diff --git a/-NE5T4oBgHgl3EQfRg5P/content/tmp_files/load_file.txt b/-NE5T4oBgHgl3EQfRg5P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..642bcce054a8f29714d3fb868a76c362cbf4f232 --- /dev/null +++ b/-NE5T4oBgHgl3EQfRg5P/content/tmp_files/load_file.txt @@ -0,0 +1,1851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf,len=1850 +page_content='AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport⋆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Dargavillea, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Smedley-Stevensonb,a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Smithc,a, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Paina aApplied Modelling and Computation Group, Imperial College London, SW7 2AZ, UK bAWE, Aldermaston, Reading, RG7 4PR, UK cANSWERS Software Service, Jacobs, Kimmeridge House, Dorset Green Technology Park, Dorchester, DT2 8ZB, UK Abstract We develop a reduction multigrid based on approximate ideal restriction (AIR) for use with asymmetric linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We use fixed-order GMRES polynomials to approximate A−1 ff and we use these polynomials to build grid transfer operators and perform F-point smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also apply a fixed sparsity to these polynomials to prevent fill-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When applied in the streaming limit of the Boltzmann Transport Equation (BTE), with a P0 angular discretisation and a low-memory spatial discretisation, this “AIRG” multigrid used as a preconditioner to an outer GMRES iteration outperforms the lAIR implementation in hypre, with two to three times less work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG is very close to scalable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we find either fixed work in the solve with slight growth in the setup, or slight growth in the solve with fixed work in the setup when using fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Using fixed sparsity we see less than 20% growth in the work of the solve with either 6 levels of spatial refinement or 3 levels of angular refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In problems with scattering AIRG performs as well as lAIR, but using the full matrix with scattering is not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then present an iterative method designed for use with scattering which uses the additive combination of two fixed-sparsity preconditioners applied to the angular flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' a single AIRG V-cycle on the streaming/removal operator and a DSA method with a CG FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We find with space or angle refinement our iterative method is very close to scalable with fixed memory use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Keywords: Asymmetric multigrid, Advection, Radiation transport, Boltzmann, AIR, GMRES polynomials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Introduction The Boltzmann transport equation (BTE) describes the distribution of particles moving through an interacting medium and is used to model radiation transport, along with spectral-waves and fluid problems through kinetic and lattice-Boltzmann methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The mono-energetic steady-state form of the Boltzmann Transport Equation (BTE), with linear scattering and straight-line propagation is written in (1) as Ω · ∇rψ(r, Ω) + σtψ(r, Ω) − � Ω′ σs(r, Ω′ → Ω)ψ(r, Ω′)dΩ′ = S e(r, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (1) Equation (1) is a 5-dimensional linear PDE, with three spatial dimensions and two angular dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we neglect the energy and time dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The angular flux, ψ(r, Ω), describes the number of particles moving in direction Ω, at spatial position r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The macroscopic total cross section of the material that the particles are moving through is given by σt, which describes particles removed either through absorption or scattering in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The source of particles coming from scattering in many radiative processes is given by an integral term, where σs is the macroscopic scatter cross-sections for this process that describes how particles scatter from direction Ω′ into direction Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Finally any external sources of particles are given by S e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' ⋆UK Ministry of Defence © Crown owned copyright 2023/AWE Email address: dargaville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='steven@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='com (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Dargaville) Preprint submitted to Elsevier January 16, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05521v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='comp-ph] 13 Jan 2023 One of the main challenges in solving (1) is that when the scattering cross-sections are large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the parti- cles are interacting with the material), the BTE tends to a diffusion equation, whereas when the scattering and total cross-sections are zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', particles are moving through a vacuum), the BTE is purely hyperbolic and hence stable discretisations must be used and the resulting linear systems are asymmetric and non-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In (1), the scattering cross-section, σs, for any given angle-to-angle scattering event is often described by a Leg- endre expansion whose coefficients we denote as σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we discretise in space/angle (with a discretisation like Sn, or FEM), we introduce, ϕ, which is the angular flux, ψ, in Legendre space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then write (1) as a 2 × 2 block system, namely � I Dm −MmΣs L � �ϕ ψ � = � 0 ˆSe � , (2) where L is the streaming/removal operator, Σs is a matrix with the scattering cross-sections for each spatial node, Dm and Mm are formed from the tensor product of the spatial mass matrices and the mapping operators which map between our angular discretisation and the moments of the Legendre space and ˆSe is the discretised source term from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When discretised in space with an upwind discretisation and an appropriate ordering of ψ is applied, L is a block- diagonal matrix, where each of the blocks is lower triangular and corresponds to the spatial coupling for each direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', each direction in L is not coupled to the others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' L represents a set of advection equations for each direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Equation (2) is typically solved by forming the Schur complement of block L, namely (I + DmL−1MmΣs)ϕ = −DmL−1 ˆSe, (3) and then a preconditioned Richardson iteration (known as a source iteration in the transport community), with pre- conditioner M−1 is applied to recover ϕn+1 = ϕn − M−1(DmL−1(MmΣsϕn + ˆSe) + ϕn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (4) Computing the solution to (2) therefore only requires the Legendre representation of the angular flux (the angular flux can easily be formed, either at a single spatial point or across the domain if needed), as each of the components of (4) are block-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This allows for a very low memory iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If there is scattering in the problem, then typically an additive preconditioner like M−1 = I + D−1 diff is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is known as diffusion-synthetic acceleration (DSA), where Ddiff is a diffusion operator, with diffusion/sink/boundary coefficients taken from an asymptotic analysis of the BTE in the diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The spatial discretisation applied to the diffusion operator in DSA can govern its effectiveness in accelerating convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' research into discretisations of the diffusion operator that are effective and/or “consistent” with transport are extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The use of a Krylov method to solve (3) instead of a Richardson method can reduce the dependence on this consistency [1] and hence “inconsistent” discretisations can be applied to the diffusion operator as part of a DSA resulting in SPD systems that can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Transport synthetic acceleration (TSA) [2] has also been used to accelerate convergence in the scattering limit, where lower order transport solutions are used instead of diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' see [3] for a review of both DSA and TSA method in transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The solution of (3) relies on the exact inversion of L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' if as mentioned above the spatial and angular discretisations result in L with lower-triangular structure then L can be inverted exactly with a single (matrix-free) Gauss-Seidel (GS) iteration, also known as a sweep in the transport community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If unstructured spatial grids are used, inverting L can become more difficult, particularly in parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' finding an ideal sweep ordering is then NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Furthermore, the introduction of different spatial or angular discretisations, time dependence or additional physics that cannot be mapped to Legendre space compounds this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similarly, if we wish to use angular adaptivity where the angular resolution differs throughout the spatial grid, L is no longer block-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our goal in the AMCG has been to investigate different discretisations, adaptive and iterative methods for solving the BTE, that have the potential to overcome these difficulties while remaining scalable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', perform a fixed amount of work with a fixed memory consumption, even with space/angle refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We instead form the Schur complement of block I from (2) and solve the system formed with the angular flux, namely (L + MmΣsDm)ψ = ˆSe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (5) There are several disadvantages to solving (5) instead of (3), the most important of which is the considerable increase in memory required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The angular flux, ψ, is much bigger than ϕ and the matrix L + MmΣsDm is not block diagonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2 instead the scattering operator MmΣsDm couples different angles together and results in dense angle-angle blocks where the nnzs scale like O(n2) with angle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These disadvantages would typically preclude the development of a practical transport algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Previously we have tackled those problems through the combination of: using a stable spatial discretisation based on static condensation that has the stencil of a CG discretisation (reducing the size of ψ but at the cost of making the blocks in L no longer lower triangular),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' using angular adaptivity to only focus angular resolution where required (reducing the size of ψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' and using a matrix-free multigrid to solve (5) that does not rely on the explicit construction of MmΣsDm or on the lower triangular structure in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We showed previously [4, 5] that these techniques perform well on many transport problems, allowing the practical use of both angular adaptivity with high levels of refinement and unstructured spatial grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These methods do not use GS/sweep smoothers however and do not scale well in the streaming limit where σs tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similarly, most multigrid methods in the literature that achieve good performance for the BTE have relied on either block- based smoothers, often on a cell/element, which do not scale with increasing angle size but which perform well in the scattering limit [6, 7, 8, 9], or GS/sweeps as smoothers which perform well in the streaming limit [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] but as mentioned can be difficult to parallelise on unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Developing multigrid methods which perform well with local smoothers is difficult, particularly as hyperbolic problems have long proved difficult to solve with multigrid due to the lack of developed theory for non-SPD matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Recently multigrid methods based on approximate ideal restrictors (AIR) have been developed [26, 27, 28, 29] that show good convergence in asymmetric problems, and in particular with the BTE when using point-based smoothers [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The aim of this work is hence to build an iterative method that solves (5) with the following features: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Compatible with the space/angle discretisations we have developed previously [4, 5] and hence does not rely on L having block diagonal and/or lower triangular structure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Does not require the explicit construction of MmΣsDm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Does not require the use of GS/sweeps 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Scalable in both the streaming and scattering limits with space/angle refinement 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Compatible with angular adaptivity 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Good strong and weak scaling in parallel with unstructured grids We examine the first four of these points in this paper, and leave investigation of adaptivity and the parallel per- formance of our method to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Here we present two contributions: the first is an algaebraic multigrid method based on combining AIR with low-order GMRES polynomials, which we call AIRG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the second is an iter- ative method where we apply additive preconditioners to an outer GMRES iteration on the angular flux, based on the streaming/removal operator and a DSA diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Both these contributions can be used independently of the other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for example we could use AIRG to invert L and/or the DSA operator as part of a typical DG FEM/source iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then compare our iterative method to the hypre implementation of lAIR and find performance advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We therefore have an iterative method on unstructured grids which never requires the assembly of the full matrix in (5), that has both good performance and fixed memory consumption across all parameter regimes with space/angle refinement for Boltzmann transport problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Discretisations We begin by presenting the spatial and angular discretisations used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' they are based on those presented in [30, 31, 32, 33, 4, 5] and hence we only discuss their key features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Angular discretisation We use a P0 DG FEM in angle (or equivalently a cell-centred FVM) with constant area azi/polar elements that we normalise so that the angular mass matrix is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The first level of our angular discretisation is denoted level 1, with one constant basis function per octant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Each subsequent level of refinement comes from splitting an angular element into four at the midpoint of the azi/cosine polar bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is structured, nested refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is equivalent to the Haar wavelet discretisations discussed in [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' indeed as part of the matrix-free iterative method in [4] an O(n) mapping to/from this P0 space to Haar space was performed during every matvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We instead choose to solve in P0 space as we can form an assembled copy of the streaming/removal matrix that has fixed sparsity with angular refinement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this property was not needed for the matrix-free methods used in our previous work, all we required was a scalable mapping between P0 and Haar spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also adapt in this P0 space in the same manner as our wavelet space [4], given the equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We investigate adapting in P0 space in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Spatial discretisation Our spatial discretisation is a sub-grid scale FEM, which represents the angular flux as ψ = φ + θ, where φ is the solution on a “coarse” scale and θ is the solution on a “fine” scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The finite element expansions for both the fine and coarse scales can be written as φ(r, Ω) ≈ ηN � i=1 Ni(r)˜φi(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' θ(r, Ω) ≈ ηQ � i=1 Qi(r)˜θi(Ω), (6) with ηN continuous basis functions, Ni, and ηQ discontinuous basis functions, Qi, with ˜φi and ˜θi the expansion co- efficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this work we use linear basis functions for both the continuous and discontinuous spatial expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1, we use a P0 discretisation, with (constant) basis functions G j(Ω), with a spatially varying number of angular elements ηi A and ηi D on the coarse and fine scales respectively (we enforce that DG nodes have the same angular expansion as their CG counterparts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The expansion coefficients ˜φi and ˜θi in (6) can then be written as space/angle expansion coefficients ˜φi, j and ˜θi,j via the expansion ˜φi(Ω) ≈ ηi A � j=1 G j(Ω)˜φi, j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' ˜θi(Ω) ≈ ηi D � j=1 G j(Ω)˜θi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (7) Following standard FEM theory the discretised form of (1) can then be written as �A B C D � �Φ Θ � = �SΦ SΘ � , (8) where Φ and Θ are vectors containing the coarse and fine scale expansion coefficients, we denote the number of unknowns in Φ as NCDOFs and in Θ as NDDOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The discretised source terms for both scales are SΦ and SΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We should note the matrices A and D are the standard CG and DG FEM matrices that result from discretising (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can then form a Schur complement of block D and recover (A − BD−1C)Φ = SΦ − BD−1SΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (9) The fine solution Θ can then be computed Θ = D−1(SΘ − CΦ), (10) and our discrete solution is the addition of both the coarse and fine solutions, namely Ψ = Φ + Θ (where the coarse solution Φ has been projected onto the fine space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In order to solve (9) and (10) efficiently/scalably, we must sparsify D and this sparsification is dependent on the angular discretisation used, see [33, 34, 35, 36, 37, 38, 4, 39] for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this work we replace D−1 in (9) and (10) with ˆD −1, which is the streaming operator with removal and self-scatter only, and vacuum conditions applied on each DG element (as this removes the jump terms that couple the DG elements, resulting in element blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can then invert this matrix element-wise and store the result, with a 4 constant nnzs with space/angle refinement, as it has the same stencil as the streaming operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Now if we consider the streaming/removal (denoted with a subscript Ω) and scattering contributions (denoted with a subscript S) in (9) separately, along with our sparsified ˆD −1 we can rearrange (9) and write � AΩ − BΩ ˆD −1CΩ � Φ + � (AS + BS(y + ˆD −1CΩ) + BΩy � Φ = SΦ − (BΩ + BS) ˆD −1SΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (11) where y = ˆD −1CS and our fine component is Θ = ˆD −1(SΘ−(CΩ+CS)Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see that (11) is now written similarly to (5), where the left term is (very close to) the sub-grid scale streaming/removal operator, with the remainder being the contribution from scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It is not exactly the streaming/removal operator as ˆD −1 contains self-scatter, but it has the same stencil as the streaming/removal operator and we call it such below for simplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' practically we can modify our stabilisation such that ˆD −1 does not contain self-scatter without any substantial differences to either the stability of our system or the preconditioning described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As mentioned in Section 1, we cannot explicitly form the scattering contribution as it has dense blocks, but given that ˆD −1 has fixed sparsity and the scatter contributions from AS, BS and CS can be formed in Legendre space and mapped back, we can scalably compute a matrix-free matvec with angular refinement in P0 space (with a fixed scatter order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Practically, there are many ways to implement a matvec for (11) (these involve further rearrangement of (11)), depending on the amount of memory available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Some key points include that away from boundaries, the element matrices in AΩ, BΩ, CΩ and ˆD −1 all share the same (block) sparsity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' in fact given matching angular resolution and the same order of basis functions on the continuous and discontinuous spatial meshes (we enforce both conditions) two of the element matrices are identical, with BΩ and CΩ related by the transpose of the spatial table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' A matvec with these components can therefore consist of performing several matvecs on small (max 3 × 3 or 4 × 4) angular blocks which can be kept in cache (similar to the work performed during a DG sweep, but without the Gauss-Seidel dependency on ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also combine a number of the maps to/from Legendre space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We describe some of the choices we make in Section 7, but note that a FLOP count shows our sub-grid scale matvec with scattering can be computed with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8x the FLOPs of an equivalent DG matvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With our sub-grid scale discretisation, we are therefore choosing to increase the number of (local) FLOPs in order to significantly decrease our memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can build iterative methods that only depend on the the coarse- scale solution, Φ, which is on a CG stencil, which in 3D has approximately 20× fewer unknowns than a DG method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', a GMRES(20) space built on Φ would take approximately the same space as a single copy of Ψ in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' One additional benefit is as noted in [4], is that with either a wavelet or non-wavelet angular discretisation, our sub-grid scale discretisation does not require interpolation between areas of different angular resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', across faces), due to the lack of DG jump terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Additively preconditioned iterative method This section details the iterative method we use to solve (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For simplicity we begin by writing the unseparated form (9) with our sparsified ˆD −1 and introduce a preconditioner M−1 on the right to give (A − B ˆD −1C)M−1u = SΦ − BD−1SΘ, u = Mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (12) We solve (12) with GMRES and use a matrix-free matvec as described above to compute the action of (A − B ˆD −1C) (and to compute the source and Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The preconditioner we apply is based on the additive combination of a two-level method in angle and the streaming/removal operator, which we denote as M−1 = M−1 angle + M−1 Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (13) The first of these uses a DSA type operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' both DSA and TSA can be thought of as additive angular multigrid preconditioners, with DSA forming a two-level multigrid with the coarse level represented by a diffusion equation, whereas TSA can form multiple coarse grids if desired through lower angular resolution transport discretisations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', see [6, 14, 16, 17, 18, 20, 40, 41, 24, 25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found both were very effective, but in this work we use a DSA method which we write as M−1 angle = RangleD−1 diffPangle, (14) 5 where the angular restrict/prolong are simply the mappings between the constant moment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the 0th order scatter term) and our P0 angular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Ddiff is a standard DSA diffusion operator with diffusion coefficient of κ = 1/3σs and Robin conditions on vacuum boundaries that we discretise with a CG FEM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this makes our DSA “inconsistent” but we find this performs well with a Krylov method as the outer iteration, as in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For further discussions we appeal to the wealth of literature on different acceleration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The second component of our additive preconditioner is based on the sub-grid scale streaming/removal operator in (11), namely M−1 Ω = � AΩ − BΩ ˆD −1CΩ �−1 (15) The additive combination of these two preconditioners is designed to achieve good performance in both the stream- ing and scattering limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For problems with both streaming/scattering regions we would apply each of the additive preconditioners only in regions which require it [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It is trivial to form the grid-transfer operators Rangle and Pangle regardless of angular refinement, but for a scalable iterative method we must be able to apply the inverses of both the diffusion matrix with a fixed amount of work given spatial refinement, and the streaming/removal operator with fixed work given space/angle refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can do so inexactly given these are applied as preconditioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is in contrast to a source iteration like (4), where [29] notes that inexact application of L−1 changes the discrete solution and they show that increasingly accurate solves are required with grid refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The next section describes a novel reduction multigrid that we can use to apply the inverses of our stream- ing/removal operator, AΩ − BΩ ˆD −1CΩ (or the streaming operator in the limit of zero total cross-section), and a CG diffusion operator, Ddiff if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For comparison purposes we also test AIRG on the full matrix, A − B ˆD −1C, in the scattering limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we cannot form this matrix scalably but it helps demonstrate that AIRG is applicable in both advective and diffuse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG multigrid We begin this section with a summary of a reduction multigrid [43, 44, 45, 28, 26, 46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we should note that reduction multigrids, block LDU factorisations/preconditioners and multi-level ILU methods all use similar building blocks (we discuss this further below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we consider a general linear system Ax = b we can form a block-system due to a coarse/fine (CF) splitting as �Aff Afc Acf Acc � �xf xc � = �bf bc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (16) If we write the prolongator and restrictor as P = �W I � , R = � Z I � , (17) then we can form a coarse grid matrix as Acoarse = RAP and a multgrid hierarchy can be built by applying the same technique to Acoarse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To see how the operators R and P are constructed we consider an exact two-grid method, where down smoothing (“pre”) occurs before restriction and up smoothing occurs after prolongation (“post”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can write the error at the ith step as ei = ¯x − xi, where ¯x is the exact solution to our linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Following [26], if we have error on the top grid after the down smooth, we can form our coarse grid residual by computing RAei and hence the error after a coarse grid solve is A−1 coarseRAei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The error on the top grid after coarse grid correction, denoted as ei+1 is hence ei+1 = (I − PA−1 coarseRA)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (18) We can consider the error, ei, to be made up of two components, one of which is in the range of interpolation and a remainder, namely ei = �ef ec � = �Wec ec � + �δei f 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (19) We can then write (18) as ei+1 = (I − PA−1 coarseRA) �δei f 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (20) 6 If δei f = 0 then F-point error is in the range of interpolation and the coarse-grid correction is exact, as ei+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Methods based on ideal restriction however choose Z to ensure that RA �δei f 0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (21) This enforces that any error on the top grid that is not in the range of interpolation does not make it down to the coarse grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' A local form of (21) is used explicitly to form the restrictor in lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Expanding (21) gives the condition (ZAff + Acf)δei f = 0, and hence Z = −AcfA−1 ff is known as the “ideal” restrictor that satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The error after coarse-grid correction (20) then becomes ei+1 = �ei f − Wei c 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (22) Post F-point smoothing to convergence then gives an exact two-grid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is one of the defining characteristics of a reduction multigrid, that ideal restriction ensures that after coarse-grid correction, the C-point error is zero and hence F-point smoothing is appropriate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' a similar statement can be used to construct an “ideal” prolongator given by W = −A−1 ff Afc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Both [26, 28] build approximations to the ideal restrictor and then build a classical one-point prolongator that interpolates each F-point from it’s strongest C-point connection with injection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' by definition this preserves the constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [26] show that this is sufficient to ensure good convergence in advection problems, and [27] prove more general criteria on the required operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With the ideal operators the near-nullspace vectors are naturally preserved and we don’t have to explicitly provide (or determine through adaptive methods [47, 48]) the near-nullspace vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' in fact any error modes (near-nullspace or otherwise) that are not in the range of interpolation stay on the top grid and are smoothed, with the rest being accurately restricted to the coarse grid by construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' if we consider near-nullspace vectors this can be easily demonstrated by considering that if A[nf, nc]T ≈ 0, where [nf, nc] is a (partitioned) near-nullspace vector, then with the ideal operators Acoarsenc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is in contrast to traditional multigrid methods, which smooth high-frequency modes preferentially on the top grid, with interpolation specifically designed to transfer low-frequency modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', near-nullspace vectors) to the coarse grid, where they become high-frequency and hence can be smoothed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [26] show that the error propagation matrix, ϵ of a reduction multigrid with F point up smooths (and no down smooths) is given by ϵ = I − M−1 LDUA, (23) where MLDU is a block LDU factorisation of A given by MLDU = � I 0 AcfA−1 ff I � �Aff 0 0 S � � I A−1 ff Afc 0 I � , (24) where S = Acc − AcfA−1 ff Afc is the Schur complement of block Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The inverse of MLDU is given by M−1 LDU = �I W 0 I � �A−1 ff 0 0 S−1 � �I 0 Z I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (25) The ideal operators give that Acoarse = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Block LDU factorisations formed with approximate ideal operators have a long history of being used as preconditioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The same factorisation can be applied to S to recover a multilevel LDU method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The benefit to using a block LDU method, instead of a reduction multigrid is that up F-point smoothing and coarse grid correction occur additively (as written in (25)), rather than F-point smoothing occuring after coarse grid correction with a reduction multigrid (as written in (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is an attractive property in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we wish to form a reduction multigrid (or an LDU method) we must approximate A−1 ff ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we denote the approxi- mation as ˆA −1 ff ≈ A−1 ff and hence our approximate ideal restrictor and prolongator are given by P = � − ˆA −1 ff Afc I � , R = � −Acf ˆA −1 ff I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (26) 7 The strength of our approximate ideal operators allows the use of zero “down” smoothing iterations, with F-point smoothing only on the “up” cycle after coarse-grid correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The authors in [26] however do perform some C-point smoothing on their “up” cycle (with F-F-C Jacobi) for robustness, but we did not find that necessary in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Equation (22) suggests a simple choice for the prolongator would be W = 0, but in a multi-level setting where we are approximating A−1 ff [28, 26] show this would require an increasingly accurate approximation with grid refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Rather than build a classical prolongator like [26, 28], as we have an approximation of A−1 ff it only costs one extra matmatmult per level to compute the ideal prolongator, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then keep only the largest (in magnitude) entry, and hence we form a one-point ideal prolongator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is like using AIR in conjunction with AIP which [27] suggest could form a scalable method for non-symmetric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is different to lAIR [28], where Z is computed directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' computing the ideal prolongator would therefore require an equivalent calculation for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We find this is a robust choice for our prolongator (as it satisfies the approximation properties required by [27]) while also being very simple as it doesn’t require knowledge of the near-nullspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To perform up F-point smoothing on each level, we use a Richardson iteration to apply ˆA −1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' On each level if we are smoothing Ae = r, where r is the residual computed after the down smooth then en+1 f = en f + ˆA −1 ff (rf − Afcen c − Affen f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (27) The coarse-grid error does not change during this process, so we can cache the result of Afcen c during multiple F-point smooths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The next section details how we construct our approximation ˆA −1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' GMRES polynomials Forming good, but sparse approximations to A−1 ff is the key to a reduction multigrid (and LDU methods as men- tioned);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is achieved in-part by ensuring a “good” CF splitting that results in a well-conditioned Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In particular Aff is often better conditioned (or more diagonally dominant) than A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this section we assume a suitable CF splitting has been performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' see Section 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The original AMGr work [44] approximated A−1 ff using the inverse diagonal of Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The nAIR method presented by [26] on advection-diffusion equations used a matrix-polynomial approximation of A−1 ff generated from a truncated Neumann series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' unfortunately this does not converge well in the diffuse limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', when Aff is not lower-triangular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The work in [28] however solved dense, local linear systems, which enforce that RA = 0 within a certain F-point sparsity pattern to compute Z directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this was denoted as lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For advection-diffusion equations, this showed good performance in both the advection and diffuse limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The work of [29] specifically examined the performance of lAIR in parallel for the BTE, but used on L in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [46] showed the use of sparse approximate inverses (SAIs) [49, 50] on diffusion equations to approximate A−1 ff (they also approximated both W and A−1 cc for C-point smoothing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The variant of SAI method used in [46] solved the minimisation problem, argminM||I − AM||F (which can be written as a separate least-squares problem for each column), to generate an approximate inverse M, while enforcing the fixed sparsity of Aff on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For LDU