
+ReLU
+�
+WC2(F S
+mask; θ1)
+�
+; θ2
+�
+⊙ F T
+clue,
+(6)
+where F T
+clue ∈ R1×1×C denotes the learned channel adaptive
+clue for the student feature. It is fused with the output
+from the generation block via a Hadamard product denoted
+as ⊙. The WL(·; θ) and WC(·; θ) are weight matrices of
+linear projection and convolution layer for SE and generation
+modules, respectively.
+Benefiting from this design, our model further explores the
+object-aware potential, resulting in a significant improvement
+over those vanilla counterparts, i.e., models with no channel-
+adaptive design. More interestingly, we observe that our AMD
+can achieve a remarkable mAP improvement in the case of
+detecting small objects, demonstrating the effectiveness of our
+proposed method. We also provide the visualization results of
+the feature map derived from different distillation models as
+shown in Fig 3. It can be easily observed that the object feature
+produced from our AMD is more distinguishable than those
+of methods.
+C. Loss function
+Based on the proposed distillation method, we design the
+following distillation loss for AMD:
+Lfea =
+C
+�
+k=1
+H
+�
+i=1
+W
+�
+j=1
+�
+F T
+k,i,j − GS(F S
+mask)
+�2
+(7)
+where C, H, and W respectively denote the channel number,
+height and width of the feature map. F S
+mask denotes the
+masked student feature map. Thus, the overall loss function
+is as follows:
+Loverall
+�
+F T , F S�
+= α · Lfea + Loriginal
+(8)
+where α is a hyper-parameter to balance distillation loss and
+original loss, and Loriginal is the original loss of the detection
+task.
+IV. EXPERIMENT
+A. Experimental Setting
+To verify the effectiveness of our AMD for object detection,
+we evaluate our method on MS COCO2017 [27] benchmark
+dataset, which contains 80 object categories and over 160k
+images. We use 120k training images for training and 5k
+validation images for testing. For performance measures, we
+use Average Precision (AP) and Average Recall (AR) to
+evaluate the performance of different object detectors. Three
+mainstream detectors including the anchor-based one-stage
+detector RetinaNet [8], the two-stage detector Faster-RCNN
+[9], and the anchor-free detector RepPoint [10] are involved
+in our comprehensive experiments. In addition, ResNeXt101
+and ResNet50 are respectively used as the backbone of the
+teacher network and its student counterpart.
+We also conduct a series of ablation studies to ex-
+plore the effects of individual components on the per-
+formance
+of
+our
+AMD
+framework.
+In
+implementation,
+
+TABLE I: Comparison of our method with other distillation methods for object detection on COCO.
+Teacher
+Student
+mAP
+APS
+APM
+APL
+mAR
+ARS
+ARM
+ARL
+RetinaNet-Res50
+37.4
+20.6
+40.7
+49.7
+53.9
+33.1
+57.7
+70.2
+FKD [28]
+39.6 (+2.2)
+22.7
+43.3
+52.5
+56.1 (+2.2)
+36.8
+60.0
+72.1
+FGD [5]
+40.7 (+3.3)
+22.9
+45.0
+54.7
+56.8 (+2.9)
+36.5
+61.4
+72.8
+MGD [6]
+41.0 (+3.6)
+23.4
+45.3
+55.7
+57.0 (+3.1)
+37.2
+61.7
+72.8
+RetinaNet
+ResNeXt101
+(41.0)
+AMD (ours)
+41.3 (+3.9)
+23.9
+45.4
+55.7
+57.4 (+3.5)
+38.2
+61.7
+73.5
+RepPoints-Res50
+38.6
+22.5
+42.2
+50.4
+55.1
+34.9
+59.4
+70.3
+FKD [28]
+40.6 (+2.0)
+23.4
+44.6
+53.0
+56.9 (+1.8)
+37.3
+60.9
+71.4
+FGD [5]
+42.0 (+3.4)
+24.0
+45.7
+55.6
+58.2 (+3.1)
+37.8
+62.2
+73.3
+MGD [6]
+42.3 (+3.7)
+24.4
+46.2
+55.9
+58.4 (+3.3)
+40.4
+62.3
+73.9
+RepPoints
+ResNeXt101
+(44.2)
+AMD (ours)
+42.7 (+4.1)
+24.8
+46.5
+56.3
+58.8 (+3.7)
+40.6
+62.4
+74.1
+Faster RCNN-Res50
+38.4
+21.5
+42.1
+50.3
+52.0
+32.6
+55.8
+66.1
+FKD [28]
+41.5 (+3.1)
+23.5
+45.0
+55.3
+54.4 (+2.4)
+34.0
+58.2
+69.9
+FGD [5]
+42.0 (+3.6)
+23.8
+46.4
+55.5
+55.4 (+3.4)
+35.5
+60.0
+70.0
+MGD [6]
+42.1 (+3.7)
+23.7
+46.4
+56.1
+55.5 (+3.5)
+35.4
+60.0
+70.5
+Cascade
+Mask RCNN
+ResNeXt101
+(47.3)
+AMD (ours)
+42.4 (+4.0)
+24.1
+46.5
+56.2
+55.8 (+3.8)
+35.3
+60.0
+70.8
+all the experiments are conducted on a server with one
+RTX3090 GPU using MMdetection toolbox [29] and Py-
+torch framework [30]. Besides, the hyper-parameters are
+empirically set to
+�
+α = 2.5 × 10−7, λ = 1, T = 0.5
+�
+and
+�
+α = 4 × 10−6, λ = 1.2, T = 0.5
+�
+for the one-stage models
+and the two-stage models respectively. During the training
+process, SGD optimizer is used for training all the detectors
+within 24 epochs. Meanwhile, momentum is set as 0.9 whilst
+weight decay is set to 0.0001. Moreover, single-scale training
+strategy is utilized in our experiments.