methods, [51] outlined many of the techniques used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' these include using diagonal approximations, or incomplete LU (often referred to as multi-level ILU [52]) and Cholesky factorisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Rather than explicitly approximate A−1 ff , many multigrid methods such as smoothed aggregation [53, 54, 55] or root-node [56] methods build their prolongation operators by minimising the energy (in an appropriate norm) of their prolongator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is equivalent to solving AP = 0 (either column-wise or in a global sense) and given the equivalent construction of (21) for the prolongator, these methods can therefore converge to the ideal prolongator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' One of the benefits of forming the ideal operators through a minimisation process is that constraints on the sparsity of the resulting operator can be applied at all points through this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In particular, if this sparsity affects the ability to preserve near-nullspace vectors, their preservation can be enforced at each step as the minimisation converges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', see both [57] for an overview and [58] for a general application to asymmetric systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this work we compute ˆA −1 ff with GMRES polynomials [59, 60, 61, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Polynomial methods have been used in multigrid for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Examples include, as mentioned, in nAIR [26] to approximate A−1 ff , in AMLI to approximate the inverse of the coarse-grid matrix [64, 65] and commonly with Chebychev polynomials used as smoothers for SPD matrices, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The aim of using GMRES polynomials in this work is to ensure good approximations to ˆA −1 ff that don’t depend on particular orderings of the unknowns, diagonal, lower-triangular or block structure, while still allowing good performance in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such AIRG should be applicable to many common 8 discretisations of advection-diffusion problems, in both limits, or any operators where a suitable splitting can produce a well-conditioned Aff that a GMRES polynomial can approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' A key difference between this work and previous work with AIR is that we also reuse these polynomials as our F-point smoothers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This allows us to build a very simple and strong multigrid method for asymmetric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We begin by summarising the GMRES method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For a general linear system Ax = b, where A is n × n, we specify an initial guess x0 = 0 and hence an initial residual r0 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The Krylov subspace of dimension m: span{b, Ab, A2b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' , Am−1b} is used in GMRES to build an approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This solution at step m can be written as xm = qm−1(A)b, where qm−1(A) is a matrix polynomial of degree m − 1 known as the GMRES polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is the polynomial that minimises the residual ||rm|| = ||p(A)r0||2 subject to p(0) = 1, where p(A) = 1 − Aqm−1(A) is known as the residual polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The coefficients of qm−1 correspond to the required linear combinations of the Krylov vectors and hence a typical GMRES algorithm can be modified to generate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [61, 62] discuss the generation and application of these polyno- mials in detail and care must be taken if high order polynomials are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Thankfully we are only concerned with low-order polynomials for use within our multigrid hierarchy and as such consider two different bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As part of a typical GMRES algorithm, we consider a set of orthonormal vectors which form a basis for our subspace, stored in the columns of the matrix Vm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the Arnoldi basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similarly the Krylov vectors b, Ab, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' make up the columns of the matrix Km (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the Krylov basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the vectors in Vm and Km span the same space we can write them as linear combinations of each other and hence Vm = KmCm, where Cm is of size m×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Typically GMRES doesn’t store Cm but a small modification can be made to store these values at each GMRES step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The approximate solution produced by GMRES is given by xm = x0 + Vmym, where ym comes from the solution of the least-squares problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The coefficients for the polynomial qm−1 can then be computed through (α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' , αm−1)T = Cmym (as in [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Equivalently (in exact arithmetic) a QR factorisation of Km+1 = QR can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we form the submatrix ˜R from R but without the first column, and note that β = R1,1 then the polynomial coefficients come from the solution of the least-squares problem (α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' , αm−1)T = argminym||βe1 − ˜Rym||2, where e1 is the first column of the m + 1 identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' GMRES methods normally don’t use the Krylov basis directly as it is poorly conditioned when m → ∞, although this is typically not a concern at low-order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for example [61] found using the Krylov basis was stable up to 10th order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In either case our GMRES polynomial of degree m − 1 is given by qm−1(A) = α0 + α1A + α2A2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' + αm−1Am−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (28) At each step of a GMRES method, the subspace size, m, grows and hence the GMRES polynomial changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Polyno- mial preconditioning methods typically freeze the size of the subspace and perform a precompute step to generate a GMRES polynomial of a fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This polynomial is then used as a stationary preconditioner, often for GMRES itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this work we want to approximate A−1 ff on each multigrid level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we use GMRES polynomials with a fixed polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To generate the coefficients for our GMRES polynomials, during our multigrid setup, on each level of our multigrid hierarchy we set m to a fixed value, assign an initial guess of zero and use a random rhs (see [66, 67, 62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can then chose to use either the Arnoldi or Krylov basis with Aff to form our coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The Arnoldi basis requires m steps of the modified GMRES described above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this costs m matvecs along with a number of dot products and norms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', reductions) on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Instead we could use the Krylov basis, Km+1, which still requires m matvecs (the setup of a matrix-power kernel may not be worth it given we only need to compute our low-order polynomial coefficients once), but in parallel a tall- skinny QR (TSQR) factorisation can be used to decrease the number of reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This relies on QR factorisations of small local blocks along with a single all-reduce to generate R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we don’t require Q to compute our polynomial coefficients and hence the small local Q blocks can be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is equivalent to modifying a communication- avoiding GMRES with s = 1 [68, 67] to generate the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We tested using both the Arnoldi and Krylov basis in this work and they generate the same polynomial coefficients to near round-off error which shows that stability is not a concern at such low orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Hence we can use the Krylov basis in parallel and the generation of the polynomial coefficients can be considered as communication-avoiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Rather than use a random vector, we could form a GMRES polynomial by using a block method and solving ZAff = −Acf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' That polynomial would be tailored to computing Z which is not desirable given we also use our polynomial to perform F-point smoothing and hence want to probe all the modes of Aff, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9 If we wish to use low-order GMRES polynomials to approximate A−1 ff , we risk not having converged our approxi- mate ideal operators sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similarly, the introduction of fixed sparsity, drop tolerances or equivalent in ˆA −1 ff , P, R or Acoarse may exacerbate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We examine the impact of both varying m and sparsifying our operators below in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fixed sparsity polynomials In the limit m → n, the GMRES polynomials converge to A−1 ff exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Practically this is impossible to achieve given the storage requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Even at low-order however, we would like to avoid the fill-in that comes from explic- itly forming our polynomials with m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To do this, we can construct our polynomials by enforcing a fixed sparsity on each of the matrix powers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for simplicity we chose the sparsity of Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is particularly suited to convection operators given that the fill-in should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For example, if we consider a third-order polynomial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', m = 4), and denoting the sparsity pattern of Aff as S ⊂ {(i, j) | (Aff)i, j � 0}, we enforce that ( ˜A2 ff)i,j = (AffAff)i,j, ( ˜A3 ff)i, j = ( ˜A2 ffAff)i, j (i, j) ∈ S, (29) and for (i, j) not in S , the entries are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is simply computing A2 ff with no fill-in, using this approximation when computing subsequent matrix-powers and again enforcing no fill-in on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our fixed-sparsity approximation to A−1 ff with m = 4 would then be given by ˆA −1 ff = α0 + α1Aff + α2 ˜A2 ff + α3 ˜A3 ff ≈ q3(Aff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (30) Fixing the sparsity of the matrix powers reduces the memory consumption of our hierarchy and also allows us to optimise the construction of our polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For example, with m = 4 it costs two matrix-matrix additions and two matmatmults to explicitly construct our polynomial, but given the shared sparsity, the additions can be performed quickly and the matmatmults with m > 2 can share the same row data required when computing A2 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In parallel this means we only require the communication of off-processor row data once with m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Computing low-order GMRES polynomials with fixed sparsity with the Krylov basis therefore only requires the communication associated with m matvecs, a single all-reduce and if m > 2 the matmatmult which produces A2 ff, regardless of the polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This has the potential to scale well, in contrast to some of the other methods which approximate A−1 ff described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We cannot use nAIR with truncated Neumann series due to the lack of lower triangular structure in our spatial discretisation, while with lAIR we found we require greater than distance 2 neighbours for scalability (we exam- ine this in Section 7), but the communication required for this becomes prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Using ILU factorisations make parallelisation difficult given the sequential nature of the underlying Gaussian elimination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' approximate ILU factori- sations have more parallelism [69, 70] but they still require triangle solves (which could then also be approximated with truncated Neumann series for better performance in parallel, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The SAIs used by [46] should scale well, given that if the fixed sparsity of Aff is used, then the formation of an approximate inverse only requires the same communication as computing A2 ff;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we must however also consider the local cost of computing a SAI and the effectiveness of its approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we examine this in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Typically with AIR the approximations ˆA −1 ff on each level are thrown away once the grid-transfer operators have been built and different F-point smoothers are used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' as mentioned we save them to use as smoothers instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The GM- RES polynomials are very strong smoothers (which don’t require the calculation of any extra dampening parameters) but storing them explicitly takes extra memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We examine the total memory consumption of the AIRG hierarchy in Section 7, but note that given the fixed sparsity of ˆA −1 ff , our experiments with pure streaming show it has approxi- mately 65% as many non-zeros as R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We could instead throw away ˆA −1 ff after our grid transfer operators are computed on each level and perform F-point smoothing by applying qm−1(Aff) matrix-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With m ≤ 2 the number of matvecs required is the same and hence ˆA −1 ff could be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With m > 2 however, applying qm−1(Aff) matrix-free would require more matvecs in order to apply the matrix-powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we believe the (constant sized) extra memory required is easily justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also investigated using Chebyshev polynomials as smoothers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We precomputed the required eigenvalue estimates with GMRES in order to build the bounding circle/ellipse given our asymmetric linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found that smoothing with these polynomials was far less efficient/robust (and practically more difficult) than using the GMRES polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' indeed using the GMRES polynomials directly is one of the key messages of [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Drop tolerance on Aff If our original linear system Ax = b is not sparse, neglecting the fill-in with the fixed sparsity polynomials in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 may not be sufficient by itself to ensure a practical multigrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also introduce a drop tolerance to Aff that is applied prior to constructing our polynomial approximations ˆA −1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In general this is not necessary in the streaming limit, but with scattering we find it helps keep the complexity low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is similar to the strong R threshold used in the hypre implementation of lAIR, which determines strong neighbours prior to the construction of Z, and in Section 7 we denote it as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can either apply the dropping after the polynomial coefficients are computed (in a similar fashion to how the fixed sparsity in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 is applied), or before such that we are forming a polynomial approximation to a sparsified Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With scattering, we did not find much of a difference in convergence so we chose to apply the dropping before, as this helps reduce the cost of the matvec used to compute the polynomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Approximations of A−1 ff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 (a) Operators formed from GMRES polynomials without fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 (b) Operators formed from fixed sparsity GMRES polynomials and relative drop tolerances applied to the resulting R and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 (c) Operators formed from fixed sparsity GMRES polynomials, relative drop tolerances applied to the resulting R and P and rela- tive drop tolerances applied to the resulting Acoarse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Figure 1: Eigenvalue distribution of A−1 coarseS on the second level of a pure streaming problem, where S is the exact Schur complement formed from the ideal restrictor and prolongator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the exact coarse-grid matrix) and Acoarse is the approximate coarse grid matrix formed from our approximate ideal restrictor and prolongator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The colours correspond to different GMRES polynomial orders for approximating A−1 ff , with m = 1, m = 2, m = 3 and m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 11 As mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1, the low-order polynomials we use and the fixed sparsity described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 may impact the operators in our multigrid hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given this we examine the impact of our approximations to A−1 ff and the resulting operators by considering the spectrum of A−1 coarseS and Acoarse, where Acoarse is the coarse matrix formed from our approximate operators and S is the exact coarse grid matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As an example, we use AIRG on a pure streaming problem with an unstructured grid, with two levels of uniform refinement in angle (giving 16 angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1 plots the the spectrum of A−1 coarseS and ideally it should approach one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1a shows the result of constructing our coarse grid matrix with increasing orders of our GMRES polynomial, but without fixing the sparsity of the matrix powers, so we are using q0(Aff), q1(Aff), q2(Aff) and q3(Aff) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see increasing the order increases the accuracy of our resulting coarse grid matrix, as would be expected, with the eigenvalues converging to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The radius of a circle that bounds the eigenvalues reduces from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1760 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0072 with m = 1 to m = 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1b shows the result of introducing both the fixed sparsity matrix powers discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 and introducing relative drop tolerances to the resulting R and P operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For the restrictor we drop any entry in a row that is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 times the maximum absolute row entry, and for the prolongator we keep only the largest entry in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see these approximations are detrimental to Acoarse compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1a, with the eigenvalues further from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The bounding circle at all orders has greater radius, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2533 for m = 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0913 for m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1c shows the results from using the same fixed sparsity matrix powers and drop tolerances on R and P while also introducing a relative drop tolerance on the resulting Acoarse, where we drop any entry in a row that is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 times the maximum absolute row entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We again see this further degrades our approximate coarse grid matrix, with the effect of increasing the order of our GMRES polynomials diminished;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the bounding circle goes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3539 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3532 with m = 1 to m = 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It is clear that introducing additional sparsity into our GMRES polynomials and resulting operators degrades our coarse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We examine this further by plotting the smallest eigenvalues of Acoarse and S in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In the limit of ideal operators we know the near-nullspace vectors are preserved, but we would like to verify that with an approximate ideal restrictor and approximate ideal prolongator this is still the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2a that the GMRES polynomials with m = 4 and fixed sparsity does an excellent job capturing the smallest eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Furthermore introducing both fixed sparsity to the GMRES polynomial and drop tolerances on R and P results in reasonable approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 2b that introducing the drop tolerances on the resulting Acoarse results in small eigenvalues that do not match the exact coarse grid matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These results indicate that our low-order GMRES polynomials with fixed sparsity and the introduction of drop tolerances to our approximate ideal R and P results in an excellent approximation for the coarse grid streaming operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It is well known that introducing additional sparsity to multigrid operators can harm the resulting operators, and it is clear from these results that introducing a drop tolerance to our coarse grid matrix has the biggest impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In a multilevel setting however, we find that doing this often result in the best performance, with the slight increase in iterations balanced by the reduced complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' care must be taken to not make the drop tolerance too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CF splitting All multigrid/multilevel methods require the formation of a hierarchy of “grids”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' LDU methods and reduction multigrids like in this work require the selection of a subset of DOFs defined as “fine” and “coarse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For asymmetric linear systems, CF splitting algorithms often result in coarse grids with directionality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', they result in a semi- coarsening), typically through heuristic methods that identify strong connections in matrix entries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g, see [71]) with algorithms like CLJP, PMIS, HMIS, etc [72]) or through compatible relaxation [73, 74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We would like the CF splitting to produce a well-conditioned Aff on each level without giving a large grid or operator complexity across the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The effectiveness of some of the approximations used in the literature (described in Section 4) for A−1 ff also clearly depend on the sparsity of Aff produced by a CF splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Previous works have used various CF splittings, including those that produce a maximally-independent set, giving a diagonal Aff that is easily inverted [76];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' or if a block-independent set is generated then Aff is block-diagonal and the blocks can be inverted directly [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With a more general CF splitting [52] used ILU factorisations to approximate A−1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [79] produced CF splittings specifically for reduction multigrids and LDU methods that are targeted at producing a diagonally dominant Aff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='042 4 3 2 1 0 1 2 3 4 #10 -3 (a) The × are eigenvalues of Acoarse with fixed sparsity GMRES polynomial, the ∗ are with fixed sparsity GMRES polynomial and relative drop tolerances applied to the resulting R and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='042 4 3 2 1 0 1 2 3 4 #10 -3 (b) The .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' are eigenvalues of Acoarse with fixed sparsity GMRES polynomial, relative drop tolerances applied to the resulting R and P and relative drop tolerances applied to the resulting Acoarse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Figure 2: Eigenvalue distribution for the smallest eigenvalues of Acoarse and S on the second level of a pure streaming problem, where S is the exact Schur complement formed from the ideal restrictor and prolongator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the exact coarse-grid matrix) whose eigenvalues are denoted with the black “o”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The red symbols are the eigenvalues of Acoarse, which is the approximate coarse grid matrix formed from our approximate ideal restrictor and prolongator given a GMRES polynomial approximation of A−1 ff with m = 4 and different sparsification applied to operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In this work we use traditional CF splitting algorithms (like in [28]) as we find they perform well enough and parallel implementations are readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Section 7 presents the results from using the lAIR implementation in hypre and we found that using the Falgout-CLJP algorithm in hypre resulted in good CF splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In order to make fair comparisons, we show results from using AIRG with the same algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Work estimates One of the key metrics we use to quantify the performance of the iterative methods tested is the number of Work Units (WUs) required to solve our linear systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is a FLOP count scaled by the number of FLOPs required to compute a matvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We present several different WU calculations, each of which is scaled by a different matvec FLOP count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is in an attempt to show fair comparisons against other multigrid methods, along with source iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To begin, we must first establish a FLOP count for all the different components of our iterative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We begin with our definition of the Cycle Complexity (CC) of AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The CC is the amount of work performed during a single V-cycle, scaled by the number of nnzs in the top-grid matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our calculation of the CC includes the work performed during smoothing and grid-transfer operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we use our definition of CC and WUs in all the results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We define the work required to compute a matvec with our matrices on each level as {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='}l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For an assembled matrix we set this as the nnzs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This assumes fused-multiply-add (FMA) instructions are available and hence the cost of multiplying by Aff for example is nnzs rather than 2×nnzs (this cost scales out of the CC anyway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we consider a general linear system, Ax = b, the FLOP count for performing a single V-cycle with lmax levels of AIRG is given by FLOPAIRG V = { ˆA −1}lmax + l=lmax−1 � l=1 vup{ ˆA −1 ff }l + vup{Aff}l + {Afc}l + {R}l + {P}l, (31) where vup = 2 is the number of up F-point smooths and we perform one application of a GMRES polynomial approx- imation of ˆA −1 as a coarse grid solve on l = lmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In Section 7, we also show the results from using lAIR in hypre with FCF-Jacobi smoothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' by default the CC output by hypre doesn’t include all work associated with smoothing, residual calculation, etc, and hence we recompute 13 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Due to the use of FCF-Jacobi, the result of Afcxc during the F smooths cannot be cached (similarly for the C-point smooths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We therefore compute FLOPhypre V = { ˆA −1}lmax + l=lmax−1 � l=1 vup(2nl F + nl C) + vup � 2 � {Aff}l + {Afc}l� + {Acf}l + {Acc}l� + {R}l + {P}l, (32) where nl F and nl C are the number of F and C-points on level l, respectively and given we only use one FCF-Jacobi as our smoother on each level vup = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cycle complexity (for either hypre or AIRG) is then given by CC = FLOPV {A}1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (33) Any matvec that involves scatter should be computed matrix-free and we denote that with an “mf” subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given our sub-grid scale discretisation, we need to account for the cost of computing the source on the rhs of (11), the fine-scale solution Θ and the addition of the coarse and fine scale solutions to form ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These are given by FLOPsource = {B}mf + { ˆD −1}, FLOPSGS = {C}mf + { ˆD −1}, FLOPψ = NDDOFs (34) The FLOP count of one iteration of our angular preconditioner (14) is given by FLOPangle = 2 × NCDOFs + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 × {Ddiff}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (35) We investigated using AIRG to invert the diffusion operator, but found it difficult to beat the default boomerAMG implementation in hypre (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', not lAIR), which is unsurprising given hypre has been heavily optimised for such elliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 comes from the cycle complexity of running boomerAMG on a heavily refined spatial grid and as might be expected we see the cycle complexity plateaus to around this value (hence we have an upper bound on work on less refined grids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We must now quantify the cost of our matrix-free matvecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 there are numerous ways we could compute such a matvec, depending on how much memory we have available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Section 3 discussed that we store the streaming/removal operator, MΩ to precondition with and hence we use that in our matvec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The additional cost therefore comes with scattering (which we assume is isotropic in this work) and hence we have {A − B ˆD −1C}mf = {MΩ} + � i∈cg nodes 2 × δ(i) × NCDOFs(i)+ � i∈dg nodes δ(i) × (2 × NDDOFs(i) + 2 × nnodes + 1) + � e∈eles 3 × δ(i) × { ˆD −1 e } (36) where nnodes is the number of spatial nodes on our DG elements (3 in 2D or 4 in 3D with tri/tets),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' { ˆD −1 e } is the number of non-zeros in our stored block approximation on a given element (the factor of 3 comes from one application of ˆD −1 and one of BΩ and CΩ which have the same sparsity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this sparsity depends on the angles present on each DG node, but is at a maximum when uniform angle is used and for each angle present on all nodes of an element we have a nnodes × nnodes block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' N*DOFS(i) is the number of DOFs on an individual (CG or DG) spatial node i and δ(i) is 1 or 0 depending on the presence of scatter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for a CG node this is zero if every element connected to node i has a zero scatter cross-section, and one otherwise, for a DG node this is zero if the element containing node i has a zero cross-section, one otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The calculation in (36) includes the work required to map to/from Legendre space on both our coarse (CG) and fine (DG) spatial meshes in order to apply the scatter component in A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similar expressions are used for the individual {B}mf and {C}mf in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We now have all the components required to calculate our WUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We begin with a simple definition, where we use AIRG as a preconditioner on the assembled matrix, A − B ˆD −1C, that could include scatter and is hence non-scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If nits is the number of outer GMRES iterations performed then the total FLOPs are FLOPfull = nits � {A − B ˆD −1C} + FLOPV � + FLOPsource + FLOPSGS + FLOPψ, (37) 14 and hence the WUs WUsfull = FLOPfull {A − B ˆD −1C} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (38) If instead we use the additively preconditioned iterative method defined in Section 3 along with our matrix-free matvec, our total FLOPs are FLOPmf = nits � {A − B ˆD −1C}mf + FLOPV + FLOPangle � + FLOPsource + FLOPSGS + FLOPψ, (39) We then scale these FLOP counts in several different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Firstly there is the WUs required to compute a matrix-free matvec of our sub-grid scale discretisation, namely WUsmf = FLOPmf {A − B ˆD −1C}mf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (40) In order to make rough comparisons with a traditional DG FEM source iteration method, we can scale by the work required to compute a matrix-free matvec with a DG FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In order to not unfairly disadvantage a DG discretisation, we assume the DG streaming operator is stored in memory, and hence the FLOPs required to compute a single DG matvec is FLOPDG = 5 3{ ˆD −1} + � i∈dg nodes δ(i) × (2 × NDDOFs(i) + nnodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (41) The factor of 5/3 comes from the jump terms (in 2D) that are not included in the nnzs of our sparsified DG matrix ˆD −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The work units scaled by this quantity are therefore WUsDG = FLOPfull FLOPDG or FLOPmf FLOPDG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (42) Again we note that with scattering we have {A − B ˆD −1C}mf ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 × FLOPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Results Outlined below are several examples problems in both the streaming and scattering limits, designed to test the per- formance of AIRG and our additively preconditioned iterative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We solve our linear systems with GMRES(30) to a relative tolerance of 1× 10-10, with an absolute tolerance of 1× 10-50 and use an initial guess of zero unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We should note that we use AIRG (and lAIR) on our matrices without relying on any (potential) block structure, for example in the streaming limit with uniform angle we could use our multigrid on each of the angle blocks separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also don’t scale our matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for example we could view a diagonal scaling as preconditioning the outer GMRES iteration and/or the GMRES polynomials in our multigrid, but we did not find it necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When using AIRG we perform zero down smooths and two up F-point smooths (our C-points remain unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' On the bottom level of our multigrid we use one Richardson iteration to apply a GMRES polynomial approximation of the coarse matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We use a 1-point prolongator which computes W and then keeps the biggest absolute entry per row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Unless otherwise noted we use 3rd order (m = 4) GMRES polynomials with fixed sparsity as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We only use isotropic scatter in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For both AIRG and lAIR, we use the row-wise infinity norm to define any drop tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When using the lAIR implementation in hypre, we use zero down smooths and one iteration of FCF-Jacobi for up smooths, while on the bottom level we use a direct-solve, as we found these options resulted in the lowest cost/best scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We searched the parameter space to try and find the best values for drop tolerances, strength of connections, etc when using lAIR and AIRG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' these searches were not exhaustive however, and it is possible there are more optimal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' All timing results are taken from compiling our code, PETSc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='15 and hypre with “-O3” optimisation flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We compare timing results between our PETSc implementation of AIRG and the hypre implementation of lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we try and limit the impact of different implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given this, when calculating the setup time we exclude the CF splitting time, the time required to drop entries from matrices (as the PETSc interfaces require us to take copies 15 of matrices), and to extract the submatrices Aff, Afc, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These should all be relatively low cost parts of the setup, are shared by both AIRG and lAIR and should scale with the nnzs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The setup time we compare is therefore that required to form the restrictors, prolongators and coarse matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Also we should note that with AIRG our setup time is an upper bound, as we have not built an optimised matmatmult for building our fixed sparsity GMRES polynomials (we know the sparsity of the matrix powers is the same as the two input matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we compute a standard matmatmult in PETSc and then drop entries (although we do provide a flop count for a matmatmult with fixed sparsity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When timing lAIR, we use the PETSc interface to hypre and hence we run two solves (and set the initial condition to zero in both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The hypre setup occurs on the 0th iteration of the first solve, so a second solve allows us to correctly measure just the solve time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We only show solve times for problems with pure streaming, as our implementation of the P0 matrix-free matvec with scattering is not well optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' All tabulations of memory used are scaled by the total NDOFs in ψ in (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', NDOFs=NCDOFs + NDDOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Included in this figure is the memory required to store the GMRES space, both additive preconditioners (if required) and hence the AIRG hierarchy, ˆA −1 ff , separate copies of Aff, Afc, Acf and Acc, the diffusion operator and temporary storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We do not report the memory use of hypre, although given the operator complexities it is similar to AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG on the matrix A − B ˆD −1C In this section, we build the matrix A − B ˆD −1C and use this matrix as a preconditioner, applied with 1 V-cycle of either AIRG or lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we do not use the iterative method described in Section 3, nor do we use the matrix-free matvec described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As mentioned using an iterative method that relies on the full operator is not practical with scattering given the nonlinear increase in nnzs with angular refinement, but we wish to show our methods are still convergent in the scattering (diffuse) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Instead, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 shows the results from using the iterative method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our test problem is a 3× 3 box with a source of strength 1 and size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 in the centre of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We apply vacuum conditions on the boundaries and discretise this problem with unstructured triangles and ensure that our grids are not semi-structured (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', we don’t refine coarse grids by splitting elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We use uniform level 1 refinement in angle, with 1 angle per octant (similar to S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Pure streaming problem For the pure streaming problem we set the total and scatter cross-sections to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To begin, we examine the performance of AIRG and lAIR with spatial refinement and a fixed uniform level 1 angular discretisation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e, with 4 angles in 2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Table 1 shows that using distance 1 lAIR in this problem, with Falgout-CLJP CF splitting results in growth in both the iteration count and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We could not find a combination of parameters that results in scalability with lAIR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' increasing the number of FCF smooths to 3 results in an iteration count with similar growth, namely 15, 14, 14, 17, 19 and 22, but with a cycle complexity at the finest spatial refinement of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 and hence 271 WUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Even using distance 2 lAIR didn’t result in scalability, as shown in Table 2 where we can see a slightly decreased iteration count, with higher cycle and operator complexities, resulting in a similar number of WUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Using both distance 2 lAIR and 3 FCF smooths still results in growth, with iteration counts of 15, 13, 14, 16, 17 and 19, with a cycle complexity of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 and hence 323 WUs at the finest spatial refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes NDOFs nits CC Op Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 106 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='77 116 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 138 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 159 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 189 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='16 225 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 Table 1: Results from using distance 1 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025 and a strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 16 CG nodes NDOFs nits CC Op Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='58 112 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='85 116 28 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='04 141 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='16 164 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='26 192 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='32 219 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 Table 2: Results from using distance 2 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025 and a strong R threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also tried decreasing the strong R threshold to 1 × 10-7 in case some neighbours were being excluded, but this resulted in very little change in the iteration count, while the number of nnzs in the restrictor (and hence the setup time) grew considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Preliminary investigation suggests we would need to include neighbours at greater distance than two (along with only F-point smoothing, rather than FCF), but this increases the setup cost considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also note that using nAIR (at several different orders) in this case results in a similar iteration count (even with diagonal scaling of our operators);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is likely due to the lack of lower triangular structure in our discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='96 83 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='48 79 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='88 70 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='16 73 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='36 82 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='48 84 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 Table 3: Results from using AIRG with m = 4 and without fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Tables 3 & 4 however shows that using AIRG with a third-order GMRES polynomial and Falgout-CLJP CF splitting results in less work with smaller growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Using GMRES polynomials without fixed sparsity requires 84 WUs at the highest level of refinement, compared to 75 with fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The work in Table 3 has plateaued, however the work with fixed sparsity is growing slightly with spatial refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Compared to lAIR, we see that AIRG with sparsity control needs three times less work to solve our pure streaming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Rather than use our one-point ideal prolongator, we also investigated using W with the same drop tolerances as applied to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With fixed-sparsity this resulted in 11, 10, 9, 9, 9 and 10 iterations, with 67, 68, 66, 68, 70, and 78 WUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The slight increase in cycle complexity is compensated by using one fewer iterations at the finest level, but overall we see similar work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In an attempt to ascertain the effect of using our one-point approximation to the ideal prolongator, we also constructed a classical one-point prolongator as in [28, 26] where each F-point is injected from its strongest C-point neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With fixed-sparsity this results in the same number of iterations and WUs as in Table 4, which confirms that the classical prolongator is not responsible for the poor performance of lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The benefit of using a classical one-point prolongator in this fashion is we require one fewer matmatmults on each level of our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For the rest of this paper we use the one-point approximation to the ideal operator, but it is clear there is a range of practical operators (with different sparsification strategies) we could use with AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Table 4 shows we can solve our streaming problem with the equivalent of approximately 18 DG matvecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also see that the plateau in the operator complexity results in almost constant memory use, at 10 copies of the angular flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We should also note that we can use AIRG as a solver, rather than as a preconditioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We see the same iteration count in pure streaming problems and the lack of the GMRES space means we only need memory equivalent to approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5 copies of the angular flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is the amount of memory that would be required to store just a DG streaming operator in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This shows that our discretisation and iterative method are low-memory in streaming 17 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='97 67 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 10 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 62 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 10 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='87 60 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='17 67 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='36 69 16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 75 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 Table 4: Results from using AIRG with m = 4 and fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cost of a solve must also be balanced by the cost of the setup of our multigrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our setup involves computing our GMRES polynomial approximations, followed by standard AMG operations, namely computing matrix-matrix products to form our restrictors/prolongators and coarse matrices on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3 begins by showing the relative amount of work required to compute Z for AIRG and distance 1 lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For AIRG this is the sum of FLOPs required to compute ˆA −1 ff and those to compute −Acf ˆA −1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For lAIR, it is more difficult to calculate a FLOP count for the small dense solves which locally enforce RA = 0, as it is dependent on the BLAS implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' instead we take the size of each of the n × n dense systems and compute the sum of n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we do not try and compare an absolute measure of the work required by both (we have timing results below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3 instead shows that the growth in work for both AIRG with fixed sparsity and lAIR begins to plateau with grid refinement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG without fixed sparsity however grows with refinement, showing the necessity of the sparsity control on the matrix powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4a shows that the cost of computing our GMRES polynomial approximations to ˆA −1 ff with fixed sparsity is constant, at around 8 FLOPs per DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The plateauing growth seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3 therefore comes from the matmatmult required to compute −Acf ˆA −1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The FLOP count with fixed sparsity relies on a custom matmatmult algorithm that assumes the same sparsity in each component of the matmatmult, as in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Computing the matrix powers with a standard matmatmult and then dropping any fill-in results in an almost constant number of FLOPs per DOF, at around 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6, which is convenient as this makes it less necessary to build a custom matmatmult implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we do not fix the sparsity of our polynomial approximations we can see growth in the cost, due to the increasingly dense matrix powers, as might be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4b shows that the cost of our setup with fixed sparsity plateaus, and the total cost of our setup plus solve grows, due to the increase in work during the solve shown in Table 4, with 29% growth from lowest to highest refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Without fixed sparsity we see less growth in Table 3 from the solve, but higher growth in the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This results in similar growth in the total work, but the fixed sparsity case uses approximately 30% fewer FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5 shows results for the solve, Z, setup and total time taken to solve our system with AIRG and lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5a that the results match those in Tables 1-5, with AIRG with Falgout-CLJP coarsening and fixed sparsity giving the lowest solve times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found a difference in the efficiency of our PETSc implementation vs the hypre implementations however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we modify the parameters used with AIRG and increase the work required to solve to roughly match that distance 1 lAIR in this problem, we found that there was a factor of almost 2 difference in the solve times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5b shows the total cost of computing the approximate ideal restrictor, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For AIRG we also show the time for computing the GMRES polynomial approximation to A−1 ff separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see for AIRG with fixed sparsity there is slight growth in the time to compute our polynomials, with much greater growth without fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the FLOP count in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4a we would expect the time with fixed sparsity to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The further growth we see in the time to compute Z comes from the matmatmult to compute −AcfA−1 ff , which Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4b suggests should plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also see growth in the time taken to compute Z with lAIR, with higher growth for distance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Again Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3 shows the work estimate plateauing with spatial refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' At the higher spatial refinements, the Z with AIRG, Falgout- CLJP coarsening and fixed sparsity costs more to compute than distance 1 lAIR, but less than distance 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' An AIRG implementation that takes advantage of the shared sparsity of the matrices when forming ˆA −1 ff could reduce this setup further, with a reduction in the FLOPs required (by approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 20% as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4a), and also in the cost of any 18 102 103 104 105 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2 CG Nodes Relative work (a) The × is for AIRG with fixed sparsity, × is for AIRG without fixed sparsity and ⊗ is for lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Figure 3: Sum of FLOPs required across all levels to compute Z, scaled to the NDOFs, and then relative to the work required on the least refined spatial grid, for AIRG with m = 4 (see Table 4) and distance 1 lAIR (see Table 1) with Falgout-CLJP in a 2D pure streaming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The relative scaling is done separately for each line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' they do not all cost the same at the coarsest resolution, we provide timings in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' symbolic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the substantially improved convergence shown in Table 4 when compared to distance 2 lAIR in Table 2, this shows AIRG is an effective and relatively cheap way to compute approximate ideal operators in convection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5c shows the setup times (which include the times to compute Z from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5b) and we can see that lAIR has the cheapest overall setup, with fixed sparsity AIRG coming in the middle, and AIRG without fixed sparsity giving the highest setup times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We expect AIRG without fixed sparsity to be increasing given the increased cost of computing the GMRES polynomial with spatial refinement, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 4a, but we see increased setup time with AIRG with fixed sparsity and distance 1 lAIR, whereas the work estimates in Figures 3 and 4b again suggests the setup times should plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found that the time taken to compute the matmatmults used to build the transfer operators and coarse matrices is increasing more than the FLOP count would imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Of course a FLOP count is not necessarily perfectly indicative of the time required in a matmatmult given the symbolic compute and memory accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It is possible more efficient matmatmult implementations are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 5d shows the total time taken to setup and solve with our multigrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Two of our AIRG results, namely using GMRES polynomials without and without fixed sparsity manage to beat lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This helps show the promise of GMRES polynomials as part of an AIR-style multigrid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' even without fixed sparsity they can be competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The fixed sparsity AIRG however results in a considerably decrease in total time, taking approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 3× less time than distance 1 lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Even if we try and remove the effect of implementation differences between lAIR and AIRG discussed above, by equating the solve time of lAIR with an AIRG result that requires similar WUs, fixed sparsity AIRG still has a total time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='65 × that of distance 1 lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also examined using SAIs to build an approximation of A−1 ff with the fixed sparsity of Aff, like in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found approximations of A−1 ff computed with SAI comparable to our fixed sparsity GMRES polynomials with m = 4, with identical convergence behavior compared to the fixed sparsity AIRG shown in Table 4 (except with the first spatial refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With spatial refinement using SAI had 11, 10, 9, 10, 10 and 11 iterations and hence the same work units and solve times as AIRG given the matching fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' However we found it was more expensive to setup SAIs on each level when compared to our GMRES polynomials, by approximately 2×;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we used the ParaSails implementation in hypre for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In particular the fixed sparsity GMRES polynomials require fewer FLOPs as the nnzs in Aff grow compared to fixed sparsity SAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If nF is the number of F-points, we require m matvecs which scale linearly with the nnzs and a single QR factorisation of size nF × (m + 1), which doesn’t depend on the nnzs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If m > 2, we must also compute m − 2 fixed sparsity matrix powers at cost (m − 2)srscnF, where sr and sc are the average number of nnzs per row and column of Aff, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we assume s = sr ≈ sc then the cost of computing 19 102 103 104 105 8 10 12 14 CG Nodes FLOPs per DOF (a) The × is the cost of computing ˆA−1 ff for AIRG with fixed spar- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the × is with fixed sparsity computed with a standard mat- matmult followed by dropping entries and the × is without fixed sparsity 102 103 104 105 50 100 150 CG Nodes FLOPs per DOF (b) The dashed × is the cost of the setup for AIRG with fixed sparsity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the × is without fixed sparsity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the × is the setup plus solve with fixed sparsity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the × is without fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Figure 4: Sum of FLOPs required across all levels during the setup and solve, scaled to the NDOFs, for AIRG with m = 4 and Falgout-CLJP in a 2D pure streaming problem (see Tables 4 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the matrix-powers can be written as s(m − 2) × nnzs(Aff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In comparison, the SAI algorithm with the fixed sparsity of Aff requires solving nF (local) least-squares problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we just consider the required nF QR factorisations of size s × s, this requires a FLOP count of s2 × nnzs(Aff) (we have dropped the constants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For many problems a low polynomial order is acceptable and hence (m − 2) ≪ s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is particularly true on lower multigrid levels where the average row or column sparsity can grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' There may be a scale, however, at which the extra local cost of setting up SAIs is balanced by the extra communication required by our fixed sparsity GMRES polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With m = 4 we require the communication associated with four matvecs, a single all-reduce and computing A2 ff, whereas SAI only requires that of A2 ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' One of the benefits of using our GMRES polynomials however is that we can decrease the amount of communication required by decreasing the polynomial order below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given this, we examine the role of changing the GMRES polynomial order with AIRG and fixed sparsity, from zero to four (m = 1 to m = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the zeroth and first order polynomials implicitly have fixed sparsity) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see the zeroth order polynomial is very cheap to construct, but results in the highest total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is because the iteration count is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The increasingly higher order polynomials take longer to setup, although the difference between successive orders decreases, due to the shared sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With increasing polynomial order, on the most refined spatial grid we have 40, 16, 12, 11 and 11 iterations to solve, with cycle complexities of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='67, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='81, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='84 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='85, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see that all the polynomials between first and third order result in similar total times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the decreased cost of setup for the lower orders is balanced by the increase in iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We see however very similar growth in iteration count with spatial refinement with the first through fourth order polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' for example with first order GMRES polynomials (m = 2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 6 we have 15, 14, 12, 14, 14 and 16 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This gives a higher overall amount of work than with m = 4 (up to approximately 100 WUs at the highest refinement), but given the similar growth in iteration count and reduced communication in parallel during the setup, requiring only two matvecs and a single all-reduce, this may be a good choice in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the results with spatial refinement above, we chose to examine the role of angular refinement with AIRG with fixed sparsity and m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Table 5 shows that with 3 levels of angular refinement on the third refined spatial grid, the iteration count increases slightly from 9 to 11, but the cycle and operator complexity are almost identical and hence we have fixed memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We do not show the timings as they scale as would be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Table 6 shows that distance 1 lAIR requires a fixed amount of work with angular refinement, but this is roughly twice that of AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 20 CG nodes Angle lvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsfull WUsDG Memory 2313 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='87 60 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2313 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='88 65 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2313 3 1× 106 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='87 70 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 Table 5: Results from using AIRG with m = 4 and fixed sparsity on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025 with different levels of angular refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes Angle lvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' NDOFs nits CC Op Complx WUsfull WUsDG Memory 2313 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 138 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 2313 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='36 133 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2313 3 1× 106 28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='35 133 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 Table 6: Results from using distance 1 lAIR in hypre on a pure streaming problem in 2D with CF splitting by the hypre implementation of Falgout- CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, drop tolerance on A of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0075 and R of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='025 and a strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='25 with different levels of angular refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Scattering problem To test the performance of AIRG with diffusion, we set the total and scattering cross-section to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Tables 7 and 8 show that both distance 1 and distance 2 lAIR, respectively, perform similarly, with approximately 3× growth in WUs from the least to most refined spatial grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Both the cycle and operator complexities have plateaued though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes NDOFs nits CC Op Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 77 49 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 112 69 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 134 82 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 153 93 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 183 110 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 238 144 Table 7: Results from using distance 1 lAIR in hypre on a pure scattering problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG performs simliarly to lAIR in this problem, as shown in Tables 9 without fixed sparsity and 10 with fixed sparsity, with around 3× growth and similar number of WUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG with fixed sparsity results in slightly lower operator complexities, but both methods result in memory use of around 20 copies of the angular flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is higher than that in the streaming limit as the full matrix with scattering and a uniform angular discretisation at one level of refinement has 4× the nnzs as that in the streaming limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' There is not a great deal of difference in the cycle complexity between AIRG with and without fixed sparsity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is because the strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 results in a very sparse version of Aff being used to construct the GMRES polynomials (and hence very little fill-in relative to the top grid matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can decrease the strong R tolerance to decrease the iteration count (in both AIRG and lAIR), but the nnzs in (or equivalently the number of neighbours used to construct) Z grows considerably, as might be expected with scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7 shows the timing results from AIRG and lAIR in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Again we see a curious implementation difference in solve times in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7a, as both lAIR and AIRG require simliar number of WUs, but the hypre imple- mentation of lAIR requires roughly twice the time to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7b shows that the time to compute the GMRES polynomial for AIRG is largely constant, with the time to compute Z again between distance 1 and distance 2 lAIR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is also true for the total setup time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7d shows that our AIRG implementation is the cheapest method overall, taking around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× the amount of time as lAIR to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we again equate the solve time be- tween lAIR and AIRG given the similar amount of work required, lAIR and AIRG perform similarly, each requiring 21 CG nodes NDOFs nits CC Op Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 80 51 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 117 72 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 133 81 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 150 91 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 178 108 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 225 136 Table 8: Results from using distance 2 lAIR in hypre on a pure scattering problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 68 43 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 88 54 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 122 74 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 155 94 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 175 106 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 232 140 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 Table 9: Results from using AIRG with m = 4 and without fixed sparsity on a pure scattering problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 µs total time per DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 8 shows the results from changing the GMRES polynomial order and similar trends to that in the streaming limit can be seen, namely the 0th order polynomial is very cheap to setup, but results in the highest total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Indeed the 0th order polynomial did not converge at the two highest spatial refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The first through fourth order polynomials all result in similar total times, with 62, 61, 60 and 64 iterations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This result indicates that the higher polynomial order does not necessarily help decrease the iteration count with scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This is because the strong R threshold is so high;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' decreasing this makes the effect of the polynomial order (and the fixed sparsity) much more pronounced, but we found the lowest overall total times by allowing heavy dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given using the full matrix is not scalable with angular refinement, we don’t show convergence results in that case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' instead the next section uses the iterative method defined in Section 3 on this scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Additively preconditioned iterative method In this section we show the performance of the iterative method from Section 3 in the scattering limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The results in this section are not designed to show the performance of a standard DSA method with different scattering ratios, optical cell lengths, etc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we appeal to the wealth of literature on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Instead we wish to show that the additive combination of our preconditioners is effective and that multigrid methods can be used to invert these operators scalably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As discussed, this means forming the streaming/removal operator MΩ, a CG diffusion operator Ddiff and the streaming/removal components BΩ and/or CΩ, all of which can be done scalably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We use 1 V-cycle of AIRG with the same drop/strong tolerances as in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 to apply M−1 Ω and 1 V-cycle of boomerAMG (with default options) to apply D−1 diff per outer GMRES iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For our iterative method to be effective, 1 V-cycle of both methods must reduce the error by a fixed amount with space/angle refinement (which is equivalent to a solve with a fixed tolerance taking a fixed amount of work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can assume that multigrid methods such as boomerAMG can invert the diffusion operator with fixed work (we also scale the diffusion operator by its inverse diagonal prior to use), but in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 we only showed that AIRG can invert the streaming operator with fixed work in the solve, rather than the streaming/removal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Thankfully the removal term results in a better conditioned matrix given extra term on the (block) diagonals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the streaming limit is the most difficult to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9 shows (part of) the spectrum of the streaming operator vs the streaming/removal 22 CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsfull WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 67 43 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 88 54 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 122 74 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 144 87 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 196 118 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 224 135 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 Table 10: Results from using AIRG with m = 4 and fixed sparsity on a pure scattering problem in 2D with CF splitting by the hypre implementation of Falgout-CLJP with a strong threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9, drop tolerance on A of 1× 10-4, R of 1× 10-2 and strong R threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' operator for the 2D source problem with the third refined grid, level one angular refinement and total cross-section of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can see that the smallest eigenvalues of the streaming/removal operator are (slightly) further from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The convergence of AIRG relies on the convergence of our GMRES polynomial approximations to A−1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We can also see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9 that the eigenvalues of Aff are more compact than that of the full operators, confirming that the CF splitting is helping produce a better conditioned Aff in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We know that our operators are non-normal, so the spectrum does not completely determine the convergence of our GMRES polynomials [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9 also plots the field of values (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', the numerical range) of our operators, given by F (A) = {x∗Ax | xx∗ = 1, x ∈ Cn}, (43) which is a convex set that contains the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' To give some insight into the convergence of the GMRES polynomials, we define µ to be the distance from the origin, or µ = min z∈F (A) |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (44) [80] show that (see also [63]) if the field of values doesn’t contain the origin, β ∈ (0, π/2) such that cos(β) = µ/||A|| and the Hermitian part of A, namely (A + A∗)/2 is positive definite then the residual at step m is bounded by ||rm|| ≤ ||r0|| � 2 + 2√ 3 � (2 + γβ)γm β , (45) where γβ = 2 sin � β 4 − 2β/π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (46) We confirmed numerically that none of our operators or their fine-fine sub-matrices touch the origin (and hence the field of values are all in the right-half of the complex plane) and that their Hermitian parts are positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For the Aff component of the streaming operator on the top grid, pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9 we found that with m = 4, (45) gives ||rm|| ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='41||r0||, while for the Aff component of the streaming/removal operator we have ||rm|| ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='40||r0|| (the disk bound in [81] gives a similar conclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These bounds are not particularly tight, but they do indicate that a 3rd order GMRES polynomial should result in a smaller residual for the streaming/removal operator and hence we would expect AIRG to perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We should note that as µ → 0, β gets closer to π/2 and the asymptotic convergence factor, γm β → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In general this means the further the minimum field of values is from the origin, the better the convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' this is also demonstrated by considering the disk bound