+B. Results
+In our comparative studies, we carry out three groups of
+experiments to evaluate different distillation methods with the
+three popular detectors involved. The corresponding experi-
+mental results are shown in Table I.
+In the first group of experiments, RetinaNet is used as the
+detection framework for both the teacher and the student.
+The corresponding experimental results demonstrate that our
+distillation method provides significant performance boosts of
+3.9% in mAP over the baseline student network by reporting
+the highest accuracy at 41.3%. This result consistently outper-
+forms the state-of-the-art methods FGD and MGD by 0.6%
+and 0.3%, while it even surpasses the teacher model achieving
+41.0% mAP. Similar performance improvement can also be ob-
+served with respect to mAR metric. The experimental setting
+in the second group is analogous to the first one except that the
+RetinaNet framework is replaced with RepPoints. Consistent
+with the results in the first group, dramatic performance gains
+of 4.1% in mAP and 3.7% in mAR are reported, and similar
+performance superiority to the competing distillation methods
+is also demonstrated. The results reveal that our method can
+adaptively learn more important information from the teacher
+and significantly contribute to the improvement of the student
+model.
+To further assess the generalization capability of our pro-
+posed method, we make use of different detection frameworks
+for the teacher and student models. To be specific, the more
+powerful detector Cascade Mask-RCNN is used as the teacher
+network while the Faster-RCNN for the student model. As
+shown in Table I, our method boosts the baseline student
+model from 38.4% to 42.4% in mAP and from 52.0% to 55.8%
+in mAR, outperforming MGD 0.3% both in mAP and mAR. It
+sufficiently suggests our method is independent of the specific
+detector and shows consistent advantages in cross-framework
+scenarios.
+C. Ablation Study
+In this section, we conduct extensive ablation experiments
+to explore the effect of different configurations on the pro-
+posed AMD. Consistent with the above setting, the ablation
+experiments with different configurations are conducted based
+on the three popular detectors, i.e., RetinaNet, Faster-RCNN,
+and RepPoint.
+As shown in Table II, when RetinaNet is used for the
+detection framework for both the teacher and the student, we
+explore two primary modules in our AMD model, namely
+the spatially adaptive masking (Ada-Mask) and the channel
+adaptive clues generation (Ada-Channel). It is observed that
+the complete AMD model including both the Ada-Mask and
+Ada-Channel components achieves the best results. Further-
+more, when we remove either component, there is a clear
+performance drop in particular in the small-object detection
+scenario (0.3%↓ w/o Ada-Mask and 0.5%↓ w/o Ada-Channel).
+This implies that our AMD method can improve object-
+awareness capability which is crucial for dense prediction
+tasks.
+When the RetinaNet is replaced with the RepPoint, similar
+results can be obtained. As displayed in Table III, both the
+Ada-Mask and Ada-Channel components play critical roles
+in our AMD model. Specifically, single Ada-Mask module
+reports 24.4%, 46.3% and 56.0% in APS, APM and APL
+scores. With the help of additional channel adaptive clues, fur-
+ther performance gains of 0.4%, 0.2% and 0.3% are reported
+for the respective metrics.
+Furthermore, we also perform ablation studies in cross-
+framework scenarios. Specifically, the Cascade Mask-RCNN
+is used as the teacher network, while the Faster-RCNN as the
+student counterpart. As shown in Table IV, the complete AMD
+model achieves the highest accuracy. In particular, the highest
+
+TABLE II: Ablation studies using RetinaNet [8] framework for both the teacher and the student. The backbone of the teacher
+network is ResNeXt-101 whilst its student counterpart is ResNet-50. Ada-Mask and Ada-channel respectively denote spatially
+adaptive masking and channel adaptive clue generation module. They constitute two main components in our proposed AMD
+model.
+Ada-Mask
+Ada-Channel
+Student: RetinaNet + Res50
+AP b
+AP b
+50
+AP b
+75
+APS
+APM
+APL
+
+
+41.3
+61.0
+44.1
+23.9
+45.4
+55.7
+
+41.0
+61.0
+43.8
+23.7
+45.3
+55.6
+
+41.2
+60.8
+44.0
+23.4
+45.2
+55.6
+TABLE III: Ablation studies using RepPoint [10] framework for both the teacher and the student. ResNeXt-101 and ResNet-50
+are respective backbones.