in [81], given by |δ/c| = (1−cos(β))/(1+cos(β)) < γβ, where δ and c are the radius and centre of a disk, respectively, that covers F (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This helps explain why using GMRES polynomials to approximate Aff can be effective even when GMRES polynomial preconditioning of the full operators may not be;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 9a shows that the field of values for the streaming operator almost touches the origin, with µ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 × 10-5 and hence single- level GMRES polynomial preconditioning would likely not be effective in this problem (this is backed by numerical experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we find considerable growth in the iteration count with refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This hints at the importance of combining GMRES polynomials with a reduction multigrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 23 CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsmf WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 33 61 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 36 67 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 38 69 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 39 72 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 39 73 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 39 73 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 Table 11: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes Angle lvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsmf WUsDG Memory 2313 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 38 69 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2313 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 40 71 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2313 3 1× 106 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 41 73 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 Table 12: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 in 2D with angle refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We see in Table 11 our additively preconditioned iterative method is effective with a total and scatter cross-section of 10, with the iteration count growing from 23 to a plateau of 26 with spatial refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The work is very close to constant, with fixed iteration count and slight growth in the cycle complexity, even though we used AIRG with fixed sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' As such we didn’t investigate using AIRG without fixed sparsity, as the fixed sparsity was sufficient to give plateauing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This helps confirm the observations above, namely that AIRG is more effective on the streaming/removal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Comparing to the results in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2, it uses roughly 73 DG WUs, compared to around 135 when using either AIRG or lAIR as a preconditioner on the full matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' It also uses less memory at approximately 18 copies of the angular flux, even though we have to store the diffusion operator and several extra temporary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Compared to the pure streaming problem, this method requires approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1× more work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' we can see in Table 11 that this work largely comes from computing the matrix-free matvec with scattering, with 26 iterations at the highest spatial refinement requiring 39 WUsmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The split of work is 26 WUsmf in the matvec required by the outer GMRES, 11 WUsmf to apply the additive preconditioners and around 2 WUsmf to compute the source and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Similarly, Table 13 shows a (lower) constant iteration count with a lower total and scatter cross-section of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the streaming/removal operator is easier to solve, lAIR performs better when used additively to invert MΩ, compared with just the streaming operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With a total and scatter cross-section of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0, we find both distance 1 and 2 lAIR give 30, 33, 31, 34, 36 and 39 iterations with spatial refinement, with similar grid and operator complexities to AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Increasing the number of FCF smooths from 1 to 3 results in an iteration count with less growth, namely 30, 25, 25, 26, 26 and 27 iterations, but the cycle complexity at the finest level of refinement is large at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7, compared with AIRG at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also know from Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 that the iteration count of lAIR grows in the streaming limit, so we do not test lAIR any further as part of our additive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Tables 12 and 14 show that the additive method with AIRG and angular refinement perform well, as the iteration count and the work required is very close to constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Importantly we can also see the memory use is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' These results show that as might be expected, using an (inconsistent) CG DSA can form an effective preconditioner in scattering problems when used with an outer GMRES iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Importantly the combination of a single V-cycle of AIRG used to apply the streaming/removal operator and a single V-cycle of a traditional multigrid on the diffusion operator can be used additively and results in almost constant work with spatial and angular refinement on unstructured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 24 CG nodes NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsmf WUsDG Memory 97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4× 103 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9 27 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 591 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6× 104 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 27 50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 2313 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 28 51 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 9166 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 29 54 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 35784 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='9× 105 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 30 55 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 150063 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2× 106 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 30 55 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 Table 13: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' CG nodes Angle lvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' NDOFs nits CC Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Complx WUsmf WUsDG Memory 2313 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3× 104 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 28 51 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 2313 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5× 105 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 27 49 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='7 2313 3 1× 106 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 29 51 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 Table 14: Results from using additive preconditioning on a pure scattering problem with total and scattering cross-section of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0 in 2D with angle refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The cycle and operator complexity listed are for AIRG on MΩ with CF splitting by Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Conclusions This paper presented a new reduction multigrid based on approximate ideal restrictors (AIR) combined with GMRES polynomials (AIRG) with excellent performance in advection-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Matrix polynomial methods have been used for many years in multilevel methods but we believe we are the first to use GMRES polynomials in this fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Reduction multigrids and LDU methods in particular benefit from using GMRES polynomials, as the improved conditioning of Aff, when compared to A, can allow the formation of good approximate inverses with low polynomial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' This allowed us to easily build both approximate ideal restrictors, approximate ideal prolongators (without the need to compute near-nullspace vectors) and perform F-point smoothing (without the need to compute additional dampening parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' GMRES polynomials share many advantages with other polynomial methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' in particular their coefficients can be computed very simply;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' low-order polynomials don’t require additional work to ensure stability (like in [61, 62]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' explicitly forming approximate matrix inverses is simple and only involves matrix-matrix products or if desired;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' the polynomials can be applied matrix-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' their application is highly parallel with their setup able to use communication- avoiding techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' and they also work well across a range of symmetric and asymmetric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' When applied to the time independent Boltzmann Transport Equation (BTE) we could solve pure streaming prob- lems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', in the pure advection limit) on unstructured spatial grids with space/angle refinement with fixed memory use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The time-independent streaming limit is the most challenging to solve and we found we could either get fixed work in the solve and growth in the setup, or by introducing fixed sparsity into the matrix-powers of our GMRES polynomials, we found fixed work in the setup with growth in the solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found good performance from using between first to fourth order GMRES polynomials on each level of our multigrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Fixing the sparsity of our third- order (m = 4) GMRES polynomials resulted in a fixed FLOP count in the setup, and building an implementation of a matmatmult A = BC where the three matrices share the same sparsity would reduce the implementation costs of our setup for second order polynomials and higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' With fixed sparsity we found at most 20% growth in the work to solve with either 6 levels of spatial refinement or three levels of uniform angular refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' A balance must be struck between the scalability of the solve vs expense of the setup, but we believe this is the first method to show scalable solves with a stable spatial discretisation that doesn’t feature lower-triangular structure in the streaming limit of the BTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We did not spend much effort tweaking parameters and we have found that that we can get better performance in these problems with standard AMG tweaks such as level specific drop tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We also compared AIRG to two different reduction multigrids and found performance advantages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' one where sparse approximation inverses (SAIs) are used to approximate A−1 ff , and the lAIR implementation in hypre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We used ParaSails in hypre to form SAIs with the fixed sparsity of Aff and found almost identical convergence behavior 25 to fixed sparsity AIRG (and hence the same solve time) in the streaming limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We found however that the setup of the SAIs took twice as long as our GMRES polynomials with m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For lAIR, we could not find a set of parameters that resulted in fixed work in the solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Our further investigations suggest the combination of distance 3 or 4 lAIR plus (only) F-point smooths are required to get scalable results with lAIR, but this is not practical given the setup/communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In comparison to distance 1 or 2 lAIR, AIRG took roughly two to three times less work to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Timing the setup showed that computing Z with our third-order GMRES polynomial approximations cost between that of computing Z with distance 1 and 2 lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The total time of our setup at the highest level of spatial refinement matched that of distance 2 lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The total time (setup plus solve) for AIRG was roughly 3× less than lAIR, though implementation differences make this comparison difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then investigated using AIRG and lAIR on the full matrix formed with scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Forming this matrix cannot be done scalably with angular refinement, but we showed that AIRG is applicable in the diffuse limit, performing about as well as lAIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' We then built an iterative method that used the additive combination of two preconditioners applied to the angular flux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 1 V-cycle of AIRG was used to invert the streaming/removal operator and 1 V-cycle of boomerAMG was used to invert a CG diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The streaming/removal operator is easier to solve than the streaming operator and hence we found the work in the solve plateaued with fixed sparsity AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Using distance 1 or 2 lAIR to invert the streaming/removal operator resulted in cycle complexities over twice that of fixed sparsity AIRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the performance shown here it would be worth investigating the use of AIRG as part of a standard DG FEM source iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' preliminary work reveals AIRG performs similarly when used with DG streaming or streaming/removal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The only remaining consideration is how we can apply this method with our previously developed angular adap- tivity and the parallel performance, which we will investigate in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' AIRG should be performant in parallel, as the entire multigrid hierarchy can be applied with only matrix-vector products (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=', no reductions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The CF split- ting algorithm used has a parallel implementation available in hypre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The GMRES polynomial coefficients on each level must be computed once during the setup and can be trivially stored for multiple solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Furthermore given our use of low-order GMRES polynomials in AIRG, we found a single step method based on a QR factorisation of the Krylov basis could be used to generate these coefficients stably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' In parallel we could therefore use a tall-skinny QR and generate the coefficients of a polynomial of order m − 1 with m matvecs and a single all-reduce on each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For zero and first order polynomials, there is no other communication required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' For second order and higher, the remainder of the GMRES polynomial setup uses m − 2 matmatmults and matmatadds to compute matrix-powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' If we impose the aforementioned fixed sparsity we only need to communicate the required off-processor rows of Aff in the matmatmults once in order to compute those matrix powers, regardless of the order of the polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Given the results in this paper, we believe the combination of our low-memory sub-grid scale discretisation, AIRG with low-order GMRES polynomials, and an iterative method that additively preconditions with the stream- ing/removal operator and an inconsistent CG DSA forms an excellent method for solving transport problems on unstructured grids in both the streaming and scattering limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Acknowledgments The authors would like to acknowledge the support of the EPSRC through the funding of the EPSRC grants EP/R029423/1 and EP/T000414/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' References References [1] J.' metadata={'source': 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+page_content='279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [79] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' MacLachlan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Saad, A Greedy Strategy for Coarse-Grid Selection, SIAM Journal on Scientific Computing 29 (2007) 1825–1853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [80] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Beckermann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Goreinov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Tyrtyshnikov, Some Remarks on the Elman Estimate for GMRES, SIAM Journal on Matrix Analysis and Applications 27 (2005) 772–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Publisher: Society for Industrial and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' [81] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Liesen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Tich´y, The field of values bound on ideal GMRES, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' ArXiv:1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5969 [cs, math].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 29 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 CG Nodes Time (µs) per DOF (a) Solve time 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 CG Nodes Time (µs) per DOF (b) Dashed are time to compute ˆA−1 ff for AIRG, solid are time to compute Z 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 CG Nodes Time (µs) per DOF (c) Setup time 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 CG Nodes Time (µs) per DOF (d) Total time Figure 5: Timings per DOF for AIRG with m = 4 and lAIR in a 2D pure streaming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The × is AIRG with fixed sparsity with Falgout-CLJP, the × is AIRG without fixed sparsity and Falgout-CLJP, the ⊗ is distance 1 lAIR with Falgout-CLJP, the ⊗ is distance 2 lAIR with Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 30 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 CG Nodes Time (µs) per DOF (a) Setup time 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 CG Nodes Time (µs) per DOF (b) Total time Figure 6: Timings per DOF for AIRG with fixed sparsity, Falgout-CLJP and with varying GMRES polynomial order in a 2D pure streaming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The × is AIRG with m = 1, o is m = 2, ⊗ is m = 3, □ is m = 4, ⋄ is m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 31 102 103 104 105 1 2 3 4 5 CG Nodes Time (µs) per DOF (a) Solve time 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 CG Nodes Time (µs) per DOF (b) Dashed are time to compute ˆA−1 ff for AIRG, solid are time to compute Z 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='8 CG Nodes Time (µs) per DOF (c) Setup time 102 103 104 105 1 2 3 4 5 6 CG Nodes Time (µs) per DOF (d) Total time Figure 7: Timings per DOF for AIRG with m = 4 and lAIR in a 2D pure scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The × is AIRG with fixed sparsity with Falgout-CLJP, the × is AIRG without fixed sparsity and Falgout-CLJP, the ⊗ is distance 1 lAIR with Falgout-CLJP, the ⊗ is distance 2 lAIR with Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' 32 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='6 CG Nodes Time (µs) per DOF (a) Setup time 102 103 104 105 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='5 3 CG Nodes Time (µs) per DOF (b) Total time Figure 8: Timings per DOF for AIRG with fixed sparsity, Falgout-CLJP and with varying GMRES polynomial order in a 2D pure scattering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' The × is AIRG with m = 1, o is m = 2, ⊗ is m = 3, □ is m = 4, ⋄ is m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' (a) Streaming operator (b) Streaming/removal operator with a total cross-section of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Figure 9: The 10 biggest and smallest eigenvalues (by real, imaginary parts and magnitude) (dots) and field of values (solid lines) of different operators, with the red the equivalent for Aff with CF splitting by Falgout-CLJP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE5T4oBgHgl3EQfRg5P/content/2301.05521v1.pdf'} +page_content=' Computed on the third refined spatial grid with level one angular refinement.' metadata={'source': 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Dynamics +in the Breathing Pyrochlore System LiGa0.95In0.05Cr4O8 +Yu Tanaka1 *, Rafal Wawrzy´nczak2, Manh Duc Le3, Tatiana Guidi3, Yoshihiko Okamoto4, Takeshi Yajima1, +Zenji Hiroi1, Masashi Takigawa1, and Gøran J. Nilsen3 +1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan +2Institut Laue-Langevin, CS 20156, Cedex 9, 38042 Grenoble, France +3ISIS Facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot OX11 0QX, United Kingdom +4Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan +The A-site ordered chromate spinels LiGa1−xInxCr4O8 host a network of size-alternating spin-3/2 Cr3+ tetrahedra +known as a “breathing” pyrochlore lattice. For the x = 0.05 composition, the complex magneto-structural ordering +observed in the parent x = 0 material is replaced by a single transition at T f = 11 K, ascribed to the collinear nematic +order caused by strong spin-lattice coupling. We present here an inelastic neutron scattering study of the spin dynamics +in this composition. Above T f , the dynamical scattering function S (Q, E) is ungapped and quasi-elastic, similar to +undoped LiGaCr4O8. Below T f , the spectral weight splits between a broad inelastic feature at 5.8 meV and toward the +elastic line. The former feature can be ascribed to spin precessions within antiferromagnetic loops, lifted to finite energy +by the effective biquadratic spin-lattice term in the spin Hamiltonian. +When magnetic frustration is combined with strong spin- +lattice coupling, a range of possible magneto-structural be- +haviors result,1) including nematic transitions,2) magnetiza- +tion plateaus, and localized spin excitations.3,4) An ideal +arena for exploring this interplay is provided by the chro- +mate spinels, A2+Cr3+ +2 O4. Here, the frustration originates from +the corner-sharing pyrochlore network of Cr3+ (S = 3/2) +tetrahedra on the B-site, while the strong spin-lattice cou- +pling arises from the direct overlap of Cr3+ d-orbitals on ad- +jacent sites. A common starting point to understand the low- +temperature physics of the chromate spinels is the so-called +bilinear-biquadratic model,5,6) +H = J +� +i,j +Si · Sj + b +� +i,j +(Si · Sj)2 +(1) +where the classical nearest-neighbor Heisenberg Hamiltonian +is extended with an effective biquadratic term, b(Si · Sj)2 +(where b is a coupling constant, and Si,j are classical spins). +This term, generated by spin-lattice coupling to local distor- +tions, lifts some of the degeneracy of the ground state mani- +fold by selecting collinear or coplanar spin configurations. Al- +though the bilinear-biquadratic model ignores the long-range +interactions which eventually cause magneto-structural order +in many chromate spinels, it is able to successfully describe +both the short-range spin correlations and the magnetic phase +diagram of virtually every member of the family. It has not, +however, yet been applied to the unusual magnetic excita- +tion spectra in the ordered phases of MgCr2O4,7) HgCr2O4,4) +and ZnCr2O4;8) these are characterized by weak spin-wave +branches and sharp, non-dispersive inelastic bands, with wave +vector dependences characteristic of small spin clusters.4,9) +While bands assigned to both hexamer and heptamer clus- +ters are observed in MgCr2O4 and HgCr2O4, only the for- +mer are seen in ZnCr2O4. Due to the complexity of the low- +temperature magneto-structural orders in these materials, the +connection between the structure and the apparent spin cluster +excitations is unclear. +This link is more evident, at least at high temperature, in +the so-called “breathing” pyrochlore spinels A+A′3+Cr4O8, +where the A-site is now populated by an ordered arrangement +of mono- and trivalent cations. Here, the term “breathing” +refers to the alternation of Cr3+-Cr3+ distances and, hence, +magnetic exchanges J and J′ between adjacent Cr3+ +4 tetrahe- +dra as a consequence of the order on the A-site. The degree +of magnetic alternation is quantified by the breathing factor +Bf = J′/J, where Bf → 0 corresponds to isolated tetrahedra +and Bf → 1 to the isotropic pyrochlore lattice. Starting from +the former limit, the excitations at T ∼ J are localized non- +dispersive triplets (and higher multiplets) separated by a spin +gap ∆ from the singlet ground state. When Bf is increased, ∆ +is suppressed, and the excitations become qualitatively simi- +lar to the isotropic case beyond Bf ∼ 0.25. This simple picture +is again complicated by the influence of the spin-lattice cou- +pling and long-ranged terms, which are responsible for the +collective excitations in the low-temperature ordered phases. +LiGaCr4O8 and LiInCr4O8 were rediscovered by Okamoto +et. al.10) as breathing pyrochlore systems, which have A+ = +Li+ and A′3+ = Ga3+/In3+. The nearest-neighbor magnetic in- +teractions on the small and large tetrahedra are estimated to +be J ∼ 50 K and J′ ∼ 30 K (Bf ∼ 0.6) for LiGaCr4O8, +and J ∼ 60 K and J′ ∼ 6 K (Bf ∼ 0.1) for LiInCr4O8.10) +For LiGaCr4O8, the upper magneto-structural transition at +Tu ∼ 20 K results in phase separation into cubic paramag- +netic and tetragonal collinear phases. The cubic phase then +undergoes another transition at Tl = 13.8 K, into a second +tetragonal phase, the structure of which has not yet been deter- +mined. As in other chromate spinels, both transitions are first- +order. However, the paramagnetic component shows a diver- +gence in the nuclear spin-lattice relaxation rate 1/T1 extracted +from 7Li-NMR, implying proximity to a tricritical point or to +a second-order transition to another phase.11) +In this letter, we describe inelastic neutron scattering mea- +surements of the spin excitation spectrum of the “breath- +1 +arXiv:2301.05064v1 [cond-mat.str-el] 12 Jan 2023 + +J. Phys. Soc. Jpn. +LETTERS +152 K +50.4 K +(a) +(b) +(c) +(d) +12.4 K +5.2 K +S (Q, E) (arb. units) +Fig. 1. +(Color online) Temperature dependence of S (Q, E) for LiGa0.95In0.05Cr4O8 recorded with 16 meV incident energy. (a)-(c) are taken at T > T f and +(d) below T f . Blank patches are due to gaps between detectors. +ing” pyrochlore chromate spinel, LiGa0.95In0.05Cr4O8, where +Bf ∼ 0.6. Previous diffraction measurements indicate that +LiGa0.95In0.05Cr4O8 undergoes a possible second-order tran- +sition to a nematic collinear ground state at T f = 11.1 K, +in accordance with predictions from the bilinear-biquadratic +model.2) The excitations at T > T f are gapless and Lorentzian +in form, as is also the case for MgCr2O4, HgCr2O4, and +ZnCr2O4. Below T f , the spectral weight shifts to the elastic +line and an inelastic feature at ∼5.8 meV. We identify the lat- +ter with spin precession within antiferromagnetic hexagonal +spin clusters created by the nematic order, and lifted to finite +energy by the biquadratic term. We thus provide, for the first +time, a plausible link between the magnetic excitations and +magnetic structure. The remaining spectral weight appears to +be consistent with collective spin-wave-like excitations. +The powder sample of LiGa0.95In0.05Cr4O8 was prepared +by sintering a stoichiometric mixture of the two end-member +compounds LiGaCr4O8 and LiInCr4O8.12) These were in turn +prepared by the standard solid-state route,10) using starting +materials enriched with 7Li to reduce neutron absorption. For +our inelastic neutron scattering measurements, 8.1 g of pow- +der was packed in an Al sachet, which was rolled into an annu- +lus and loaded into an Al can with � = 45 mm. The measure- +ments were performed on the MARI direct-geometry time-of- +flight chopper spectrometer at the ISIS facility, UK, using in- +cident energies Ei = 10, 16, 25, and 35 meV. For all values of +Ei, the elastic energy resolution was close to ∆E/E ∼ 4.5%. +Temperatures between 5 and 300 K were accessed using a +closed-cycle refrigerator. The diffraction measurements re- +ported in Ref. 2 were performed on the same sample. +The dynamical structure factors S (Q, E = Ei − E f ) mea- +sured at four selected temperatures between 5.2 K and 152 K +are shown in Fig. 1. We begin with an analysis of the data +taken above T f : at T = 152 K≫ T f , a quasi-elastic rod +of scattering extending up to ∼15 meV is observed. This +is characteristic of diffusive spin excitations in the corre- +lated paramagnetic state,13) and resembles the S (Q, E) of both +LiInCr4O814) and LiGaCr4O815) at similar temperatures. The +intensity of the rod is centered around 1.6 Å−1, corresponding +approximately to the reciprocal space position of the Cr-Cr +nearest-neighbor distance. Upon cooling to 50.4 K, intensity +builds up near the elastic line, again as in LiGaCr4O8, but in +contrast to LiInCr4O8, where the scattering becomes inelas- +tic.14) The modulation of the quasi-elastic scattering is also +enhanced, indicating the development of longer-ranged spin- +spin correlations ⟨S (0) · S (r)⟩, as may be seen in Fig. 2. +To determine the spatial extent of the correlations, we +fit the Q-dependence of the scattering integrated between 2 +and 7 meV (≃ S (Q) at high temperature) to a shell model +[Fig. 2(a)]: +S (Q) = f(Q)2 � +i +⟨S (0) · S (ri)⟩ Ni +sin(Qri) +Qri +, +(2) +where f(Q) is the magnetic form factor for Cr3+, and Ni is +the coordination number of the ith shell at radial distance ri. +For simplicity, r1 is approximated as the mean of the r1 and r′ +1 +distances. +The summation in the fitting function was extended to the +third neighboring shell, at which point the fit quality did +not increase. The extracted parameters reveal antiferromag- +2 + +MAR20556Reduced SQW +T= 50.4 K +Energy transfer (me V) +10 +0 +1 +2 +3 +4 +5 +Q(A-1)MAR20554Reduced SQW +T= 12.4 K +Energy transfer (me V) +10 +0 +1 +2 +3 +5 +Q(A-1)MAR20551Reduced SQW +T= 5.2 K +Energy transfer (me V) +10 +0 +1 +2 +3 +4 +Q(A-1)MAR20560Reduced SQW +T= 152 K +Energy transfer (me V) +.10 +0 +1 +2 +3 +4 +Q(A-1)J. Phys. Soc. Jpn. +LETTERS +1 +2 +3 +4 +Q ( ˚A−1) +S(Q) (a.u.) +152 K +75 K +50 K +24 K +18 K +14 K +12.4 K +5.2 K +(a) +E = 2−7 meV +0 +2 +4 +6 +8 +10 +12 +r ( ˚A) +−0.3 +−0.2 +−0.1 +0.0 +0.1 +0.2 +⟨S(0)·S(r)⟩/S(S+1) +(b) +RMC +1.5 K +30 K +5.2 K +24 K +152 +Hexamer cluster +Hexamer cluster +Fig. 2. +(Color online) (a) Q dependence of the magnetic scattering I(Q), +integrated over the energy range 2-7 meV at different temperatures. Dashed +lines are fitting curves calculated from the shell model (Eq. (2)) with the first +three nearest neighbors and flat backgrounds. Solid red lines show the results +of structure factor calculations for hexagonal chromium rings at T < 20 K. +(b) Real space spin-spin correlation functions ⟨S (0) · S (ri)⟩ versus r. Solid +circles are obtained from the fits to the shell model in Fig. 2(a), and the open +triangles are obtained by the reverse Monte Carlo (RMC) simulation on the +magnetic diffuse scattering observed in the elastic ND measurement.2) Green +stars mark the correlations for an isolated hexagonal antiferromagnetic loop +(Fig. 4). +netic nearest-neighbor spin-spin correlations, with progres- +sively weaker alternating ferro- and antiferromagnetic cor- +relations for the second and third nearest neighbors, respec- +tively [Fig. 2(b)]. The extracted correlations are thus con- +sistent with the reverse Monte Carlo results presented in +Ref. 2, where energy-integrated data from a diffractometer +were used; i.e., the true S (Q) was reflected. The temperature +dependence of the parameters indicates smooth growth of the +spin-spin correlations in the entire temperature range, as ex- +pected. +To analyze the temperature dependence of the quasi-elastic +feature further, the imaginary part of the magnetic dynamic +susceptibility χ′′ was calculated by applying the fluctuation +dissipation theorem16) χ′′(Q, E) = π(1 − e− +E +kBT )S (Q, E) to the +E dependence of the intensity integrated over the Q range +1.1 − 1.9 Å−1 [Fig. 3(a)]. At T > T f , the contribution of elas- +tic scattering is subtracted by approximating it with a sharp +Gaussian centered around E = 0. The obtained χ′′(ω) are well +fit by a quasi-elastic Lorentzian χ′′(ω) = χ′ωΓ/(ω2 + Γ2), +20 +15 +10 +5 +0 +χ" (arb. unit) +10 +8 +6 +4 +2 +0 +E transfer (meV) + 5.2 K + 12.4 K + 14 K + 18 K + 24 K + 50 K + 75 K + 152 K +12 +10 +8 +6 +4 +2 +0 +Γ (meV) +160 +120 +80 +40 +0 +T (K) +30 +20 +10 +0 + χ' (arb. units) +(a) +(b) +Fig. 3. +(Color online) (a) Energy dependence of the dynamic susceptibil- +ity χ′′(ω), integrated over the Q range 1.1-1.9 Å−1 for all measured tem- +peratures. The elastic line was subtracted from each dataset. The solid lines +are resolution-broadened quasi-elastic Lorentzian fits. (b) Temperature de- +pendence of the inverse relaxation rate Γ and the static susceptibility χ’ as +determined by quasi-elastic Lorentzian fitting. The solid and dotted lines are +a linear and power-law curve fits to Γ. +which is the time-Fourier transform of an exponential de- +cay exp(−t/τ), with τ ∝ 1/Γ and χ′ the static susceptibil- +ity. On cooling below 18 K, the Lorentzian fits become poor +at E < 2 meV, indicating that the scattering is no longer de- +scribed by a single relaxation process. This coincides with the +appearance of a stretching exponent β < 1 in fits of the T1 re- +laxation process,17) and thus is likely connected with the onset +of critical fluctuations above T f . +Figure 3(b) shows the temperature dependence of the in- +verse relaxation time Γ and the static susceptibility χ′ ex- +tracted from the fits described above. Γ decreases smoothly +in the temperature range 18 K, and is well described by a +power law Γ ∝ T γ with γ = 0.66 (dashed line). For Heisen- +berg spins on the isotropic pyrochlore lattice, theory predicts +Γ ∝ T (γ = 1);13,18,19) however, a linear fit to the data (solid +line) is poor at high temperature, even permitting a nonzero +intercept Γ0 = 1.09 meV. Although a similar reduction of γ +has also been observed in ZnCr2O4 (γ = 0.81), the cause re- +mains unclear.3) Aside from this, χ′ is consistent with the bulk +susceptibility. +Turning now to the form of S (Q, E) below T f shown in +Fig. 1(d), most of the high-temperature quasi-elastic scatter- +ing shifts either towards the elastic line or to an inelastic fea- +ture centered around 5.8 meV. The latter is similar to the “res- +onance” observed in LiGaCr4O815) and other spinels, but is +considerably broader in energy. Like the resonance, however, +its structure factor suggests local modes on small antiferro- +magnetic spin loops. An analysis of the reverse Monte Carlo +spin configurations derived from fits to S (Q) in our previ- +ous publication2) identifies these with a large number of six- +membered hexagonal antiferromagnetic spin loops, as well +as a few with eight or more members. Indeed, the calculated +structure factor for the hexagonal rings (Fig. 4) agrees almost +perfectly with that of the energy-integrated data in Figure +2(a), also accounting for the variation of ⟨S (0) · S (ri)⟩ versus +r from the model-independent fits above. As shown in Fig. 4, +hexagonal antiferromagnetic spin loops are only possible in +the presence of three types (colors) of collinear state on the +Cr3+ tetrahedra.5) +By analogy with the coplanar nematic state in the kagome +lattice antiferromagnet,20–22) collinear nematic states on the +3 + +J. Phys. Soc. Jpn. +LETTERS +Fig. 4. +(Color online) Hexamer loop determined within the breathing py- +rochlore lattice (cyan bonds). Spheres represent Cr3+ ions. RBG coloring of +the bonds and vertices of the tetrahedra corresponds to the bond ordering de- +scribed in Refs. 2 and 5. Cyan arrows represent antiferromagnetically coupled +spins on the nodes of hexagonal cluster precessing around the easy direction +of nematic phase (dashed black lines). +pyrochlore lattice support two types of loop excitations: (i) +loop flips, which invert the moment directions around the +loop, hence transforming one nematic ground state configura- +tion to another, and (ii) “weathervane” modes, small displace- +ments of the moment direction about the equilibrium direction +[Fig. 4]. The former, related to the diffusive high-temperature +excitations, is expected to produce a quasi-elastic signal with +a temperature-dependent width, and thus cannot account for +the inelastic feature. As such, we tentatively assign the fea- +ture to weathervane modes on the hexagonal loops. Consid- +ering only the bilinear term, the ground state criterion of two +spins up and two down on each tetrahedron results in a zero +net exchange field for the spins around the hexagon, and the +weathervane modes therefore carry no energy cost. When the +biquadratic (magneto-elastic) and other long-ranged terms are +included, however, they are lifted to finite energy. In particu- +lar, inserting the bilinear-biquadratic Hamiltonian (1) into the +classical equation of motion +dSi(t) +dt += −1 +ℏSi(t) × ∇Si(t)H +(3) +results in an energy gap ∆E ≃ 8bavS 3, where bav is the aver- +age bilinear-biquadratic coupling constant between the small +and large tetrahedra. In deriving this expression, we assumed +that there is no coupling between the loops and S z +i(t) ≃ S ; i.e., +the spin displacements are small. From Jav = (J + J′)/2 = +45 K estimated from the magnetic susceptibility and the ex- +perimental excitation energy, we obtain bav ∼ 0.05Jav, which +is close to the b reported for related materials.23) In addi- +tion, using T f ≃ bS 4 for the isotropic pyrochlore lattice,24) +bav ∼ 0.05Jav yields T f ∼ 12 K, which is in excellent agree- +ment with experiment. +Now we address the large width of the feature relative to the +much sharper features observed in other spinels: could this +be due to the disorder inherent to the nematic state? Below +T f , the Cr-Cr bond lengths, and hence the biquadratic bond +energies, are expected to follow a Gaussian distribution (is +indeed found for the d-spacings in [2]). The resulting spec- +trum is then broadened by σ(bav), the FWHM of the Gaus- +sian. The experimental feature at 5.8 meV is approximately +Gaussian, with an FWHM of ∼2 meV. To reproduce this, the +distribution of mean Cr-Cr bond lengths around a spin loop +is required to be ∼0.1 Å wide, assuming a linear relationship +between the exchange and the Cr-Cr distance. This is larger +by approximately a factor of 4 than the distribution estimated +from Rietveld refinements, which, however, ignore any local +structure. +The significant amount of inelastic and quasi-elastic spec- +tral weight at energies above and below the 5.8 meV feature, +may be associated with other excitations (also observed on +the kagome lattice), including the loop flips mentioned above +and longer-ranged spin-wave-like excitations (which may ex- +tend to much higher energies), perhaps belonging to the short- +range magnetic order superimposed on the nematic state. The +long high-energy tail of the inelastic scattering, extending to +∼15 meV, is certainly compatible with the latter. Loop flips, +on the other hand, are expected to give a quasi-elastic signal of +width ∝ 1/ exp(−b/T). Ultimately, single-crystal studies and +spin dynamics simulations of the bilinear-biquadratic model +with disorder on the present lattice will be required to disen- +tangle all the contributions to the excitation spectrum in the +nematic phase. +Looking beyond the breathing pyrochlores, many features +of the LiGa1−xInxCr4O8 series are shared with the undistorted +ZnxCd1−xCr2O4 family.8,25) Starting with the x − T phase di- +agrams, the introduction of bond disorder by even vestigial +doping is found to lead to the suppression of the N´eel phase +and adoption of a disordered frozen state at small x in both +cases, as also observed in Monte Carlo simulations.24) The +persistence of a sharp phase transition in the specific heat, +despite glassy behavior in the magnetic susceptibility, is also +common to both systems. These commonalities suggest the +intriguing possibility that ZnxCd1−xCr2O4 with x < 0.1 and +other similar systems also exhibit nematic transitions.24) +Comparing the x = 0.05 compositions of both families, +In0.05 and Cd0.05, the form of the scattering is at first glance +nearly identical above and below the transitions at T f . How- +ever, the dynamic susceptibility χ′′(E) of In0.05 is describable +using only one relaxation rate down to 18 K ∼ 1.6T f , while +that of Cd0.05 requires a distribution of relaxation rates already +below 4T f .8) This is indicative of a stronger doping effect in +the latter case. In regard to the gap in S (Q, E) at T < T f , ∆E is +4.5 meV in Cd0.05 versus 5.8 meV in In0.05, giving a ratio close +to that of the exchange couplings in the two systems. Given +the similar b/J, this could point to a similar physical origin for +the gap. On the other hand, non-collinearity or strong further +neighbor couplings could also generate a nonzero exchange +field around a hexagon, and the former is thought to be fa- +vored by bond disorder.26) Indeed, flat features in the inelastic +scattering are also observed in Y2Ru2O7 and ZnCr2O4, where +non-collinear orders have been proposed. +We finally note that although inelastic resonances have +been interpreted as quantum two-level excitations in the +past,4) they should not be considered as such in the present +case. This is because the singlet-triplet gap is rapidly sup- +pressed by both further neighbor couplings and a negative bi- +quadratic exchange. In addition to this, none of the expected +higher multiplets are observed at any temperature. +We have presented an inelastic neutron scattering study +of the spin dynamics in the classical spin nematic mate- +rial LiGa0.95In0.05Cr4O8. The high-temperature dynamics are +4 + +J. Phys. Soc. Jpn. +LETTERS +quasi-elastic and resemble those observed in other pyrochlore +systems, while the excitation spectrum below the transition at +T f = 11 K is dominated by a broad, non-dispersive inelastic +feature at 5.8 meV. A plausible origin for this feature mode is +the so-called weathervane modes on hexagonal antiferromag- +netic loops (abundant in the nematic state), which are lifted +to finite energy by the biquadratic term that induces the ne- +matic order. Possible collective excitations with a bandwidth +of 15 meV are also observed. In order to verify this interpreta- +tion, more detailed spin dynamics simulations of the bilinear- +biquadratic model on the breathing pyrochlore lattice will be +required. +Acknowledgments +We thank Y. Motome, H. Shinaoka and M. Gingras +for fruitful discussions. This work was supported by JSPS KAKENHI (Grant +Nos. 25287083 and 16J01077). Y.T. was supported by the JSPS through the +Program for Leading Graduate Schools (MERIT). +1) R. 