+Ada-Mask
+Ada-Channel
+Student: RepPoint + Res50
+AP b
+AP b
+50
+AP b
+75
+APS
+APM
+APL
+
+
+42.7
+63.5
+46.5
+24.8
+46.5
+56.3
+
+42.4
+63.2
+46.4
+24.6
+46.5
+56.1
+
+42.4
+63.3
+46.2
+24.4
+46.3
+56.0
+TABLE IV: Ablation studies in a cross-framework scenario. The Cascade Mask-RCNN [31] is employed for the teacher
+framework, while the Faster R-CNN is for the student counterpart.
+Ada-Mask
+Ada-Channel
+Student: Faster-RCNN + Res50
+AP b
+AP b
+50
+AP b
+75
+APS
+APM
+APL
+
+
+42.4
+63.1
+46.2
+24.1
+46.5
+56.2
+
+42.1
+62.8
+46.0
+23.8
+46.4
+56.3
+
+42.3
+63.0
+46.2
+23.6
+46.6
+56.3
+APS score 24.1% is reported, outperforming the other settings
+w/o either Ada-Mask or Ada-Channel. This indicates that our
+AMD model benefits small-object detection with improved
+object-awareness capability.
+TABLE V: Comparison of different generation blocks. For
+MBConv [32], we use 5 × 5 depthwise convolution.
+Student: RetinaNet-Res50
+Generation Block
+MBConv
+3 × 3 Dense Conv * 1
+3 × 3 Dense Conv * 2
+mAP
+41.0
+41.2
+41.3
+In addition to the above ablation studies, we also discuss the
+effect of different generation blocks on the performance of our
+method. As illustrated in Table V, three different generation
+blocks are compared within the RetinaNet framework. The
+results reveal that a slightly inferior performance is reported
+by the advanced MBConv [32]. In contrast, a better result
+is achieved by simply stacking two vanilla convolutional
+layers. We assume that the channel adaptive clues learned
+from the teacher network is not compatible with MBConv
+block, because MBConv somewhat encodes the channel clues
+from the student model. This incompatibility results from the
+difference of the channel clues between the teacher and the
+student network.
+To gain a deeper insight into the effect of the Ada-Channel
+module on feature generation, we explore the following two
+TABLE VI: Comparison of different locations of Ada-channel.
+After and Within denote that we apply the channel adaptive
+clues after the generation block and between the two convo-
+lution layers, respectively.
+Student: Faster-RCNN + Res50
+Location
+After
+Within
+mAP
+42.4
+42.2
+cases with Cascade Mask-RCNN and Faster-RCNN respec-
+tively used as the teacher and the student. In the first case,
+Ada-Channel follows the generation block, and the two com-
+ponents function separately. In the other case, Ada-Channel
+is embedded within two consecutive convolution layers of the
+generation block, which implies that two modules are coupled.
+As shown in Table VI, decoupling the two components brings
+an improvement of 0.2% in mAP, suggesting that the genera-
+tion process working on the masked feature of the student is
+repulsive with other exotic clues, even the informative ones.
+D. Parameter Analysis
+In our AMD method, the hyper-parameter λ in Eq. 4
+controls the coverage of feature mask. A larger λ value
+indicates that only the points with higher attention scores of
+the teacher model are masked, and most of the pixel points
+are in the object-specific ground-truth region. In contrast, it
+
+is likely that masked points appear in the background region
+when decreasing λ. In our experiments, we discuss the effect
+of λ using RepPoints as the detection framework. It is observed
+from Fig. 4 that the highest mAP 42.7% and mAR 58.8%
+are reported when λ = 1.0, suggesting it helps the model to
+better compromise between encoding low-score and high-score
+regions.
+40
+45
+50
+55
+60
+0.7
+1
+1.5
+42.5
+42.7
+42.3
+58.4
+58.8
+58.5
+λ
+Accuracy(%)
+mAR
+mAP
+Fig. 4: Parameter λ analysis on one-stage RepPoints frame-
+work.
+V. CONCLUSION
+In this paper, we focus on the topic of feature-based masked
+distillation and propose spatial-channel adaptive masked dis-
+tillation termed AMD for object detection. On the one hand,
+we perform spatially adaptive feature masking to encode
+the region-specific importance, such that more important and
+expressive features can be learned from the teacher net-
+work. On the other hand, to improve the object-awareness
+capability, we utilize the simple and efficient SE block to
+generate informative channel-adaptive clues for the student
+model. Extensive experiments demonstrate the superiority and
+effectiveness of our method, showing that the proposed AMD
+model not only significantly boosts the performance of the
+baseline student model but also outperforms the other state-
+of-the-art distillation approaches.
+In our proposed AMD, the spatial attention map generated
+from the feature of the teacher model lacks information
+interaction. Our future work will focus on exploring alternative
+strategies to enhance the interaction among different locations
+on the attention map.
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