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Moessner: +Canadian Journal of Physics 79 (2001) 1365. +5 + diff --git a/2NE4T4oBgHgl3EQfagwY/content/tmp_files/load_file.txt b/2NE4T4oBgHgl3EQfagwY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6d8e524ec27987ff3cf8ace84bc4ebc41c1871f --- /dev/null +++ b/2NE4T4oBgHgl3EQfagwY/content/tmp_files/load_file.txt @@ -0,0 +1,497 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf,len=496 +page_content='Journal of the Physical Society of Japan LETTERS Inelastic Neutron Scattering Study of the Spin Dynamics in the Breathing Pyrochlore System LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8 Yu Tanaka1 *, Rafal Wawrzy´nczak2, Manh Duc Le3, Tatiana Guidi3, Yoshihiko Okamoto4, Takeshi Yajima1, Zenji Hiroi1, Masashi Takigawa1, and Gøran J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Nilsen3 1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 2Institut Laue-Langevin, CS 20156, Cedex 9, 38042 Grenoble, France 3ISIS Facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot OX11 0QX, United Kingdom 4Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan The A-site ordered chromate spinels LiGa1−xInxCr4O8 host a network of size-alternating spin-3/2 Cr3+ tetrahedra known as a “breathing” pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' For the x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 composition, the complex magneto-structural ordering observed in the parent x = 0 material is replaced by a single transition at T f = 11 K, ascribed to the collinear nematic order caused by strong spin-lattice coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We present here an inelastic neutron scattering study of the spin dynamics in this composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Above T f , the dynamical scattering function S (Q, E) is ungapped and quasi-elastic, similar to undoped LiGaCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Below T f , the spectral weight splits between a broad inelastic feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV and toward the elastic line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The former feature can be ascribed to spin precessions within antiferromagnetic loops, lifted to finite energy by the effective biquadratic spin-lattice term in the spin Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' When magnetic frustration is combined with strong spin- lattice coupling, a range of possible magneto-structural be- haviors result,1) including nematic transitions,2) magnetiza- tion plateaus, and localized spin excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='3,4) An ideal arena for exploring this interplay is provided by the chro- mate spinels, A2+Cr3+ 2 O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Here, the frustration originates from the corner-sharing pyrochlore network of Cr3+ (S = 3/2) tetrahedra on the B-site, while the strong spin-lattice cou- pling arises from the direct overlap of Cr3+ d-orbitals on ad- jacent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' A common starting point to understand the low- temperature physics of the chromate spinels is the so-called bilinear-biquadratic model,5,6) H = J � i,j Si · Sj + b � i,j (Si · Sj)2 (1) where the classical nearest-neighbor Heisenberg Hamiltonian is extended with an effective biquadratic term, b(Si · Sj)2 (where b is a coupling constant, and Si,j are classical spins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This term, generated by spin-lattice coupling to local distor- tions, lifts some of the degeneracy of the ground state mani- fold by selecting collinear or coplanar spin configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Al- though the bilinear-biquadratic model ignores the long-range interactions which eventually cause magneto-structural order in many chromate spinels, it is able to successfully describe both the short-range spin correlations and the magnetic phase diagram of virtually every member of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' It has not, however, yet been applied to the unusual magnetic excita- tion spectra in the ordered phases of MgCr2O4,7) HgCr2O4,4) and ZnCr2O4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8) these are characterized by weak spin-wave branches and sharp, non-dispersive inelastic bands, with wave vector dependences characteristic of small spin clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4,9) While bands assigned to both hexamer and heptamer clus- ters are observed in MgCr2O4 and HgCr2O4, only the for- mer are seen in ZnCr2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Due to the complexity of the low- temperature magneto-structural orders in these materials, the connection between the structure and the apparent spin cluster excitations is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This link is more evident, at least at high temperature, in the so-called “breathing” pyrochlore spinels A+A′3+Cr4O8, where the A-site is now populated by an ordered arrangement of mono- and trivalent cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Here, the term “breathing” refers to the alternation of Cr3+-Cr3+ distances and, hence, magnetic exchanges J and J′ between adjacent Cr3+ 4 tetrahe- dra as a consequence of the order on the A-site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The degree of magnetic alternation is quantified by the breathing factor Bf = J′/J, where Bf → 0 corresponds to isolated tetrahedra and Bf → 1 to the isotropic pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Starting from the former limit, the excitations at T ∼ J are localized non- dispersive triplets (and higher multiplets) separated by a spin gap ∆ from the singlet ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' When Bf is increased, ∆ is suppressed, and the excitations become qualitatively simi- lar to the isotropic case beyond Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This simple picture is again complicated by the influence of the spin-lattice cou- pling and long-ranged terms, which are responsible for the collective excitations in the low-temperature ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' LiGaCr4O8 and LiInCr4O8 were rediscovered by Okamoto et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='10) as breathing pyrochlore systems, which have A+ = Li+ and A′3+ = Ga3+/In3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The nearest-neighbor magnetic in- teractions on the small and large tetrahedra are estimated to be J ∼ 50 K and J′ ∼ 30 K (Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='6) for LiGaCr4O8, and J ∼ 60 K and J′ ∼ 6 K (Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1) for LiInCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='10) For LiGaCr4O8, the upper magneto-structural transition at Tu ∼ 20 K results in phase separation into cubic paramag- netic and tetragonal collinear phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The cubic phase then undergoes another transition at Tl = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 K, into a second tetragonal phase, the structure of which has not yet been deter- mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' As in other chromate spinels, both transitions are first- order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' However, the paramagnetic component shows a diver- gence in the nuclear spin-lattice relaxation rate 1/T1 extracted from 7Li-NMR, implying proximity to a tricritical point or to a second-order transition to another phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='11) In this letter, we describe inelastic neutron scattering mea- surements of the spin excitation spectrum of the “breath- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05064v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='str-el] 12 Jan 2023 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' LETTERS 152 K 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K (a) (b) (c) (d) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K S (Q, E) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' units) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (Color online) Temperature dependence of S (Q, E) for LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8 recorded with 16 meV incident energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (a)-(c) are taken at T > T f and (d) below T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Blank patches are due to gaps between detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' ing” pyrochlore chromate spinel, LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8, where Bf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Previous diffraction measurements indicate that LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8 undergoes a possible second-order tran- sition to a nematic collinear ground state at T f = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 K, in accordance with predictions from the bilinear-biquadratic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2) The excitations at T > T f are gapless and Lorentzian in form, as is also the case for MgCr2O4, HgCr2O4, and ZnCr2O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Below T f , the spectral weight shifts to the elastic line and an inelastic feature at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We identify the lat- ter with spin precession within antiferromagnetic hexagonal spin clusters created by the nematic order, and lifted to finite energy by the biquadratic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We thus provide, for the first time, a plausible link between the magnetic excitations and magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The remaining spectral weight appears to be consistent with collective spin-wave-like excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The powder sample of LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8 was prepared by sintering a stoichiometric mixture of the two end-member compounds LiGaCr4O8 and LiInCr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='12) These were in turn prepared by the standard solid-state route,10) using starting materials enriched with 7Li to reduce neutron absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' For our inelastic neutron scattering measurements, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 g of pow- der was packed in an Al sachet, which was rolled into an annu- lus and loaded into an Al can with � = 45 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The measure- ments were performed on the MARI direct-geometry time-of- flight chopper spectrometer at the ISIS facility, UK, using in- cident energies Ei = 10, 16, 25, and 35 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' For all values of Ei, the elastic energy resolution was close to ∆E/E ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Temperatures between 5 and 300 K were accessed using a closed-cycle refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The diffraction measurements re- ported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2 were performed on the same sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The dynamical structure factors S (Q, E = Ei − E f ) mea- sured at four selected temperatures between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K and 152 K are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We begin with an analysis of the data taken above T f : at T = 152 K≫ T f , a quasi-elastic rod of scattering extending up to ∼15 meV is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This is characteristic of diffusive spin excitations in the corre- lated paramagnetic state,13) and resembles the S (Q, E) of both LiInCr4O814) and LiGaCr4O815) at similar temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The intensity of the rod is centered around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='6 Å−1, corresponding approximately to the reciprocal space position of the Cr-Cr nearest-neighbor distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Upon cooling to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K, intensity builds up near the elastic line, again as in LiGaCr4O8, but in contrast to LiInCr4O8, where the scattering becomes inelas- tic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='14) The modulation of the quasi-elastic scattering is also enhanced, indicating the development of longer-ranged spin- spin correlations ⟨S (0) · S (r)⟩, as may be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' To determine the spatial extent of the correlations, we fit the Q-dependence of the scattering integrated between 2 and 7 meV (≃ S (Q) at high temperature) to a shell model [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2(a)]: S (Q) = f(Q)2 � i ⟨S (0) · S (ri)⟩ Ni sin(Qri) Qri , (2) where f(Q) is the magnetic form factor for Cr3+, and Ni is the coordination number of the ith shell at radial distance ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' For simplicity, r1 is approximated as the mean of the r1 and r′ 1 distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The summation in the fitting function was extended to the third neighboring shell, at which point the fit quality did not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The extracted parameters reveal antiferromag- 2 MAR20556Reduced SQW T= 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K Energy transfer (me V) 10 0 1 2 3 4 5 Q(A-1)MAR20554Reduced SQW T= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K Energy transfer (me V) 10 0 1 2 3 5 Q(A-1)MAR20551Reduced SQW T= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K Energy transfer (me V) 10 0 1 2 3 4 Q(A-1)MAR20560Reduced SQW T= 152 K Energy transfer (me V) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='10 0 1 2 3 4 Q(A-1)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' LETTERS 1 2 3 4 Q ( ˚A−1) S(Q) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=') 152 K 75 K 50 K 24 K 18 K 14 K 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='4 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K (a) E = 2−7 meV 0 2 4 6 8 10 12 r ( ˚A) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 ⟨S(0)·S(r)⟩/S(S+1) (b) RMC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='5 K 30 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K 24 K 152 Hexamer cluster Hexamer cluster Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (Color online) (a) Q dependence of the magnetic scattering I(Q), integrated over the energy range 2-7 meV at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Dashed lines are fitting curves calculated from the shell model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (2)) with the first three nearest neighbors and flat backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Solid red lines show the results of structure factor calculations for hexagonal chromium rings at T < 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (b) Real space spin-spin correlation functions ⟨S (0) · S (ri)⟩ versus r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Solid circles are obtained from the fits to the shell model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2(a), and the open triangles are obtained by the reverse Monte Carlo (RMC) simulation on the magnetic diffuse scattering observed in the elastic ND measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2) Green stars mark the correlations for an isolated hexagonal antiferromagnetic loop (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' netic nearest-neighbor spin-spin correlations, with progres- sively weaker alternating ferro- and antiferromagnetic cor- relations for the second and third nearest neighbors, respec- tively [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The extracted correlations are thus con- sistent with the reverse Monte Carlo results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2, where energy-integrated data from a diffractometer were used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=', the true S (Q) was reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The temperature dependence of the parameters indicates smooth growth of the spin-spin correlations in the entire temperature range, as ex- pected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' To analyze the temperature dependence of the quasi-elastic feature further, the imaginary part of the magnetic dynamic susceptibility χ′′ was calculated by applying the fluctuation dissipation theorem16) χ′′(Q, E) = π(1 − e− E kBT )S (Q, E) to the E dependence of the intensity integrated over the Q range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='9 Å−1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' At T > T f , the contribution of elas- tic scattering is subtracted by approximating it with a sharp Gaussian centered around E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The obtained χ′′(ω) are well fit by a quasi-elastic Lorentzian χ′′(ω) = χ′ωΓ/(ω2 + Γ2), 20 15 10 5 0 χ" (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' unit) 10 8 6 4 2 0 E transfer (meV) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='2 K 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content="4 K 14 K 18 K 24 K 50 K 75 K 152 K 12 10 8 6 4 2 0 Γ (meV) 160 120 80 40 0 T (K) 30 20 10 0 χ' (arb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' units) (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (Color online) (a) Energy dependence of the dynamic susceptibil- ity χ′′(ω), integrated over the Q range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='9 Å−1 for all measured tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The elastic line was subtracted from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The solid lines are resolution-broadened quasi-elastic Lorentzian fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (b) Temperature de- pendence of the inverse relaxation rate Γ and the static susceptibility χ’ as determined by quasi-elastic Lorentzian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The solid and dotted lines are a linear and power-law curve fits to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' which is the time-Fourier transform of an exponential de- cay exp(−t/τ), with τ ∝ 1/Γ and χ′ the static susceptibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' On cooling below 18 K, the Lorentzian fits become poor at E < 2 meV, indicating that the scattering is no longer de- scribed by a single relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This coincides with the appearance of a stretching exponent β < 1 in fits of the T1 re- laxation process,17) and thus is likely connected with the onset of critical fluctuations above T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Figure 3(b) shows the temperature dependence of the in- verse relaxation time Γ and the static susceptibility χ′ ex- tracted from the fits described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Γ decreases smoothly in the temperature range 18 K, and is well described by a power law Γ ∝ T γ with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='66 (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' For Heisen- berg spins on the isotropic pyrochlore lattice, theory predicts Γ ∝ T (γ = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='13,18,19) however, a linear fit to the data (solid line) is poor at high temperature, even permitting a nonzero intercept Γ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='09 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Although a similar reduction of γ has also been observed in ZnCr2O4 (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='81), the cause re- mains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='3) Aside from this, χ′ is consistent with the bulk susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Turning now to the form of S (Q, E) below T f shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 1(d), most of the high-temperature quasi-elastic scatter- ing shifts either towards the elastic line or to an inelastic fea- ture centered around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The latter is similar to the “res- onance” observed in LiGaCr4O815) and other spinels, but is considerably broader in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Like the resonance, however, its structure factor suggests local modes on small antiferro- magnetic spin loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' An analysis of the reverse Monte Carlo spin configurations derived from fits to S (Q) in our previ- ous publication2) identifies these with a large number of six- membered hexagonal antiferromagnetic spin loops, as well as a few with eight or more members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Indeed, the calculated structure factor for the hexagonal rings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 4) agrees almost perfectly with that of the energy-integrated data in Figure 2(a), also accounting for the variation of ⟨S (0) · S (ri)⟩ versus r from the model-independent fits above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 4, hexagonal antiferromagnetic spin loops are only possible in the presence of three types (colors) of collinear state on the Cr3+ tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='5) By analogy with the coplanar nematic state in the kagome lattice antiferromagnet,20–22) collinear nematic states on the 3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' LETTERS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' (Color online) Hexamer loop determined within the breathing py- rochlore lattice (cyan bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Spheres represent Cr3+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' RBG coloring of the bonds and vertices of the tetrahedra corresponds to the bond ordering de- scribed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Cyan arrows represent antiferromagnetically coupled spins on the nodes of hexagonal cluster precessing around the easy direction of nematic phase (dashed black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' pyrochlore lattice support two types of loop excitations: (i) loop flips, which invert the moment directions around the loop, hence transforming one nematic ground state configura- tion to another, and (ii) “weathervane” modes, small displace- ments of the moment direction about the equilibrium direction [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The former, related to the diffusive high-temperature excitations, is expected to produce a quasi-elastic signal with a temperature-dependent width, and thus cannot account for the inelastic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' As such, we tentatively assign the fea- ture to weathervane modes on the hexagonal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Consid- ering only the bilinear term, the ground state criterion of two spins up and two down on each tetrahedron results in a zero net exchange field for the spins around the hexagon, and the weathervane modes therefore carry no energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' When the biquadratic (magneto-elastic) and other long-ranged terms are included, however, they are lifted to finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' In particu- lar, inserting the bilinear-biquadratic Hamiltonian (1) into the classical equation of motion dSi(t) dt = −1 ℏSi(t) × ∇Si(t)H (3) results in an energy gap ∆E ≃ 8bavS 3, where bav is the aver- age bilinear-biquadratic coupling constant between the small and large tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' In deriving this expression, we assumed that there is no coupling between the loops and S z i(t) ≃ S ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=', the spin displacements are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' From Jav = (J + J′)/2 = 45 K estimated from the magnetic susceptibility and the ex- perimental excitation energy, we obtain bav ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Jav, which is close to the b reported for related materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='23) In addi- tion, using T f ≃ bS 4 for the isotropic pyrochlore lattice,24) bav ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Jav yields T f ∼ 12 K, which is in excellent agree- ment with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Now we address the large width of the feature relative to the much sharper features observed in other spinels: could this be due to the disorder inherent to the nematic state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Below T f , the Cr-Cr bond lengths, and hence the biquadratic bond energies, are expected to follow a Gaussian distribution (is indeed found for the d-spacings in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The resulting spec- trum is then broadened by σ(bav), the FWHM of the Gaus- sian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The experimental feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV is approximately Gaussian, with an FWHM of ∼2 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' To reproduce this, the distribution of mean Cr-Cr bond lengths around a spin loop is required to be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 Å wide, assuming a linear relationship between the exchange and the Cr-Cr distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This is larger by approximately a factor of 4 than the distribution estimated from Rietveld refinements, which, however, ignore any local structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The significant amount of inelastic and quasi-elastic spec- tral weight at energies above and below the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV feature, may be associated with other excitations (also observed on the kagome lattice), including the loop flips mentioned above and longer-ranged spin-wave-like excitations (which may ex- tend to much higher energies), perhaps belonging to the short- range magnetic order superimposed on the nematic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The long high-energy tail of the inelastic scattering, extending to ∼15 meV, is certainly compatible with the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Loop flips, on the other hand, are expected to give a quasi-elastic signal of width ∝ 1/ exp(−b/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Ultimately, single-crystal studies and spin dynamics simulations of the bilinear-biquadratic model with disorder on the present lattice will be required to disen- tangle all the contributions to the excitation spectrum in the nematic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Looking beyond the breathing pyrochlores, many features of the LiGa1−xInxCr4O8 series are shared with the undistorted ZnxCd1−xCr2O4 family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8,25) Starting with the x − T phase di- agrams, the introduction of bond disorder by even vestigial doping is found to lead to the suppression of the N´eel phase and adoption of a disordered frozen state at small x in both cases, as also observed in Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='24) The persistence of a sharp phase transition in the specific heat, despite glassy behavior in the magnetic susceptibility, is also common to both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' These commonalities suggest the intriguing possibility that ZnxCd1−xCr2O4 with x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='1 and other similar systems also exhibit nematic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='24) Comparing the x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 compositions of both families, In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 and Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05, the form of the scattering is at first glance nearly identical above and below the transitions at T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' How- ever, the dynamic susceptibility χ′′(E) of In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 is describable using only one relaxation rate down to 18 K ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='6T f , while that of Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 requires a distribution of relaxation rates already below 4T f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8) This is indicative of a stronger doping effect in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' In regard to the gap in S (Q, E) at T < T f , ∆E is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='5 meV in Cd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05 versus 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV in In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05, giving a ratio close to that of the exchange couplings in the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Given the similar b/J, this could point to a similar physical origin for the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' On the other hand, non-collinearity or strong further neighbor couplings could also generate a nonzero exchange field around a hexagon, and the former is thought to be fa- vored by bond disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='26) Indeed, flat features in the inelastic scattering are also observed in Y2Ru2O7 and ZnCr2O4, where non-collinear orders have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We finally note that although inelastic resonances have been interpreted as quantum two-level excitations in the past,4) they should not be considered as such in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This is because the singlet-triplet gap is rapidly sup- pressed by both further neighbor couplings and a negative bi- quadratic exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' In addition to this, none of the expected higher multiplets are observed at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' We have presented an inelastic neutron scattering study of the spin dynamics in the classical spin nematic mate- rial LiGa0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='95In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='05Cr4O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' The high-temperature dynamics are 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' LETTERS quasi-elastic and resemble those observed in other pyrochlore systems, while the excitation spectrum below the transition at T f = 11 K is dominated by a broad, non-dispersive inelastic feature at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' A plausible origin for this feature mode is the so-called weathervane modes on hexagonal antiferromag- netic loops (abundant in the nematic state), which are lifted to finite energy by the biquadratic term that induces the ne- matic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Possible collective excitations with a bandwidth of 15 meV are also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' In order to verify this interpreta- tion, more detailed spin dynamics simulations of the bilinear- biquadratic model on the breathing pyrochlore lattice will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Acknowledgments We thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Motome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Shinaoka and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Gingras for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' This work was supported by JSPS KAKENHI (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 25287083 and 16J01077).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' was supported by the JSPS through the Program for Leading Graduate Schools (MERIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 1) R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Yoshida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Okamoto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Manuel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Casati, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Hiroi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Takigawa, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} 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Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Kousaka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Bewley, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Guidi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Akimitsu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 9) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Tomiyasu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Suzuki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Toki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Itoh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Matsuura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Aso, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Yamada: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 101 (2008) 177401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 10) Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Okamoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Conlon and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Chalker: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfagwY/content/2301.05064v1.pdf'} +page_content=' 102 (2009) 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infimum of convex-concave saddle functions. Saddle problems arise in a wide range +of applications, including game theory, machine learning, and finance. It is well known +that a saddle problem can be reduced to a single convex optimization problem by dual- +izing either the convex (min) or concave (max) objectives, reducing a min-max problem +into a min-min (or max-max) problem. Carrying out this conversion by hand can be +tedious and error prone. In this paper we introduce disciplined saddle programming +(DSP), a domain specific language (DSL) for specifying saddle problems, for which the +dualizing trick can be automated. The language and methods are based on recent work +by Juditsky and Nemirovski [JN22], who developed the idea of conic-representable sad- +dle point programs, and showed how to carry out the required dualization automatically +using conic duality. Juditsky and Nemirovski’s conic representation of saddle problems +extends Nesterov and Nemirovski’s earlier development of conic representable convex +problems; DSP can be thought of as extending disciplined convex programming (DCP) +to saddle problems. Just as DCP makes it easy for users to formulate and solve com- +plex convex problems, DSP allows users to easily formulate and solve saddle problems. +Our method is implemented in an open-source package, also called DSP. +∗Equal contribution. +1 + +Contents +1 +Introduction +3 +1.1 +Previous and related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.2 +Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Saddle programming +5 +2.1 +Saddle functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.2 +Saddle point problems +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +Saddle extremum functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.4 +Saddle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.5 +Solving saddle problems +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.6 +Dual reduction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3 +Applications +10 +3.1 +Robust bond portfolio construction . . . . . . . . . . . . . . . . . . . . . . . +11 +3.2 +Model fitting robust to data weights . . . . . . . . . . . . . . . . . . . . . . . +11 +3.3 +Robust production problem with worst case prices . . . . . . . . . . . . . . . +12 +3.4 +Robust Markowitz portfolio construction . . . . . . . . . . . . . . . . . . . . +13 +4 +Disciplined saddle point programming +14 +4.1 +Saddle function calculus +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4.2 +Conically representable saddle functions +. . . . . . . . . . . . . . . . . . . . +14 +5 +Implementation +16 +5.1 +Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +5.2 +Calculus rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +5.3 +Saddle point problems +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.4 +Saddle extremum functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +5.5 +Saddle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +6 +Examples +23 +6.1 +Robust bond portfolio construction . . . . . . . . . . . . . . . . . . . . . . . +23 +6.2 +Model fitting robust to data weights . . . . . . . . . . . . . . . . . . . . . . . +25 +6.3 +Robust Markowitz portfolio construction . . . . . . . . . . . . . . . . . . . . +27 +2 + +1 +Introduction +We consider saddle problems, by which we mean convex-concave saddle point problems or, +more generally, convex optimization problems that include the partial supremum or infimum +of convex-concave saddle functions. Saddle problems arise in various fields such as game +theory, robust and minimax optimization, machine learning, and finance. +While there are algorithms specifically designed to solve some types of saddle point or +minimax problems, another approach is to convert them into standard convex optimization +problems using a trick based on duality that can be traced back to at least the 1920s. The +idea is to express the infima or suprema that appear in the saddle problem via their duals, +which converts them to suprema or infima, respectively. Roughly speaking, this turns a min- +max problem into a min-min (or max-max) problem, which can then be solved by standard +methods. Specific cases of this trick are well known; the classical example is converting a +matrix game, a specific saddle point problem, into a linear program (LP) [MVN53]. While +the dualizing trick has been known and used for almost 100 years, it has always been done +by hand, for specific problems. It can only be carried out by those who have a working +knowledge of duality in convex optimization, and are aware of the trick. +In this paper we propose an automated method for carrying out the dualizing trick. Our +method is based on the theory of conic representation of saddle point problems, developed +recently by Juditsky and Nemirovski [JN22]. Based on this development, we have designed +a domain specific language (DSL) for describing saddle problems, which we refer to as dis- +ciplined saddle programming (DSP). When a problem description complies with the syntax +rules, i.e., is DSP-compliant, it is easy to verify that it is a valid saddle problem, and more +importantly, automatically carry out the dualizing trick. We have implemented the DSL in +an open source software package, also called DSP, which works with CVXPY [DB16], a DSL +for specifying and solving convex optimization problems. DSP makes it easy to specify and +solve saddle problems, without any expertise in (or even knowledge of) convex duality. Even +for those with the required expertise to carry out the dualizing trick by hand, DSP is less +tedious and error prone. +DSP is disciplined, meaning it is based on a small number of syntax rules that, if followed, +guarantee that the specified problem is a valid saddle problem. It is analogous to disciplined +convex programming (DCP) [GBY06], which is a DSL for specifying convex optimization +problems. When a problem specification follows these syntax rules, i.e., is DCP-compliant, it +is a valid convex optimization problem, and more importantly can be automatically converted +to an equivalent cone program, and then solved. As a practical matter, DCP allows a large +number of users to specify and solve even complex convex optimization problems, with no +knowledge of the reduction to cone form. Indeed, most DCP users are blissfully unaware +of how their problems are solved, i.e., a reduction to cone form. DCP was based on the +theory of conic representations of convex functions and problems, pioneered by Nesterov +and Nemirovski [NN92]. Widely used implementations of DCP include CVXPY [DB16], +Convex.jl [Ude+14], CVXR [FNB20], YALMIP [Lof04], and CVX [GB14]. Like DCP did for +convex problems, DSP makes it easy to specify and solve saddle problems, with most users +unaware of the dualization trick and reduction used to solve their problems. +3 + +1.1 +Previous and related work +Saddle problems. +Studying saddle problems is a long-standing area of research, resulting +in many theoretical insights, numerous algorithms for specific classes of problems, and a +large number of applications. +Saddle problems are often studied in the context of minimax or maximin optimization +[DM90; DP95], which, while dating back to the 1920s and the work of von Neumann and +Morgenstern on game theory [MVN53], continue to be active areas of research, with many +recent advancements for example in machine learning [Goo+14]. A variety of methods have +been developed for solving saddle point problems, including interior point methods [HT03; +Nem99], first-order methods [Kor76; Nem04; Nes07; NO09; CLO13], and second-order meth- +ods [NP06; Nes08], where many of these methods are specialized to specific classes of saddle +problems. Depending on the class of saddle problem, the methods differ in convergence rate. +For example, for the subset of smooth minimax problems, an overview of rates for different +curvature assumptions is given in [The+19]. Due to their close relation to Lagrange duality, +saddle problems are commonly studied in the context of convex analysis (see, for example, +[BV04, §5.4], [Roc70, §33–37], [RW09, §11.J], [BL06, §4.3]), with an analysis via monotone +operators given in [RY22]. +The practical usefulness of saddle programming in many applications is also increas- +ingly well known. Many applications of saddle programming are robust optimization prob- +lems [BBC11; BTEGN09]. For example, in statistics, distributionally robust models can be +used when the true distribution of the data generating process is not known [DA19]. Another +common area of application is in finance, with [CPT18, §19.3–4] describing a range of finan- +cial applications that can be characterized as saddle problems. Similarly, [Boy+17; GI03; +LB00] describe variations of the classical portfolio optimization problem as saddle problems. +Disciplined convex programming. +DCP is a grammar for constructing optimization +problems that are provably convex, meaning that they can be solved globally, efficiently +and accurately. +It is based on the rule that the convexity of a function f is preserved +under composition if all inner expressions in arguments where f is nondecreasing are convex, +and all expressions where f is nonincreasing are concave, and all other expressions are +affine. A detailed description of the composition rule is given in [BV04, §3.2.4]. Using this +rule, functions can be composed from a small set of primitives, called atoms, where each +atom has known curvature, sign, and monotonicity. Every function that can be constructed +from these atoms according to the composition rule is convex, but the converse is not true. +The DCP framework has been implemented in many programming languages, including +MATLAB [GB14; Lof04], Python [DB16], R [FNB20], and Julia [Ude+14], and is used by +researchers and practitioners in a wide range of fields. +Well-structured convex-concave saddle point problems. +As mentioned earlier, dis- +ciplined saddle programming is based on Juditsky and Nemirovski’s recent work on well- +structured convex-concave saddle point problems [JN22]. +4 + +1.2 +Outline +In §2 we describe saddle programming, which includes the classical saddle point problem, as +well as convex problems that include functions described via partial minimization or maxi- +mization of a saddle function. We describe some typical applications of saddle programming +in §3. In §4 we describe disciplined saddle programming, which is a way to specify saddle +programs in such a way that validity is easy to verify, and the reduction to an equivalent +cone program can be automated. We describe our implementation in §5, showing how sad- +dle functions, saddle extremum functions, saddle point problems, and saddle problems are +specified. We present numerical examples in §6. +2 +Saddle programming +2.1 +Saddle functions +A saddle function (also referred to as a convex-concave saddle function) f : X × Y → R +is one for which f(·, y) is convex for any fixed y ∈ Y, and f(x, ·) is concave for any fixed +x ∈ X . The argument domains X ⊆ Rn and Y ⊆ Rm must be nonempty closed convex. We +refer to x as the convex variable, and y as the concave variable, of the saddle function f. +Examples. +• Functions of x or y alone. A convex function of x, or a concave function of y, are +trivial examples of saddle functions. +• Lagrangian of a convex optimization problem. The convex optimization problem +minimize +f0(x) +subject to +Ax = b, +fi(x) ≤ 0, +i = 1, . . . , m, +with variable x ∈ Rn, where f0, . . . , fm are convex and A ∈ Rp×n, has Lagrangian +L(x, ν, λ) = f(x) + νT (Ax − b) + λ1f1(x) + · · · + λnfm(x), +for λ ≥ 0 (elementwise). It is convex in x and affine (and therefore also concave) in +y = (ν, λ), so it is a saddle function with +X = +� +i=0,...,m +dom fi, +Y = Rm ++ × Rp, +• Bi-affine function. The function f(x, y) = (Ax + b)T(Cy + d), with X = Rp and +Y = Rq, is evidently a saddle function. The inner product xT y is a special case of +a bi-affine function. For a bi-affine function, either variable can serve as the convex +variable, with the other serving as the concave variable. +5 + +• Convex-concave inner product. The function f(x, y) = F(x)TG(y), where F : Rp → +Rn is a nonnegative elementwise convex function and G : Rq → Rn is a nonnegative +elementwise concave function. +• Weighted ℓ2 norm. The function +f(x, y) = +� n +� +i=1 +yix2 +i +�1/2 +, +with X = Rn and Y = Rn ++, is a saddle function. +• Weighted log-sum-exp. The function +f(x, y) = log +� n +� +i=1 +yi exp xi +� +, +with X = Rn and Y = Rn ++, is a saddle function. +• Weighted geometric mean. The function f(x, y) = �n +i=1 yxi +i , with X = Rn ++ and Y = +Rn ++, is a saddle function. +• Quadratic form with quasi-semidefinite matrix. The function +f(x, y) = +� +x +y +�T � +P +S +ST +Q +� � +x +y +� +, +where the matrix is quasi-semidefinite, i.e., P ∈ Sn ++ (the set of symmetric positive +semidefinite matrices) and −Q ∈ Sn ++. +• Quadratic form. The function f(x, Y ) = xTY x, with X = Rn and Y = Sn ++ (the set of +symmetric positive semidefinite n × n matrices), is a saddle function. +• As more esoteric example, the function f(x, Y ) = xTY 1/2x, with X = Rn and Y = Sn ++, +is a saddle function. +Combination rules. +Saddle functions can be combined in several ways to yield saddle +functions. For example the sum of two saddle functions is a saddle function, provided the +domains have nonempty intersection. A saddle function scaled by a nonnegative scalar is +a saddle function. Scaling a saddle function with a nonpositive scalar, and swapping its +arguments, yields a saddle function: g(x, y) = −f(y, x) is a saddle function provided f is. +Saddle functions are preserved by pre-composition of the convex and concave variables with +an affine function, i.e., if f is a saddle function, so is f(Ax+ b, Cx+ d). Indeed, the bi-affine +function is just the inner product with an affine pre-composition for each of the convex and +concave variables. +6 + +2.2 +Saddle point problems +A saddle point (x⋆, y⋆) ∈ X × Y is any point that satisfies +f(x⋆, y) ≤ f(x⋆, y⋆) ≤ f(x, y⋆) for all x ∈ X , y ∈ Y. +(1) +In other words, x⋆ minimizes f(x, y⋆) over x ∈ X , and y⋆ maximizes f(x⋆, y) over y ∈ Y. +The basic saddle point problem is to find such a saddle point, +find x⋆, y⋆ which satisfy (1). +(2) +The value of the saddle point problem is f(x⋆, y⋆). +Existence of a saddle point for a saddle function is guaranteed, provided some technical +conditions hold. For example, Sion’s theorem [Sio58] guarantees the existence of a saddle +point when Y is compact. There are many other cases. +Examples. +• Matrix game. In a matrix game, player one chooses i ∈ {1, . . . , m}, and player two +chooses j ∈ {1, . . . , n}, resulting in player one paying player two the amount Cij. Player +one wants to minimize this payment, while player two wishes to maximize it. In a mixed +strategy, player one makes choices at random, from probabilities given by x and player +two makes independent choices with probabilities given by y. The expected payment +from player one to player two is then f(x, y) = xT Cy. With X = {x | x ≥ 0, 1Tx = 1}, +and similarly for Y, a saddle point corresponds to an equilibrium, where no player can +improve her position by changing (mixed) strategy. The saddle point problem consists +of finding a stable equilibrium, i.e., an optimal mixed strategy for each player. +• Lagrangian. A saddle point of a Lagrangian of a convex optimization problem is a +primal-dual optimal pair for the convex optimization problem. +2.3 +Saddle extremum functions +Suppose f is a saddle function. The function G : X → R ∪ {∞} defined by +G(x) = sup +y∈Y +f(x, y), +x ∈ X , +(3) +is called a saddle max function. Similarly, the function H : Y → R ∪ {−∞} defined by +H(x) = inf +x∈X f(x, y), +y ∈ Y, +(4) +is called a saddle min function. Saddle max functions are convex, and saddle min functions +are concave. We will use the term saddle extremum (SE) functions to refer to saddle max +or saddle min functions. Which is meant is clear from context, i.e., whether it is defined by +minimization (infimum) or maximization (supremum), or its curvature (convex or concave). +Note that in SE functions, we always maximize (or take supremum) over the concave variable, +and minimize (or take infimum) over the convex variable. This means that evaluating G(x) +os H(y) involves solving a convex optimization problem. +7 + +Examples. +• Dual function. Minimizing a Lagrangian L(x, ν, λ) over x gives the dual function of +the original convex optimization problem. +• Maximizing a Lagrangian L(x, ν, λ) over y = (ν, λ) gives the objective function re- +stricted to the feasible set. +• Conjugate of a convex function. Suppose f is convex. Then g(x, y) = f(x) − xTy is a +saddle function, the Lagrangian of the problem of minimizing f subject to x = 0. Its +saddle min is the negative conjugate function: infx g(x, y) = −f ∗(y). +• Sum of k largest entries. Consider f(x, y) = xTy, with Y = {y | 0 ≤ y ≤ 1, 1Ty = k}. +The associated saddle max function G is the sum of the k largest entries of x. +Saddle points via SE functions. +A pair (x⋆, y⋆) is a saddle point of a saddle function f +if and only if x⋆ minimizes the convex SE function G in (3) over x ∈ X , and y⋆ maximizes +the concave SE function H defined in (4) over y ∈ Y. This means that we can find saddle +points, i.e., solve the saddle point problem (2), by solving the convex optimization problem +minimize +G(x) +subject to +x ∈ X , +(5) +with variable x, and the convex optimization problem +maximize +H(y) +subject to +y ∈ Y, +(6) +with variable y. The problem (5) is called a minimax problem, since we are minimizing a +function defined as the maximum over another variable. The problem (6) is called a maximin +problem. +While the minimax problem (5) and maximin problem (6) are convex, they cannot be +directly solved by conventional methods, since the objectives themselves are defined by max- +imization and minimization, respectively. There are solution methods specifically designed +for minimax and maximin problems [LJJ20; MB09], but as we will see minimax problems +involving SE functions can be transformed to equivalent forms that can be directly solved +using conventional methods. +2.4 +Saddle problems +In this paper we consider convex optimization problems that include SE functions in the +objective or constraints, which we refer to as saddle problems. The convex problems that +solve the basic saddle point problem (5) and (6) are special cases, where the objective is an +8 + +SE function. As another example consider the problem of minimizing a convex function φ +subject to the convex SE constraint H(y) ≤ 0, which can be expressed as +minimize +φ(x) +subject to +f(x, y) ≤ 0 for all y ∈ Y, +(7) +with variable x. The constraint here is called a semi-infinite constraint, since (when Y is not +a singleton) it can be thought of as an infinite collection of convex constraints, one for each +y ∈ Y [HK93]. +Saddle problems include the minimax and maximin problems (that can be used to solve +the saddle point problem), and semi-infinite problems that involve SE functions. There are +many other examples of saddle problems, where SE functions can appear in expressions that +define the objective and constraints. +Robust cost LP. +As a more specific example of a saddle problem consider the linear +program with robust cost, +minimize +supc∈C cTx +subject to +Ax = b, +x ≥ 0, +(8) +with variable x ∈ Rn, with C = {c | Fc ≤ g}. This is an LP with worst case cost over the +polyhedron C [BBC11; BTEGN09]. This is a saddle problem with convex variable x, concave +variable y, and an objective which is a saddle max function. +2.5 +Solving saddle problems +Special cases with tractable analytical expressions. +There are cases where an SE +function can be worked out analytically. An example is the max of a linear function over a +box, +sup +l≤y≤u +yTx, = (1/2)(u + l)Tx + (1/2)(u − l)T|x|, +where the absolute value is elementwise. We will see other cases in our examples. +Subgradient methods. +We can readily compute a subgradient of a saddle max function +(or a supergradient of a saddle min function), by simply maximizing over the concave variable +(minimizing over the convex variable), which is itself a convex optimization problem. We can +then use any method to solve the saddle problem using these subgradients, e.g., subgradient- +type methods, ellipsoid method, or localization methods such as the analytic center cutting +plane method. In [MB09] such an approach is used for general minimax problems. +Methods for specific forms. +Many methods have been developed for finding saddle +points of saddle functions with the special form +f(x, y) = xTKy + φ(x) + ψ(y), +9 + +where φ is convex, ψ is concave, and K is a matrix [BS15; Con13; CP11; Nes05a; Nes05b; +CP16]. Beyond this example, there are many other special forms of saddle functions, with +different methods adapted to properties such as smoothness, separability, and strong-convex- +strong-concavity. +2.6 +Dual reduction +A well-known trick can be used to transform a saddle point problem into an equivalent prob- +lem that does not contain SE functions. This method of transforming an inner minimization +is not new; it has been used since the 1950s when Von Neumann proved the minimax the- +orem using strong duality in his work with Morgenstern on game theory [MVN53]. Using +this observation, he showed that the minimax problem of a two player game is equivalent +to an LP. Duality allows us to express the convex (concave) SE function as an infimum +(supremum), which facilitates the use of standard convex optimization. We think of this as +a reduction to an equivalent problem that removes the SE functions from the objective and +constraints. +Robust cost LP. +We illustrate the dualization method for the robust cost LP (8). The +key is to express the robust cost or saddle max function supc∈C cTx as an infimum. We first +observe that this saddle max function is the optimal value of the LP +maximize +xT c +subject to +Fc ≤ g, +with variable c. Its dual is +minimize +gTλ +subject to +F Tλ = x, +λ ≥ 0, +with variable λ. Assuming that C is nonempty, this dual problem has the same optimal value +as the primal, i.e., +sup +c∈C +cTx = +inf +λ≥0, F T λ=x gTλ +Substituting this into (8) we obtain the problem +minimize +gTλ +subject to +Ax = b, +x ≥ 0, +F Tλ = x, +λ ≥ 0, +(9) +with variables x and λ. This simple LP is equivalent to the original robust LP (8), in the +sense that if (x⋆, λ⋆) is a solution of (9), then x⋆ is a solution of the robust LP (8). +We will see this dualization trick in a far more general setting in §4. +3 +Applications +In this section we describe a few applications of saddle programming. +10 + +3.1 +Robust bond portfolio construction +We describe here a simplified version of the problem described in much more detail in +[LSB22]. +Our goal is to construct a portfolio of n bonds, giving by its holdings vector +h ∈ Rn ++, where hi is the number of bond i held in the portfolio. Each bond produces a cash +flow, i.e., a sequence of payments to the portfolio holder, up to some period T. Let ci,t be +the payment from bond i in time period t. Let y ∈ RT be the yield curve, which gives the +time value of cash: A payment of one dollar at time t is worth exp(−tyt) current dollars, +assuming continuously compounded returns. The bond portfolio value, which is the present +value of the total cash flow, can be expressed as +V (h, y) = +n +� +i=1 +T +� +t=1 +hici,t exp(−tyt). +This function is convex in the yields y and concave (in fact, linear) in the holdings vector h. +Now suppose we do not know the yield curve, but instead have a convex set Y of possible +values, with y ∈ Y. The worst case value of the bond portfolio, over this set of possible yield +curves, is +V wc(h) = inf +y∈Y V (h, y). +We recognize this as a saddle min function. (In this application, y is the convex variable +of the saddle function V , whereas elsewhere in this paper we use y to denote the concave +variable.) +We consider a robust bond portfolio construction problem of the form +minimize +φ(h) +subject to +h ∈ H, +V wc(h) ≥ V lim, +(10) +where φ is a convex objective, typically a measure of return and risk, H is a convex set +of portfolio constraints (for example, imposing h ≥ 0 and a total budget), and V lim is a +specified limit on worst case value of the portfolio over the yield curve set Y, which has a +saddle min as a constraint. +For some simple choices of Y the worst case value can be found analytically. One example +is when Y has a maximum element. +In this special case, the maximum element is the +minimizer of the value over Y (since V is a monotone decreasing function of y). For other +cases, however, we need to solve the saddle problem (10). +3.2 +Model fitting robust to data weights +We wish to fit a model parametrized by θ ∈ Θ ⊆ Rn to m observed data points. We do this +by minimizing a weighted loss over the observed data, plus a regularizer, +m +� +i=1 +wiℓi(θ) + r(θ), +11 + +where ℓi is the convex loss function for observed data point i, r is a convex regularizer +function, and the weights wi are nonnegative. The weights can be used to adjust a data +sample that was not representative, as in [BAB21], or to ignore some of the data points (by +taking wi = 0), as in [BGM20]. Evidently the weighted loss is a saddle function, with convex +variable θ and concave variable w. +We consider the case when the weights are unknown, but lie in a convex set, w ∈ W. The +robust fitting problem is to choose θ to minimize the worst case loss over the set of possible +weights, plus the regularizer, +max +w∈W +m +� +i=1 +wiℓi(θ) + r(θ). +We recognize the first term, i.e., the worst case loss over the set of possible weights, as a +saddle max function. +For some simple choices of W the worst case loss can be expressed analytically. For +example with +W = {w | 0 ≤ w ≤ 1, 1Tw = k}, +(with k ∈ [0, n]), the worst case loss is given by +max +w∈W +m +� +i=1 +wiℓi(θ) = φ(ℓ1, . . . , ℓm), +where φ is the sum-of-k-largest entries [BV04, §3.2.3]. (Our choice of symbol k suggests that +k is an integer, but it need not be.) In this case we judge the model parameter θ by its worst +loss on any subset of k of data points. Put another way, we judge θ by dropping the m − k +data points on which it does best (i.e., has the smallest loss) [BGM20]. +CVXPY directly supports the sum-of-k-largest function, so the robust fitting problem +can be formulated and solved without using DSP. To support this function, CVXPY carries +out a transformation very similar to the one that DSP does. +The difference is that the +transformation in CVXPY is specific to this one function, whereas the one carried out in +DSP is general, and would work for other convex weight sets. +3.3 +Robust production problem with worst case prices +We consider the choice of a vector of quantities q ∈ Q ⊆ Rn. Positive entries indicate goods +we buy, and negative quantities are goods we sell. The set of possible quantities Q is our +production set, which is convex. In addition, we have a manufacturing cost associated with +the choice q, given by φ(q), where φ is a convex function. The total cost is the manufacturing +cost plus the cost of goods (which includes revenue), φ(q) + pTq, where p ∈ Rn is vector of +prices. +We consider the situation when we do not know the prices, but we have a convex set +they lie in, p ∈ P. The worst case cost of the goods is maxp∈P pTq. The robust production +problem is +minimize +φ(q) + maxp∈P pTq +subject to +q ∈ Q, +(11) +12 + +with variable q. Here too we can work out analytical expressions for simple choices of P, +such as a range for each component, in which case the worst case price is the upper limit +for goods we buy, and the lower limit for goods we sell. In other cases, we solve the saddle +problem (11). +3.4 +Robust Markowitz portfolio construction +Markowitz portfolio construction [Mar52] chooses a set of weights (the fraction of the total +portfolio value held in each asset) by solving the convex problem +maximize +µTw − γwTΣw +subject to +1Tw = 1, +w ∈ W, +where the variable is the vector of portfolio weights w ∈ Rn, µ ∈ Rn is a forecast of the +asset returns, γ > 0 is the risk aversion parameter, Σ ∈ Sn +++ is a forecast of the asset return +covariance matrix, and W is a convex set of feasible portfolios. The objective is called the +risk adjusted (mean) return. +Markowitz portfolio construction is known to be fairly sensitive to the (forecasts) µ and +Σ, which have to be chosen with some care; see, e.g., [BL91]. One approach is to specify +a convex uncertainty set U that (µ, Σ) must lie in, and replace the objective with its worst +case (smallest) value over this uncertainty set. This gives the robust Markowitz portfolio +construction problem +maximize +inf(µ,Σ)∈U +� +µTw − γwTΣw +� +subject to +1T w = 1, +w ∈ W, +with variable w. This is described in, e.g., in [Boy+17; GI03; LB00]. We observe that this +is directly a saddle problem, with a saddle min objective, i.e., a maximin problem. +For some simple versions of the problem we can work out the saddle min function explic- +itly. One example, given in [Boy+17], uses U = M × S, where +M += +{µ + δ | |δ| ≤ ρ}, +S += +{Σ + ∆ | Σ + ∆ ⪰ 0, |∆ij| ≤ η(ΣiiΣjj)1/2, i, j = 1, . . . , n}, +where ρ > 0 is a vector of uncertainties in the forecast returns, and η ∈ (0, 1) is a parameter +that scales the perturbation to the forecast covariance matrix. (We interpret δ and ∆ as +perturbations of the nominal mean and covariance µ and Σ, respectively.) We can express +the worst case risk adjusted return analytically as +inf +(µ,Σ)∈U +� +µTw − γwTΣw +� += µTw − γwTΣw − ρT |w| − γη +� n +� +i=1 +Σ1/2 +ii |wi| +�2 +. +The first two terms are the nominal risk adjusted return; the last two terms (which are +nonpositive) represent the cost of uncertainty. +13 + +4 +Disciplined saddle point programming +4.1 +Saddle function calculus +We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function with concave +variables x and convex variables y. The set of operations that, when performed on saddle +functions, preserves the saddle property are called the saddle function calculus. The calculus +is quite simple, and consists of the following operations: +1. Conic combination of saddle functions. Let φi(xi, yi), i = 1, . . . , k be saddle functions. +Let θi ≥ 0 for each i. Then the conic combination, φ(x, y) = �k +i=1 θiφi(xi, yi), is a +saddle function. +2. Affine precomposition of saddle functions. Let φ(x, y) be a saddle function, with x ∈ Rn +and y ∈ Rm. Let A ∈ Rn×q, b ∈ Rn, C ∈ Rm×p, and d ∈ Rm. Then, with u ∈ Rq and +v ∈ Rp, the affine precomposition, φ(Au + b, Cv + d), is a saddle function. +3. Precomposition of saddle functions. Let φ(x, y) : X × Y ⊆ Rn×m → R be a saddle +function, with x ∈ Rn and y ∈ Rm. The precomposition with a function f : Rp → Rn, +φ(f(u), y), is a saddle function if for each i = 1, . . . , n one of the following holds: +• fi(u) is convex and φ is nondecreasing in xi for all y ∈ Y and all x ∈ X . +• fi(u) is concave and φ is nonincreasing in xi for all y ∈ Y and all x ∈ X . +Similarly, the precomposition with a function g : Rq → Rm, φ(x, g(v)), is a saddle +function if for each j = 1, . . . , m one of the following holds: +• gj(v) is convex and φ is nonincreasing in yj for all x ∈ X and all y ∈ Y. +• gj(v) is concave and φ is nondecreasing in yj for all x ∈ X and all y ∈ Y. +4.2 +Conically representable saddle functions +Nemirovski and Juditsky propose a class of conic representable saddle functions which fa- +cilitate the automated dualization of saddle problems [JN22]. We will first introduce some +terminology and notation, and then describe the class of conic representable saddle functions. +Notation. +We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function +which is convex in x and concave in y. Let Kx, Ky and K be members of a collection K +of closed, convex, and pointed cones with nonempty interiors in Euclidean spaces such that +K contains a nonnegative ray, is closed with respect to taking finite direct products of its +members, and is closed with respect to passing from a cone to its dual. We denote conic +membership z ∈ K by z ⪰K 0. We call a set X ⊆ Rn K-representable if there exist constant +matrices A and B, a constant vector c, and a cone K ∈ K such that +X = {x | ∃u : Ax + Bu ⪯K c}. +14 + +CVXPY [DB16] can implement a function f exactly when its epigraph {(x, u) | f(x) ≤ u} +is K-representable. +Conically representable saddle functions. +Let X and Y be nonempty and possessing +K-representations +X = {x | ∃u : Ax + Bu ⪯K c}, +Y = {y | ∃v : Cy + Dv ⪯K e}. +A saddle function φ(x, y) : X × Y → R is K-representable if there exist constant matrices +P, Q, R, constant vectors p and s and a cone K ∈ K such that for each x ∈ X and y ∈ Y, +φ(x, y) = inf{f Ty + t | Pf + tp + Qu + Rx ⪯K s}. +This definition generalizes simple class of bilinear saddle functions. See [JN22] for much +more detail. +Automated dualization. +Suppose we have a K-representable saddle function φ as above. +The power of the conic form is that the saddle extremum +Φ(x) = sup +y∈Y +φ(x, y) +admits a tractable conic form, meaning that it can be implemented in a DSL like CVXPY. +Specifically, +Φ(x) += +sup +y∈Y +φ(x, y) += +sup +y∈Y +inf +f,t,u +� +f Ty + t +�� Pf + tp + Qu + Rx ⪯K s +� += +inf +f,t,u +� +sup +y∈Y +� +f Ty + t +� ���� Pf + tp + Qu + Rx ⪯K s +� +(12) += +inf +f,t,u +� +sup +y∈Y +� +f Ty +� ++ t +���� Pf + tp + Qu + Rx ⪯K s +� += +inf +f,t,u +� +inf +λ +� +λTe +���� +CTλ = f, DTλ = 0 +λ ⪰K∗ 0 +� ���� Pf + tp + Qu + Rx ⪯K s +� +(13) +where in (12) we use Sion’s minimax theorem [Sio58] to reverse the inf and sup, and in (13) +we invoke strong duality to replace the supremum over y with an infimum over λ. The final +line implies a conic representation of the epigraph of Φ(x), +{(x, u) | Φ(x) ≤ u} = + + +(x, u) +������ +∃λ, f, t, u : +λTe + t ≤ u +CTλ = f, DTλ = 0, λ ⪰K∗ 0 +Pf + tp + Qu + Rx ⪯K s + + + , +which is tractable and can be implemented in a DSL like CVXPY. +15 + +A mathematical nuance. +Switching the inf and sup in (12) requires Sion’s theorem to +hold. A sufficient condition for Sion’s theorem to hold is that the set Y is compact. However, +the min and max can be exchanged even if Y is not compact. Then, due to the max-min +inequality +max +y∈Y min +x∈X f(x, y) ≤ min +x∈X max +y∈Y f(x, y), +the equality in (13) is replaced with a less than or equal to, and we obtain a convex restriction. +Thus, if a user creates a problem involving an SE function (as opposed to a saddle point +problem only containing saddle functions in the objective), then DSP guarantees that the +problem generated is a restriction. This means that the variables returned are feasible and +the returned optimal value is an upper bound on the optimal value for the user’s problem. +In our implementation, a saddle problem is solved by applying the above automatic +dualization to both the objective f and −f and then solving each resulting convex problem, +with the latter having the role of convex and concave variables switched. We do so in order +to obtain both the convex and concave components of the saddle point, since the dualization +removes the concave variable. The saddle problem is only reported as solved if the optimal +value of the problem with objective f is within a numerical tolerance of the negated optimal +value of the problem with objective −f. If this holds, this actually implies that +max +y∈Y min +x∈X f(x, y) = min +x∈X max +y∈Y f(x, y), +i.e., (12) was valid, even if for example Y is not compact. Thus, a user need not concern +themselves with the compactness of Y (or any other sufficient condition for Sion’s theorem) +when using DSP to find a saddle point; if a saddle point problem is solved, then the saddle +point is guaranteed to exist. +5 +Implementation +In this section we describe our open source Python implementation of the concepts and +methods described in §4, which we also call DSP. DSP works with CVXPY [DB16], an +implementation of a DSL for convex optimization based on DCP. We use the term DSP in +two different ways. We use it to refer to the mathematical concept of disciplined saddle +programming, and also our specific implementation; which is meant should be clear from +the context. The term DSP-compliant refers to a function or expression that is constructed +according to the DSP composition rules given in §5.2. It can also refer to a problem that +is constructed according to these rules. In the code snippets below, we use the prefix cp +to indicate functions and classes from CVXPY. (We give functions and classes from DSP +without prefix, whereas they would likely have a prefix such as dsp in real code.) +5.1 +Atoms +Saddle functions in DSP are created from fundamental building blocks or atoms. These +building blocks extend the atoms from CVXPY [DB16]. In CVXPY, atoms are either jointly +16 + +convex or concave in all their variables, but in DSP, atoms are (jointly) convex in a subset +of the variables and (jointly) concave in the remaining variables. We describe some DSP +atoms below. +Inner product. +The atom inner(x,y) represents the inner product xTy. Since either +x or y could represent the convex variable, we adopt the convention in DSP that the first +argument of inner is the convex variable. According to the DSP rules, both arguments to +inner must be affine, and the variables they depend on must be disjoint. +Saddle inner product. +The atom saddle_inner(F, G) corresponds to the function +F(x)TG(y), where F and G are vectors of nonnegative and respectively elementwise convex +and concave functions. It is DSP-compliant if F is DCP convex and nonnegative and G is +DCP concave. If the function G is not DCP nonnegative, then the DCP constraint G >= 0 +is attached to the expression. This is analogous to how the DCP constraint x >= 0 is added +to the expression cp.log(x). As an example consider +f = saddle_inner(cp.square(x), cp.log(y)). +This represents the saddle function +f(x, y) = x2 log y − I(y ≥ 1), +where I is the {0, ∞} indicator function of its argument. +Weighted ℓ2 norm. +The weighted_norm2(x, y) atom represents the saddle function +(�n +i=1 yix2 +i )1/2, with y ≥ 0. It is DSP-compliant if x is either DCP affine or both convex and +nonnegative, and y is DCP concave. Here too, the constraint y >= 0 is added if y is not +DCP nonnegative. +Weighted log-sum-exp. +The weighted_log_sum_exp(x, y) atom represents the saddle +function log (�n +i=1 yi exp xi), with y ≥ 0. It is DSP-compliant if x is DCP convex, and y is +DCP concave. The constraint y >= 0 is added if y is not DCP nonnegative. +Quasi-semidefinite quadratic form. +The quasidef_quad_form(x, y, P, Q, S) atom +represents the function +f(x, y) = +� x +y +�T � P +S +ST +Q +� � x +y +� +, +where the matrix is quasi-semidefinite, i.e., P ∈ Sn ++ and −Q ∈ Sn ++. It is DSP-compliant if x +is DCP affine and y is DCP affine. +Quadratic form. +The saddle_quad_form(x, Y) atom represents the function xTY x, +where Y is a PSD matrix. It is DSP-compliant if x is DCP affine, and Y is DCP PSD. +17 + +5.2 +Calculus rules +The atoms can be combined according to the calculus described below to form expressions +that are DSP-compliant. +For example, saddle functions can be added or scaled. +DCP- +compliant convex and concave expressions are promoted to saddle functions with no concave +or convex variables, respectively. For example, with variables x, y, and z, the expression +f = 2.5 * saddle_inner(cp.square(x), cp.log(y)) + cp.minimum(y,1) - z +is DSP-compliant, with convex variable x, concave variable y, and affine variable z. +Calling the is_dsp method of an expression returns True if the expression is DSP- +compliant. The methods convex_variables, concave_variables, and affine_variables, +list the convex, concave, and affine variables, respectively. The convex variables are those +that could only be convex, and similarly for concave variables. +We refer to the convex +variables as the unambiguously convex variables, and similarly for the concave variables. +The three lists of variables gives a partition of all the variables the expression depends on. +For the expression above, f.is_dsp() evaluates as True, f.convex_variables() returns the +list [x], f.concave_variables() returns the list [y], and f.affine_variables() returns +the list [z]. Note that the role of z is ambiguous in the expression, since it could be either +a convex or concave variable. +No mixing variables rule. +The DSP rules prohibit mixing of convex and concave vari- +ables. For example if we add two saddle expressions, no variable can appear in both its +convex and concave variable lists. +DSP-compliance is sufficient but not necessary to be a saddle function. +Re- +call that if an expression is DCP convex (concave), then it is convex (concave), but the +converse is false. +For example, the expression cp.sqrt(1 + cp.square(x)) represents +the convex function +√ +1 + x2, but is not DCP. But we can express the same function as +cp.norm2(cp.hstack([1, x])), which is DCP. The same holds for DSP and saddle func- +tion: If an expression is DSP-compliant, then it represents a saddle function; but it can +represent a saddle function and not be DSP-compliant. As with DCP, such an expression +would need to be rewritten in DSP-compliant form, to use any of the other features of DSP +(such as a solution method). As an example, the expression x.T @ C @ y represents the +saddle function xT Cy, but is not DSP-compliant. The same function can be expressed as +inner(x, C @ y), which is DSP-compliant. +When there are affine variables in a DSP-compliant expression, it means that those +variables could be considered either convex or concave; either way, the function is a saddle +function. +Example. +The code below defines the bi-linear saddle function f(x, y) = xT Cy, the ob- +jective of a matrix game, with x the convex variable and y the concave variable. +18 + +Creating a saddle function. +1 from dsp import * +# notational convenience +2 import cvxpy as cp +3 import numpy as np +4 +5 x = cp.Variable(2) +6 y = cp.Variable(2) +7 C = np.array([[1, 2], [3, 1]]) +8 +9 f = inner(x, C @ y) +10 +11 f.is_dsp() +# True +12 +13 f.convex_variables() +# [x] +14 f.concave_variables() +# [y] +15 f.affine_variables() +# [] +Lines 1–3 import the necessary packages (which we will use but not show in the sequel). +In lines 5–7, we create two CVXPY variables and a constant matrix. In line 9 we construct +the saddle function f using the DSP atom inner. Both its arguments are affine, so this +matches the DSP rules. In line 11 we check if saddle_function is DSP-compliant, which +it is. In lines 13–15 we call functions that return lists of the convex, concave, and affine +variables, respectively. The results of lines 13–15 might seem odd, but recall that inner +marks its first argument as convex and its second as concave. +5.3 +Saddle point problems +Saddle point problem objective. +To construct a saddle point problem, we first create +an objective using +obj = MinimizeMaximize(f), +where f is a CVXPY expression. +The objective obj is DSP-compliant if the expression +f is DSP-compliant. This is analogous to the CVXPY contructors cp.Minimize(f) and +cp.Maximize(f), which create objectives from expressions. +Saddle point problem. +A saddle point problem is constructed using +prob = SaddlePointProblem(obj, constraints, cvx_vars, ccv_vars) +Here, obj is a MinimizeMaximize objective, constraints is a list of constraints, cvx_vars +is a list of convex variables and ccv_vars is a list of concave variables. The objective must +be DSP-compliant for the problem to be DSP-compliant. We now describe the remaining +conditions under which the constructed problem is DSP-compliant. +19 + +Each constraint in the list must be DCP, and can only involve convex variables or concave +variables; convex and concave variables cannot both appear in any one constraint. The list +of convex and concave variables partitions all the variables that appear in the objective or +the constraints. In cases where the role of a variable is unambiguous, it is inferred, and does +not need to be in either list. For example with the objective +MinimizeMaximize(weighted_log_sum_exp(x, y) + cp.exp(u) + cp.log(v) + z), +x and u must be convex variables, and y and v must be concave variables, and so do not need +to appear in the lists used to construct a saddle point problem. The variable z, however, +could be either a convex or concave variable, and so must appear in one of the lists. +The role of a variable can also be inferred from the constraints: Any variable that appears +in a constraint with convex (concave) variables must also be convex (concave). With the +objective above, the constraint z + v <= 1 would serve to classify z as a concave variable. +With this constraint, we could pass empty variable lists to the saddle point constructor, since +the roles of all variables can be inferred. When the roles of all variables are unambiguous, +the lists are optional. +The roles of the variables in a saddle point problem prob can be found by calling +prob.convex_variables() and prob.concave_variables(), which return lists of vari- +ables, and is a partition of all the variables appearing in the objective or constraints. This is +useful for debugging, to be sure that DSP agrees with you about the roles of all variables. A +DSP-compliant saddle point problem must have an empty list of affine variables. (If it did +not, the problem would be ambiguous.) +Solving a saddle point problem. +The solve() method of a SaddlePointProblem object +solves the problem. The solve method returns the optimal saddle value, i.e., the value of +the objective at the saddle point. As in CVXPY, the solve method has the side effect of +writing all variables’ .value attribute. +Example. +Here we create and solve a matrix game, continuing the example above where +f was defined. We do not need to pass in lists of variables since their roles can be inferred. +Creating and solving a matrix game. +1 obj = MinimizeMaximize(f) +2 constraints = [x >= 0, cp.sum(x) == 1, y >= 0, cp.sum(y) == 1] +3 prob = SaddlePointProblem(obj, constraints) +4 +5 prob.is_dsp() +# True +6 prob.convex_variables() +# [x] +7 prob.concave_variables() +# [y] +8 prob.affine_variables() +# [] +9 +10 prob.solve() +# solves the problem +20 + +11 prob.value +# 1.6666666666666667 +12 x.value +# array([0.66666667, 0.33333333]) +13 y.value +# array([0.33333333, 0.66666667]) +5.4 +Saddle extremum functions +Local variables. +An SE function has one of the forms +G(x) = sup +y∈Y +f(x, y) +or +H(y) = inf +x∈X f(x, y), +where f is saddle function. Note that y in the definition of G, and x in the definition of +H, are local or dummy variables, understood to have no connection to any other variable. +Their scope extends only to the definition, and not beyond. +To express this subtlety in DSP, we use the class LocalVariable to represent these +dummy variables. The variables that are maximized over (in a saddle max function) or +minimized over (in a saddle min function) must be declared using the LocalVariable() +constructor. Any LocalVariable in an SE function cannot appear in any other SE function. +Constructing SE functions. +We construct SE functions in DSP using +saddle_max(f, constraints) +or +saddle_min(f, constraints). +Here, f is a CVXPY scalar expression, and constraints is a list of constraints. We now +describe the rules for constructing a DSP-compliant SE function. +If a saddle_max is being constructed, f must be DSP-compliant, and the function’s con- +cave variables, and all variables appearing in the list of constraints, must be LocalVariables, +while the function’s convex variables must all be regular Variables. A similar rule applies +for saddle_min. +The list of constraints is used to specify the set over which the sup or inf is taken. Each +constraint must be DCP-compliant, and can only contain LocalVariables. +With x a Variable, y_loc a LocalVariable, z_loc a LocalVariable, and z a Variable, +consider the following two SE functions: +1 f_1 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1]) +2 f_2 = saddle_max(inner(x, y_loc) + z_loc, [y_loc <= 1, z_loc <= 1]) +Both are DSP-compliant. For the first, calling f_1.convex_variables() would return +[x, z], and calling f_1.concave_variables() would return [y_loc]. +For the second, +calling f_2.convex_variables() would return [x], and f_2.concave_variables() return +[y_loc, z_loc]. +Let y be a Variable. Both of the following are not DSP-compliant: +1 f_3 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1, z <= 1]) +2 f_4 = saddle_max(inner(x, y) + z_loc, [y_loc <= 1, z_loc <= 1]) +21 + +The first is not DSP-compliant because z is not a LocalVariable, but appears in the +constraints. +The second is not DSP-compliant because y is not a LocalVariable, but +appears as a concave variable in the saddle function. +SE functions are DCP. +When they are DSP-compliant, a saddle_max is a convex func- +tion, and a saddle_min is a concave function. They can be used anywhere in CVXPY that a +convex or concave function is appropriate. You can add them, compose them (in appropriate +ways), use them in the objective or either side of constraints (in appropriate ways). +Examples. +Now we provide full examples demonstrating construction of a saddle_max, +which we can use to solve the matrix game described in §5.3 as a saddle problem involving +an SE function. +Creating a saddle max. +1 # Creating variables +2 x = cp.Variable(2) +3 +4 # Creating local variables +5 y_loc = LocalVariable(2) +6 +7 # Convex in x, concave in y_loc +8 f = saddle_inner(C @ x, y_loc) +9 +10 # maximizes over y_loc +11 G = saddle_max(f, [y_loc >= 0, cp.sum(y_loc) == 1]) +Note that G is a CVXPY expression. Constructing a saddle_min works exactly the same +way. +5.5 +Saddle problems +A saddle problem is a convex problem that uses SE functions. To be DSP-compliant, the +problem must be DCP (which implies all SE functions are DSP-compliant). +When you +call the solve method on a saddle problem involving SE functions, and the solve is suc- +cessful, then all variables’ .value fields are overwritten with optimal values. This includes +LocalVariables that the SE functions maximized or minimized over; they are assigned to +the value of a particular maximizer or minimizer of the SE function at the value of the +non-local variables, with no further guarantees. +Example. +We continue our example from §5.4 and solve the matrix game using either a +saddle max. +22 + +Creating and solving a saddle problem using a saddle max to solve the matrix game. +1 prob = cp.Problem(cp.Minimize(G), [x >= 0, cp.sum(x) == 1]) +2 +3 prob.is_dsp() +# True +4 +5 prob.solve() +# solving the problem +6 prob.value +# 1.6666666666666667 +7 x.value +# array([0.66666667, 0.33333333]) +6 +Examples +In this section we give numerical examples, taken from §3, showing how to create DSP- +compliant problems. The specific problem instances we take are small, since our main point +is to show how easily the problems can be specified in DSP. But DSP will scale to far larger +problem instances. +6.1 +Robust bond portfolio construction +Our first example is the robust bond portfolio construction problem described in §3.1. We +consider portfolios of n = 20 bonds, over a period T = 60 half-years, i.e., 30 years. The +bonds are taken as representative ones in a global investment grade bond portfolio; for more +detail, see [LSB22]. The payments from the bonds are given by C ∈ R20×60, with cash flow +of bond i in period t denoted ci,t. The portfolio constraint set H is given by +H = {h | h ≥ 0, pTh = B}, +i.e., the investments must be nonnegative and have a total value (budget) B, which we take +to be $100. Here p ∈ R20 ++ denotes the price of the bonds on September 12, 2022. The +portfolio objective is +φ(h) = 1 +2∥(h − hmkt) ◦ p∥1, +where hmkt is the market portfolio scaled to a value of $100, and ◦ denotes Hadamard or +elementwise multiplication. This is called the turn-over distance, since it tells us how much +we would need to buy and sell to convert our portfolio to the market portfolio. +The yield curve set Y is described in terms of perturbations to the nominal or current +yield curve ynom, which is the yield curve on September 12, 2022. We take +Y = +� +ynom + δ +����� ∥δ∥∞ ≤ δmax, ∥δ∥1 ≤ κ, +T−1 +� +t=1 +(δt+1 − δt)2 ≤ ω +� +. +We interpret δ as a shock to the yield curve, which we limit elementwise, in absolute sum, +and in smoothness. The specific parameter values are given by +δmax = 0.02, +κ = 0.9, +ω = 10−6. +23 + +In the robust bond portfolio problem (10) we take V lim = 90, that is, the worst case value +of the portfolio cannot drop below $90 for any y ∈ Y. +We solve the problem using the following code, where we assume the cash flow matrix +C, the price vector p, the nominal yield curve y_nom, and the market portfolio h_mkt are +defined. +Robust bond portfolio construction. +1 # Constants and parameters +2 n, T = C.shape +3 delta_max, kappa, omega = 0.02, 0.9, 1e-6 +4 B = 100 +5 V_lim = 90 +6 +7 # Creating variables +8 h = cp.Variable(n, nonneg=True) +9 +10 delta = LocalVariable(T) +11 y = y_nom + delta +12 +13 # Objective +14 phi = 0.5 * cp.norm1(cp.multiply(h, p) - cp.multiply(h_mkt, p)) +15 +16 # Creating saddle min function +17 V = 0 +18 for i in range(n): +19 +t_plus_1 = np.arange(T) + 1 +# Account for zero-indexing +20 +V += saddle_inner(cp.exp(cp.multiply(-t_plus_1, y)), h[i] * C[i]) +21 +22 Y = [ +23 +cp.norm_inf(delta) <= delta_max, +24 +cp.norm1(delta) <= kappa, +25 +cp.sum_squares(delta[1:] - delta[:-1]) <= omega, +26 ] +27 +28 V_wc = saddle_min(V, Y) +29 +30 # Creating and solving the problem +31 problem = cp.Problem(cp.Minimize(phi), [h @ p == B, V_wc >= V_lim]) +32 problem.solve() +# 15.32 +We first define the constants and parameters in lines 2–5, before creating the variable +for the holdings h in line 8, and the LocalVariable delta, which gives the yield curve +24 + +Nominal portfolio +Robust portfolio +Turn-over distance +$0.00 +$15.32 +Worst-case value +$86.99 +$90.00 +Table 1: +Turn-over distance and worst-case value for the nominal (market) portfolio and the +robust portfolio. The nominal portfolio does not meet our requirement that the worst-case value +be at least $90. +perturbation, in line 10. In line 11 we define y as the sum of the current yield curve y_nom +and the perturbation delta. The objective function is defined in line 14. Lines 17–20 define +the saddle function V via the saddle_inner atom. The yield uncertainty set Y is defined in +lines 22–26, and the worst case portfolio value is defined in line 25 using saddle_min. We use +the concave expression saddle_min to create and solve a CVXPY problem in lines 31–32. +Table 1 summarizes the results. The nominal portfolio is the market portfolio, which +has zero turn-over distance to the market portfolio, i.e., zero objective value. This nominal +portfolio, however, does not satisfy the worst-case portfolio value constraint, since there are +yield curves in Y that cause the portfolio value to drop to around $87, less than our limit +of $90. The solution of the robust problem has turn-over distance $15.32, and satisfies the +constraint that the worst-case value be at least $90. +6.2 +Model fitting robust to data weights +We consider an instance of the model fitting problem described in §3.2. We use the well +known Titanic data set [HC17], which gives several attributes for each passenger on the ill- +fated Titanic voyage, including whether they survived. A classifier is fit to predict survival +based on the features sex, age (binned into three groups, 0–26, 26–53, and 53–80), and class +(1, 2, or 3). These features are encoded as a Boolean vector ai ∈ R7. The label yi = 1 means +passenger i survived, and yi = −1 otherwise. There are 1046 examples, but we fit our model +using only the m = 50 passengers who embarked from Queenstown, one of three ports of +embarkation. This is a somewhat non-representative sample; for example, the survival rate +among Queenstown departures is 26%, whereas the overall survival rate is 40.8%. +We seek a linear classifier ˆyi = sign(aT +i θ + β0), where θ ∈ R7 is the classifier parameter +vector and β0 ∈ R is the bias. The hinge loss and ℓ2 regularization are used, given by +ℓi(θ) = max(0, 1 − yiaT +i θ), +r(θ) = η∥θ∥2 +2, +with η = 0.05. +The data is weighted to partially correct for the different survival rates for our training +set (26%) and the whole data set (40.8%). To do this we set wi = z1 when yi = 1 and +wi = z2 when yi = −1. We require w ≥ 0 and 1Tw = 1, and +0.408 − 0.05 ≤ +� +yi=1 +wi ≤ 0.408 + 0.05. +25 + +Thus W consists of weights on the Queenstown departure samples that correct the survival +rate to within 5% of the overall survival rate. +The code shown below solves this problem, where we assume the data matrix is already +defined as A_train (with rows aT +i ), the survival label vector is defined as y_train, and the +indicator of survival in the training set is defined as surv. +Model fitting robust to data weights. +1 # Constants and parameters +2 m, n = A_train.shape +3 inds_0 = surv == 0 +4 inds_1 = surv == 1 +5 eta = 0.05 +6 +7 # Creating variables +8 theta = cp.Variable(n) +9 beta_0 = cp.Variable() +10 weights = cp.Variable(m, nonneg=True) +11 surv_weight_0 = cp.Variable() +12 surv_weight_1 = cp.Variable() +13 +14 # Defining the loss function and the weight constraints +15 y_hat = A_train @ theta + beta_0 +16 loss = cp.pos(1 - cp.multiply(y_train, y_hat)) +17 objective = MinimizeMaximize(saddle_inner(loss, weights) +18 ++ eta * cp.sum_squares(theta)) +19 +20 constraints = [ +21 +cp.sum(weights) == 1, +22 +0.408 - 0.05 <= weights @ surv, +23 +weights @ surv <= 0.408 + 0.05, +24 +weights[inds_0] == surv_weight_0, +25 +weights[inds_1] == surv_weight_1, +26 ] +27 +28 # Creating and solving the problem +29 problem = SaddlePointProblem(objective, constraints) +30 problem.solve() +After defining the constants and parameters in lines 2–5, we specify the variables for the +model coefficient and the weights in lines 8–9 and 10–12, respectively. The loss function +and regularizer which make up the objective are defined next in lines 15–18. The weight +constraints are defined in lines 20–26. The saddle point problem is created and solved in +26 + +Nominal classifier +Robust classifier +Train accuracy +82.0% +80.0% +Test accuracy +76.0% +78.6% +Table 2: Nominal and worst-case objective values for classification and robust classification models. +lines 29 and 30. +The results are shown in table 2. We report the test accuracy on all samples in the +dataset with a different port of embarkation than Queenstown (996 samples). We see that +while the robust classification model has slightly lower training accuracy than the nominal +model, it achieves a higher test accuracy, generalizing from the non-representative training +data better than the nominal classifier, which uses uniform weights. +6.3 +Robust Markowitz portfolio construction +We consider the robust Markowitz portfolio construction problem described in §3.4. We take +n = 6 assets, which are the (five) Fama-French factors [FF15] plus a risk-free asset. The +data is obtained from the Kenneth R. French data library [Fre22], with monthly return data +available from July 1963 to October 2022. The nominal return and risk are the empirical +mean and covariance of the returns. (These obviously involve look-ahead, but the point of +the example is how to specify and solve the problem with DSP, not the construction of a +real portfolio.) We take parameters ρ = 0.02, η = 0.2, and risk aversion parameter γ = 1. +In the code, we use mu and Sigma for the mean and covariance estimates, respectively, +and the parameters are denoted rho, eta, and gamma. +Robust Markowitz portfolio construction. +1 # Constants and parameters +2 n = len(mu) +3 rho, eta, gamma = 0.2, 0.2, 1 +4 +5 # Creating variables +6 w = cp.Variable(n, nonneg=True) +7 +8 delta_loc = LocalVariable(n) +9 Sigma_perturbed = LocalVariable((n, n), PSD=True) +10 Delta_loc = LocalVariable((n, n)) +11 +12 # Creating saddle min function +13 f = w @ mu + saddle_inner(delta_loc, w) \ +14 +- gamma * saddle_quad_form(w, Sigma_perturbed) +15 +16 Sigma_diag = Sigma.diagonal() +27 + +Nominal portfolio +Robust portfolio +Nominal objective +.295 +.291 +Robust objective +.065 +.076 +Table 3: Nominal and worst-case objective for the nominal and robust portfolios. +17 local_constraints = [ +18 +cp.abs(delta_loc) <= rho, Sigma_perturbed == Sigma + Delta_loc, +19 +cp.abs(Delta_loc) <= eta * np.sqrt(np.outer(Sigma_diag, Sigma_diag)) +20 ] +21 +22 G = saddle_min(f, local_constraints) +23 +24 # Creating and solving the problem +25 problem = cp.Problem(cp.Maximize(G), [cp.sum(w) == 1]) +26 problem.solve() +# 0.076 +We first define the constants and parameters, before creating the weights variable in +line 6, and the local variables for the perturbations in lines 8–10. 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Cambridge University Press, 2022. +32 + diff --git a/3dFQT4oBgHgl3EQf3TZi/content/tmp_files/load_file.txt b/3dFQT4oBgHgl3EQf3TZi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f213896a64a94dea712638ee068b0a44ec4aa91a --- /dev/null +++ b/3dFQT4oBgHgl3EQf3TZi/content/tmp_files/load_file.txt @@ -0,0 +1,1768 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf,len=1767 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='13427v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='OC] 31 Jan 2023 Disciplined Saddle Programming Philipp Schiele∗1, Eric Luxenberg∗2, and Stephen Boyd2 1Department of Statistics, Ludwig-Maximilians-Universit¨at M¨unchen 2Department of Electrical Engineering, Stanford University February 1, 2023 Abstract We consider convex-concave saddle point problems, and more generally convex opti- mization problems we refer to as saddle problems, which include the partial supremum or infimum of convex-concave saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle problems arise in a wide range of applications, including game theory, machine learning, and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is well known that a saddle problem can be reduced to a single convex optimization problem by dual- izing either the convex (min) or concave (max) objectives, reducing a min-max problem into a min-min (or max-max) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Carrying out this conversion by hand can be tedious and error prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In this paper we introduce disciplined saddle programming (DSP), a domain specific language (DSL) for specifying saddle problems, for which the dualizing trick can be automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The language and methods are based on recent work by Juditsky and Nemirovski [JN22], who developed the idea of conic-representable sad- dle point programs, and showed how to carry out the required dualization automatically using conic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Juditsky and Nemirovski’s conic representation of saddle problems extends Nesterov and Nemirovski’s earlier development of conic representable convex problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DSP can be thought of as extending disciplined convex programming (DCP) to saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Just as DCP makes it easy for users to formulate and solve com- plex convex problems, DSP allows users to easily formulate and solve saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Our method is implemented in an open-source package, also called DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Previous and related work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 22 6 Examples 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Robust bond portfolio construction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Model fitting robust to data weights .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 Robust Markowitz portfolio construction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 27 2 1 Introduction We consider saddle problems, by which we mean convex-concave saddle point problems or, more generally, convex optimization problems that include the partial supremum or infimum of convex-concave saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle problems arise in various fields such as game theory, robust and minimax optimization, machine learning, and finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' While there are algorithms specifically designed to solve some types of saddle point or minimax problems, another approach is to convert them into standard convex optimization problems using a trick based on duality that can be traced back to at least the 1920s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The idea is to express the infima or suprema that appear in the saddle problem via their duals, which converts them to suprema or infima, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Roughly speaking, this turns a min- max problem into a min-min (or max-max) problem, which can then be solved by standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Specific cases of this trick are well known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' the classical example is converting a matrix game, a specific saddle point problem, into a linear program (LP) [MVN53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' While the dualizing trick has been known and used for almost 100 years, it has always been done by hand, for specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It can only be carried out by those who have a working knowledge of duality in convex optimization, and are aware of the trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In this paper we propose an automated method for carrying out the dualizing trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Our method is based on the theory of conic representation of saddle point problems, developed recently by Juditsky and Nemirovski [JN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Based on this development, we have designed a domain specific language (DSL) for describing saddle problems, which we refer to as dis- ciplined saddle programming (DSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When a problem description complies with the syntax rules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', is DSP-compliant, it is easy to verify that it is a valid saddle problem, and more importantly, automatically carry out the dualizing trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We have implemented the DSL in an open source software package, also called DSP, which works with CVXPY [DB16], a DSL for specifying and solving convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DSP makes it easy to specify and solve saddle problems, without any expertise in (or even knowledge of) convex duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Even for those with the required expertise to carry out the dualizing trick by hand, DSP is less tedious and error prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DSP is disciplined, meaning it is based on a small number of syntax rules that, if followed, guarantee that the specified problem is a valid saddle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is analogous to disciplined convex programming (DCP) [GBY06], which is a DSL for specifying convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When a problem specification follows these syntax rules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', is DCP-compliant, it is a valid convex optimization problem, and more importantly can be automatically converted to an equivalent cone program, and then solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As a practical matter, DCP allows a large number of users to specify and solve even complex convex optimization problems, with no knowledge of the reduction to cone form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Indeed, most DCP users are blissfully unaware of how their problems are solved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', a reduction to cone form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DCP was based on the theory of conic representations of convex functions and problems, pioneered by Nesterov and Nemirovski [NN92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Widely used implementations of DCP include CVXPY [DB16], Convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='jl [Ude+14], CVXR [FNB20], YALMIP [Lof04], and CVX [GB14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Like DCP did for convex problems, DSP makes it easy to specify and solve saddle problems, with most users unaware of the dualization trick and reduction used to solve their problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Previous and related work Saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Studying saddle problems is a long-standing area of research, resulting in many theoretical insights, numerous algorithms for specific classes of problems, and a large number of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle problems are often studied in the context of minimax or maximin optimization [DM90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DP95], which, while dating back to the 1920s and the work of von Neumann and Morgenstern on game theory [MVN53], continue to be active areas of research, with many recent advancements for example in machine learning [Goo+14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A variety of methods have been developed for solving saddle point problems, including interior point methods [HT03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nem99], first-order methods [Kor76;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nem04;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nes07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' NO09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' CLO13], and second-order meth- ods [NP06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nes08], where many of these methods are specialized to specific classes of saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Depending on the class of saddle problem, the methods differ in convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, for the subset of smooth minimax problems, an overview of rates for different curvature assumptions is given in [The+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Due to their close relation to Lagrange duality, saddle problems are commonly studied in the context of convex analysis (see, for example, [BV04, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4], [Roc70, §33–37], [RW09, §11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='J], [BL06, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3]), with an analysis via monotone operators given in [RY22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The practical usefulness of saddle programming in many applications is also increas- ingly well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Many applications of saddle programming are robust optimization prob- lems [BBC11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' BTEGN09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, in statistics, distributionally robust models can be used when the true distribution of the data generating process is not known [DA19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Another common area of application is in finance, with [CPT18, §19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3–4] describing a range of finan- cial applications that can be characterized as saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Similarly, [Boy+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' GI03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' LB00] describe variations of the classical portfolio optimization problem as saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Disciplined convex programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DCP is a grammar for constructing optimization problems that are provably convex, meaning that they can be solved globally, efficiently and accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is based on the rule that the convexity of a function f is preserved under composition if all inner expressions in arguments where f is nondecreasing are convex, and all expressions where f is nonincreasing are concave, and all other expressions are affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A detailed description of the composition rule is given in [BV04, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Using this rule, functions can be composed from a small set of primitives, called atoms, where each atom has known curvature, sign, and monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Every function that can be constructed from these atoms according to the composition rule is convex, but the converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The DCP framework has been implemented in many programming languages, including MATLAB [GB14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Lof04], Python [DB16], R [FNB20], and Julia [Ude+14], and is used by researchers and practitioners in a wide range of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Well-structured convex-concave saddle point problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As mentioned earlier, dis- ciplined saddle programming is based on Juditsky and Nemirovski’s recent work on well- structured convex-concave saddle point problems [JN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Outline In §2 we describe saddle programming, which includes the classical saddle point problem, as well as convex problems that include functions described via partial minimization or maxi- mization of a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We describe some typical applications of saddle programming in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In §4 we describe disciplined saddle programming, which is a way to specify saddle programs in such a way that validity is easy to verify, and the reduction to an equivalent cone program can be automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We describe our implementation in §5, showing how sad- dle functions, saddle extremum functions, saddle point problems, and saddle problems are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We present numerical examples in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2 Saddle programming 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Saddle functions A saddle function (also referred to as a convex-concave saddle function) f : X × Y → R is one for which f(·, y) is convex for any fixed y ∈ Y, and f(x, ·) is concave for any fixed x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The argument domains X ⊆ Rn and Y ⊆ Rm must be nonempty closed convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We refer to x as the convex variable, and y as the concave variable, of the saddle function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Functions of x or y alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A convex function of x, or a concave function of y, are trivial examples of saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Lagrangian of a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The convex optimization problem minimize f0(x) subject to Ax = b, fi(x) ≤ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , m, with variable x ∈ Rn, where f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , fm are convex and A ∈ Rp×n, has Lagrangian L(x, ν, λ) = f(x) + νT (Ax − b) + λ1f1(x) + · · · + λnfm(x), for λ ≥ 0 (elementwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is convex in x and affine (and therefore also concave) in y = (ν, λ), so it is a saddle function with X = � i=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=',m dom fi, Y = Rm + × Rp, Bi-affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = (Ax + b)T(Cy + d), with X = Rp and Y = Rq, is evidently a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The inner product xT y is a special case of a bi-affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For a bi-affine function, either variable can serve as the convex variable, with the other serving as the concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 5 Convex-concave inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = F(x)TG(y), where F : Rp → Rn is a nonnegative elementwise convex function and G : Rq → Rn is a nonnegative elementwise concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Weighted ℓ2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = � n � i=1 yix2 i �1/2 , with X = Rn and Y = Rn +, is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Weighted log-sum-exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = log � n � i=1 yi exp xi � , with X = Rn and Y = Rn +, is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Weighted geometric mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = �n i=1 yxi i , with X = Rn + and Y = Rn +, is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Quadratic form with quasi-semidefinite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, y) = � x y �T � P S ST Q � � x y � , where the matrix is quasi-semidefinite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', P ∈ Sn + (the set of symmetric positive semidefinite matrices) and −Q ∈ Sn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function f(x, Y ) = xTY x, with X = Rn and Y = Sn + (the set of symmetric positive semidefinite n × n matrices), is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As more esoteric example, the function f(x, Y ) = xTY 1/2x, with X = Rn and Y = Sn +, is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Combination rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle functions can be combined in several ways to yield saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example the sum of two saddle functions is a saddle function, provided the domains have nonempty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A saddle function scaled by a nonnegative scalar is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Scaling a saddle function with a nonpositive scalar, and swapping its arguments, yields a saddle function: g(x, y) = −f(y, x) is a saddle function provided f is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle functions are preserved by pre-composition of the convex and concave variables with an affine function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', if f is a saddle function, so is f(Ax+ b, Cx+ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Indeed, the bi-affine function is just the inner product with an affine pre-composition for each of the convex and concave variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Saddle point problems A saddle point (x⋆, y⋆) ∈ X × Y is any point that satisfies f(x⋆, y) ≤ f(x⋆, y⋆) ≤ f(x, y⋆) for all x ∈ X , y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (1) In other words, x⋆ minimizes f(x, y⋆) over x ∈ X , and y⋆ maximizes f(x⋆, y) over y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The basic saddle point problem is to find such a saddle point, find x⋆, y⋆ which satisfy (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (2) The value of the saddle point problem is f(x⋆, y⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Existence of a saddle point for a saddle function is guaranteed, provided some technical conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, Sion’s theorem [Sio58] guarantees the existence of a saddle point when Y is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' There are many other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Matrix game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In a matrix game, player one chooses i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , m}, and player two chooses j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , n}, resulting in player one paying player two the amount Cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Player one wants to minimize this payment, while player two wishes to maximize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In a mixed strategy, player one makes choices at random, from probabilities given by x and player two makes independent choices with probabilities given by y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The expected payment from player one to player two is then f(x, y) = xT Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' With X = {x | x ≥ 0, 1Tx = 1}, and similarly for Y, a saddle point corresponds to an equilibrium, where no player can improve her position by changing (mixed) strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The saddle point problem consists of finding a stable equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', an optimal mixed strategy for each player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A saddle point of a Lagrangian of a convex optimization problem is a primal-dual optimal pair for the convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 Saddle extremum functions Suppose f is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The function G : X → R ∪ {∞} defined by G(x) = sup y∈Y f(x, y), x ∈ X , (3) is called a saddle max function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Similarly, the function H : Y → R ∪ {−∞} defined by H(x) = inf x∈X f(x, y), y ∈ Y, (4) is called a saddle min function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle max functions are convex, and saddle min functions are concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We will use the term saddle extremum (SE) functions to refer to saddle max or saddle min functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Which is meant is clear from context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', whether it is defined by minimization (infimum) or maximization (supremum), or its curvature (convex or concave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Note that in SE functions, we always maximize (or take supremum) over the concave variable, and minimize (or take infimum) over the convex variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This means that evaluating G(x) os H(y) involves solving a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 7 Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Dual function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Minimizing a Lagrangian L(x, ν, λ) over x gives the dual function of the original convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Maximizing a Lagrangian L(x, ν, λ) over y = (ν, λ) gives the objective function re- stricted to the feasible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Conjugate of a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Suppose f is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Then g(x, y) = f(x) − xTy is a saddle function, the Lagrangian of the problem of minimizing f subject to x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Its saddle min is the negative conjugate function: infx g(x, y) = −f ∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Sum of k largest entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Consider f(x, y) = xTy, with Y = {y | 0 ≤ y ≤ 1, 1Ty = k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The associated saddle max function G is the sum of the k largest entries of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle points via SE functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A pair (x⋆, y⋆) is a saddle point of a saddle function f if and only if x⋆ minimizes the convex SE function G in (3) over x ∈ X , and y⋆ maximizes the concave SE function H defined in (4) over y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This means that we can find saddle points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', solve the saddle point problem (2), by solving the convex optimization problem minimize G(x) subject to x ∈ X , (5) with variable x, and the convex optimization problem maximize H(y) subject to y ∈ Y, (6) with variable y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The problem (5) is called a minimax problem, since we are minimizing a function defined as the maximum over another variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The problem (6) is called a maximin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' While the minimax problem (5) and maximin problem (6) are convex, they cannot be directly solved by conventional methods, since the objectives themselves are defined by max- imization and minimization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' There are solution methods specifically designed for minimax and maximin problems [LJJ20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' MB09], but as we will see minimax problems involving SE functions can be transformed to equivalent forms that can be directly solved using conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4 Saddle problems In this paper we consider convex optimization problems that include SE functions in the objective or constraints, which we refer to as saddle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The convex problems that solve the basic saddle point problem (5) and (6) are special cases, where the objective is an 8 SE function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As another example consider the problem of minimizing a convex function φ subject to the convex SE constraint H(y) ≤ 0, which can be expressed as minimize φ(x) subject to f(x, y) ≤ 0 for all y ∈ Y, (7) with variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The constraint here is called a semi-infinite constraint, since (when Y is not a singleton) it can be thought of as an infinite collection of convex constraints, one for each y ∈ Y [HK93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle problems include the minimax and maximin problems (that can be used to solve the saddle point problem), and semi-infinite problems that involve SE functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' There are many other examples of saddle problems, where SE functions can appear in expressions that define the objective and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Robust cost LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As a more specific example of a saddle problem consider the linear program with robust cost, minimize supc∈C cTx subject to Ax = b, x ≥ 0, (8) with variable x ∈ Rn, with C = {c | Fc ≤ g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is an LP with worst case cost over the polyhedron C [BBC11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' BTEGN09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is a saddle problem with convex variable x, concave variable y, and an objective which is a saddle max function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='5 Solving saddle problems Special cases with tractable analytical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' There are cases where an SE function can be worked out analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' An example is the max of a linear function over a box, sup l≤y≤u yTx, = (1/2)(u + l)Tx + (1/2)(u − l)T|x|, where the absolute value is elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We will see other cases in our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Subgradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We can readily compute a subgradient of a saddle max function (or a supergradient of a saddle min function), by simply maximizing over the concave variable (minimizing over the convex variable), which is itself a convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We can then use any method to solve the saddle problem using these subgradients, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', subgradient- type methods, ellipsoid method, or localization methods such as the analytic center cutting plane method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In [MB09] such an approach is used for general minimax problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Methods for specific forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Many methods have been developed for finding saddle points of saddle functions with the special form f(x, y) = xTKy + φ(x) + ψ(y), 9 where φ is convex, ψ is concave, and K is a matrix [BS15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Con13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' CP11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nes05a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nes05b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' CP16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Beyond this example, there are many other special forms of saddle functions, with different methods adapted to properties such as smoothness, separability, and strong-convex- strong-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='6 Dual reduction A well-known trick can be used to transform a saddle point problem into an equivalent prob- lem that does not contain SE functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This method of transforming an inner minimization is not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' it has been used since the 1950s when Von Neumann proved the minimax the- orem using strong duality in his work with Morgenstern on game theory [MVN53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Using this observation, he showed that the minimax problem of a two player game is equivalent to an LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Duality allows us to express the convex (concave) SE function as an infimum (supremum), which facilitates the use of standard convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We think of this as a reduction to an equivalent problem that removes the SE functions from the objective and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Robust cost LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We illustrate the dualization method for the robust cost LP (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The key is to express the robust cost or saddle max function supc∈C cTx as an infimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We first observe that this saddle max function is the optimal value of the LP maximize xT c subject to Fc ≤ g, with variable c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Its dual is minimize gTλ subject to F Tλ = x, λ ≥ 0, with variable λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Assuming that C is nonempty, this dual problem has the same optimal value as the primal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', sup c∈C cTx = inf λ≥0, F T λ=x gTλ Substituting this into (8) we obtain the problem minimize gTλ subject to Ax = b, x ≥ 0, F Tλ = x, λ ≥ 0, (9) with variables x and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This simple LP is equivalent to the original robust LP (8), in the sense that if (x⋆, λ⋆) is a solution of (9), then x⋆ is a solution of the robust LP (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We will see this dualization trick in a far more general setting in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3 Applications In this section we describe a few applications of saddle programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Robust bond portfolio construction We describe here a simplified version of the problem described in much more detail in [LSB22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Our goal is to construct a portfolio of n bonds, giving by its holdings vector h ∈ Rn +, where hi is the number of bond i held in the portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Each bond produces a cash flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', a sequence of payments to the portfolio holder, up to some period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let ci,t be the payment from bond i in time period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let y ∈ RT be the yield curve, which gives the time value of cash: A payment of one dollar at time t is worth exp(−tyt) current dollars, assuming continuously compounded returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The bond portfolio value, which is the present value of the total cash flow, can be expressed as V (h, y) = n � i=1 T � t=1 hici,t exp(−tyt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This function is convex in the yields y and concave (in fact, linear) in the holdings vector h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Now suppose we do not know the yield curve, but instead have a convex set Y of possible values, with y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The worst case value of the bond portfolio, over this set of possible yield curves, is V wc(h) = inf y∈Y V (h, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We recognize this as a saddle min function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (In this application, y is the convex variable of the saddle function V , whereas elsewhere in this paper we use y to denote the concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') We consider a robust bond portfolio construction problem of the form minimize φ(h) subject to h ∈ H, V wc(h) ≥ V lim, (10) where φ is a convex objective, typically a measure of return and risk, H is a convex set of portfolio constraints (for example, imposing h ≥ 0 and a total budget), and V lim is a specified limit on worst case value of the portfolio over the yield curve set Y, which has a saddle min as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For some simple choices of Y the worst case value can be found analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' One example is when Y has a maximum element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In this special case, the maximum element is the minimizer of the value over Y (since V is a monotone decreasing function of y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For other cases, however, we need to solve the saddle problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Model fitting robust to data weights We wish to fit a model parametrized by θ ∈ Θ ⊆ Rn to m observed data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We do this by minimizing a weighted loss over the observed data, plus a regularizer, m � i=1 wiℓi(θ) + r(θ), 11 where ℓi is the convex loss function for observed data point i, r is a convex regularizer function, and the weights wi are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The weights can be used to adjust a data sample that was not representative, as in [BAB21], or to ignore some of the data points (by taking wi = 0), as in [BGM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Evidently the weighted loss is a saddle function, with convex variable θ and concave variable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We consider the case when the weights are unknown, but lie in a convex set, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The robust fitting problem is to choose θ to minimize the worst case loss over the set of possible weights, plus the regularizer, max w∈W m � i=1 wiℓi(θ) + r(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We recognize the first term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', the worst case loss over the set of possible weights, as a saddle max function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For some simple choices of W the worst case loss can be expressed analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example with W = {w | 0 ≤ w ≤ 1, 1Tw = k}, (with k ∈ [0, n]), the worst case loss is given by max w∈W m � i=1 wiℓi(θ) = φ(ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , ℓm), where φ is the sum-of-k-largest entries [BV04, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (Our choice of symbol k suggests that k is an integer, but it need not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') In this case we judge the model parameter θ by its worst loss on any subset of k of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Put another way, we judge θ by dropping the m − k data points on which it does best (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', has the smallest loss) [BGM20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' CVXPY directly supports the sum-of-k-largest function, so the robust fitting problem can be formulated and solved without using DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' To support this function, CVXPY carries out a transformation very similar to the one that DSP does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The difference is that the transformation in CVXPY is specific to this one function, whereas the one carried out in DSP is general, and would work for other convex weight sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 Robust production problem with worst case prices We consider the choice of a vector of quantities q ∈ Q ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Positive entries indicate goods we buy, and negative quantities are goods we sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The set of possible quantities Q is our production set, which is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In addition, we have a manufacturing cost associated with the choice q, given by φ(q), where φ is a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The total cost is the manufacturing cost plus the cost of goods (which includes revenue), φ(q) + pTq, where p ∈ Rn is vector of prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We consider the situation when we do not know the prices, but we have a convex set they lie in, p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The worst case cost of the goods is maxp∈P pTq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The robust production problem is minimize φ(q) + maxp∈P pTq subject to q ∈ Q, (11) 12 with variable q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Here too we can work out analytical expressions for simple choices of P, such as a range for each component, in which case the worst case price is the upper limit for goods we buy, and the lower limit for goods we sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In other cases, we solve the saddle problem (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4 Robust Markowitz portfolio construction Markowitz portfolio construction [Mar52] chooses a set of weights (the fraction of the total portfolio value held in each asset) by solving the convex problem maximize µTw − γwTΣw subject to 1Tw = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' w ∈ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' where the variable is the vector of portfolio weights w ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' µ ∈ Rn is a forecast of the asset returns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' γ > 0 is the risk aversion parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Σ ∈ Sn ++ is a forecast of the asset return covariance matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' and W is a convex set of feasible portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The objective is called the risk adjusted (mean) return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Markowitz portfolio construction is known to be fairly sensitive to the (forecasts) µ and Σ, which have to be chosen with some care;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', [BL91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' One approach is to specify a convex uncertainty set U that (µ, Σ) must lie in, and replace the objective with its worst case (smallest) value over this uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This gives the robust Markowitz portfolio construction problem maximize inf(µ,Σ)∈U � µTw − γwTΣw � subject to 1T w = 1, w ∈ W, with variable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is described in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', in [Boy+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' GI03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' LB00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We observe that this is directly a saddle problem, with a saddle min objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', a maximin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For some simple versions of the problem we can work out the saddle min function explic- itly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' One example, given in [Boy+17], uses U = M × S, where M = {µ + δ | |δ| ≤ ρ}, S = {Σ + ∆ | Σ + ∆ ⪰ 0, |∆ij| ≤ η(ΣiiΣjj)1/2, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , n}, where ρ > 0 is a vector of uncertainties in the forecast returns, and η ∈ (0, 1) is a parameter that scales the perturbation to the forecast covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (We interpret δ and ∆ as perturbations of the nominal mean and covariance µ and Σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') We can express the worst case risk adjusted return analytically as inf (µ,Σ)∈U � µTw − γwTΣw � = µTw − γwTΣw − ρT |w| − γη � n � i=1 Σ1/2 ii |wi| �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The first two terms are the nominal risk adjusted return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' the last two terms (which are nonpositive) represent the cost of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 13 4 Disciplined saddle point programming 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Saddle function calculus We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function with concave variables x and convex variables y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The set of operations that, when performed on saddle functions, preserves the saddle property are called the saddle function calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The calculus is quite simple, and consists of the following operations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Conic combination of saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let φi(xi, yi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , k be saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let θi ≥ 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Then the conic combination, φ(x, y) = �k i=1 θiφi(xi, yi), is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Affine precomposition of saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let φ(x, y) be a saddle function, with x ∈ Rn and y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let A ∈ Rn×q, b ∈ Rn, C ∈ Rm×p, and d ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Then, with u ∈ Rq and v ∈ Rp, the affine precomposition, φ(Au + b, Cv + d), is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Precomposition of saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let φ(x, y) : X × Y ⊆ Rn×m → R be a saddle function, with x ∈ Rn and y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The precomposition with a function f : Rp → Rn, φ(f(u), y), is a saddle function if for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , n one of the following holds: fi(u) is convex and φ is nondecreasing in xi for all y ∈ Y and all x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' fi(u) is concave and φ is nonincreasing in xi for all y ∈ Y and all x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Similarly, the precomposition with a function g : Rq → Rm, φ(x, g(v)), is a saddle function if for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' , m one of the following holds: gj(v) is convex and φ is nonincreasing in yj for all x ∈ X and all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' gj(v) is concave and φ is nondecreasing in yj for all x ∈ X and all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Conically representable saddle functions Nemirovski and Juditsky propose a class of conic representable saddle functions which fa- cilitate the automated dualization of saddle problems [JN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We will first introduce some terminology and notation, and then describe the class of conic representable saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We use the notation φ(x, y) : X × Y ⊆ Rn×m → R to denote a saddle function which is convex in x and concave in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let Kx, Ky and K be members of a collection K of closed, convex, and pointed cones with nonempty interiors in Euclidean spaces such that K contains a nonnegative ray, is closed with respect to taking finite direct products of its members, and is closed with respect to passing from a cone to its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We denote conic membership z ∈ K by z ⪰K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We call a set X ⊆ Rn K-representable if there exist constant matrices A and B, a constant vector c, and a cone K ∈ K such that X = {x | ∃u : Ax + Bu ⪯K c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 14 CVXPY [DB16] can implement a function f exactly when its epigraph {(x, u) | f(x) ≤ u} is K-representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Conically representable saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let X and Y be nonempty and possessing K-representations X = {x | ∃u : Ax + Bu ⪯K c}, Y = {y | ∃v : Cy + Dv ⪯K e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A saddle function φ(x, y) : X × Y → R is K-representable if there exist constant matrices P, Q, R, constant vectors p and s and a cone K ∈ K such that for each x ∈ X and y ∈ Y, φ(x, y) = inf{f Ty + t | Pf + tp + Qu + Rx ⪯K s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This definition generalizes simple class of bilinear saddle functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' See [JN22] for much more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Automated dualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Suppose we have a K-representable saddle function φ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The power of the conic form is that the saddle extremum Φ(x) = sup y∈Y φ(x, y) admits a tractable conic form, meaning that it can be implemented in a DSL like CVXPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Φ(x) = sup y∈Y φ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' y) = sup y∈Y inf f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='u � f Ty + t �� Pf + tp + Qu + Rx ⪯K s � = inf f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='u � sup y∈Y � f Ty + t � ���� Pf + tp + Qu + Rx ⪯K s � (12) = inf f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='u � sup y∈Y � f Ty � + t ���� Pf + tp + Qu + Rx ⪯K s � = inf f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='u � inf λ � λTe ���� CTλ = f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DTλ = 0 λ ⪰K∗ 0 � ���� Pf + tp + Qu + Rx ⪯K s � (13) where in (12) we use Sion’s minimax theorem [Sio58] to reverse the inf and sup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' and in (13) we invoke strong duality to replace the supremum over y with an infimum over λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The final line implies a conic representation of the epigraph of Φ(x), {(x, u) | Φ(x) ≤ u} = \uf8f1 \uf8f2 \uf8f3(x, u) ������ ∃λ, f, t, u : λTe + t ≤ u CTλ = f, DTλ = 0, λ ⪰K∗ 0 Pf + tp + Qu + Rx ⪯K s \uf8fc \uf8fd \uf8fe , which is tractable and can be implemented in a DSL like CVXPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 15 A mathematical nuance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Switching the inf and sup in (12) requires Sion’s theorem to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A sufficient condition for Sion’s theorem to hold is that the set Y is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' However, the min and max can be exchanged even if Y is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Then, due to the max-min inequality max y∈Y min x∈X f(x, y) ≤ min x∈X max y∈Y f(x, y), the equality in (13) is replaced with a less than or equal to, and we obtain a convex restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Thus, if a user creates a problem involving an SE function (as opposed to a saddle point problem only containing saddle functions in the objective), then DSP guarantees that the problem generated is a restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This means that the variables returned are feasible and the returned optimal value is an upper bound on the optimal value for the user’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In our implementation, a saddle problem is solved by applying the above automatic dualization to both the objective f and −f and then solving each resulting convex problem, with the latter having the role of convex and concave variables switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We do so in order to obtain both the convex and concave components of the saddle point, since the dualization removes the concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The saddle problem is only reported as solved if the optimal value of the problem with objective f is within a numerical tolerance of the negated optimal value of the problem with objective −f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' If this holds, this actually implies that max y∈Y min x∈X f(x, y) = min x∈X max y∈Y f(x, y), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', (12) was valid, even if for example Y is not compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Thus, a user need not concern themselves with the compactness of Y (or any other sufficient condition for Sion’s theorem) when using DSP to find a saddle point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' if a saddle point problem is solved, then the saddle point is guaranteed to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 5 Implementation In this section we describe our open source Python implementation of the concepts and methods described in §4, which we also call DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DSP works with CVXPY [DB16], an implementation of a DSL for convex optimization based on DCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We use the term DSP in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We use it to refer to the mathematical concept of disciplined saddle programming, and also our specific implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' which is meant should be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The term DSP-compliant refers to a function or expression that is constructed according to the DSP composition rules given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It can also refer to a problem that is constructed according to these rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In the code snippets below, we use the prefix cp to indicate functions and classes from CVXPY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (We give functions and classes from DSP without prefix, whereas they would likely have a prefix such as dsp in real code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Atoms Saddle functions in DSP are created from fundamental building blocks or atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' These building blocks extend the atoms from CVXPY [DB16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In CVXPY, atoms are either jointly 16 convex or concave in all their variables, but in DSP, atoms are (jointly) convex in a subset of the variables and (jointly) concave in the remaining variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We describe some DSP atoms below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The atom inner(x,y) represents the inner product xTy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Since either x or y could represent the convex variable, we adopt the convention in DSP that the first argument of inner is the convex variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' According to the DSP rules, both arguments to inner must be affine, and the variables they depend on must be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The atom saddle_inner(F, G) corresponds to the function F(x)TG(y), where F and G are vectors of nonnegative and respectively elementwise convex and concave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is DSP-compliant if F is DCP convex and nonnegative and G is DCP concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' If the function G is not DCP nonnegative, then the DCP constraint G >= 0 is attached to the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is analogous to how the DCP constraint x >= 0 is added to the expression cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='log(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As an example consider f = saddle_inner(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='square(x), cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='log(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This represents the saddle function f(x, y) = x2 log y − I(y ≥ 1), where I is the {0, ∞} indicator function of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Weighted ℓ2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The weighted_norm2(x, y) atom represents the saddle function (�n i=1 yix2 i )1/2, with y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is DSP-compliant if x is either DCP affine or both convex and nonnegative, and y is DCP concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Here too, the constraint y >= 0 is added if y is not DCP nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Weighted log-sum-exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The weighted_log_sum_exp(x, y) atom represents the saddle function log (�n i=1 yi exp xi), with y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is DSP-compliant if x is DCP convex, and y is DCP concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The constraint y >= 0 is added if y is not DCP nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Quasi-semidefinite quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The quasidef_quad_form(x, y, P, Q, S) atom represents the function f(x, y) = � x y �T � P S ST Q � � x y � , where the matrix is quasi-semidefinite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', P ∈ Sn + and −Q ∈ Sn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is DSP-compliant if x is DCP affine and y is DCP affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The saddle_quad_form(x, Y) atom represents the function xTY x, where Y is a PSD matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' It is DSP-compliant if x is DCP affine, and Y is DCP PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Calculus rules The atoms can be combined according to the calculus described below to form expressions that are DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, saddle functions can be added or scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DCP- compliant convex and concave expressions are promoted to saddle functions with no concave or convex variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, with variables x, y, and z, the expression f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='5 * saddle_inner(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='square(x), cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='log(y)) + cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='minimum(y,1) - z is DSP-compliant, with convex variable x, concave variable y, and affine variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Calling the is_dsp method of an expression returns True if the expression is DSP- compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The methods convex_variables, concave_variables, and affine_variables, list the convex, concave, and affine variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The convex variables are those that could only be convex, and similarly for concave variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We refer to the convex variables as the unambiguously convex variables, and similarly for the concave variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The three lists of variables gives a partition of all the variables the expression depends on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For the expression above, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='is_dsp() evaluates as True, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() returns the list [x], f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables() returns the list [y], and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='affine_variables() returns the list [z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Note that the role of z is ambiguous in the expression, since it could be either a convex or concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' No mixing variables rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The DSP rules prohibit mixing of convex and concave vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example if we add two saddle expressions, no variable can appear in both its convex and concave variable lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' DSP-compliance is sufficient but not necessary to be a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Re- call that if an expression is DCP convex (concave), then it is convex (concave), but the converse is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example, the expression cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sqrt(1 + cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='square(x)) represents the convex function √ 1 + x2, but is not DCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' But we can express the same function as cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='norm2(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='hstack([1, x])), which is DCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The same holds for DSP and saddle func- tion: If an expression is DSP-compliant, then it represents a saddle function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' but it can represent a saddle function and not be DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As with DCP, such an expression would need to be rewritten in DSP-compliant form, to use any of the other features of DSP (such as a solution method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As an example, the expression x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='T @ C @ y represents the saddle function xT Cy, but is not DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The same function can be expressed as inner(x, C @ y), which is DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When there are affine variables in a DSP-compliant expression, it means that those variables could be considered either convex or concave;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' either way, the function is a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The code below defines the bi-linear saddle function f(x, y) = xT Cy, the ob- jective of a matrix game, with x the convex variable and y the concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 18 Creating a saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 from dsp import * # notational convenience 2 import cvxpy as cp 3 import numpy as np 4 5 x = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(2) 6 y = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(2) 7 C = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='array([[1, 2], [3, 1]]) 8 9 f = inner(x, C @ y) 10 11 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='is_dsp() # True 12 13 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() # [x] 14 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables() # [y] 15 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='affine_variables() # [] Lines 1–3 import the necessary packages (which we will use but not show in the sequel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In lines 5–7, we create two CVXPY variables and a constant matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In line 9 we construct the saddle function f using the DSP atom inner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Both its arguments are affine, so this matches the DSP rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In line 11 we check if saddle_function is DSP-compliant, which it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In lines 13–15 we call functions that return lists of the convex, concave, and affine variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The results of lines 13–15 might seem odd, but recall that inner marks its first argument as convex and its second as concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 Saddle point problems Saddle point problem objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' To construct a saddle point problem, we first create an objective using obj = MinimizeMaximize(f), where f is a CVXPY expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The objective obj is DSP-compliant if the expression f is DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is analogous to the CVXPY contructors cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Minimize(f) and cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Maximize(f), which create objectives from expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Saddle point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A saddle point problem is constructed using prob = SaddlePointProblem(obj, constraints, cvx_vars, ccv_vars) Here, obj is a MinimizeMaximize objective, constraints is a list of constraints, cvx_vars is a list of convex variables and ccv_vars is a list of concave variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The objective must be DSP-compliant for the problem to be DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We now describe the remaining conditions under which the constructed problem is DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 19 Each constraint in the list must be DCP, and can only involve convex variables or concave variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' convex and concave variables cannot both appear in any one constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The list of convex and concave variables partitions all the variables that appear in the objective or the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In cases where the role of a variable is unambiguous, it is inferred, and does not need to be in either list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For example with the objective MinimizeMaximize(weighted_log_sum_exp(x, y) + cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='exp(u) + cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='log(v) + z), x and u must be convex variables, and y and v must be concave variables, and so do not need to appear in the lists used to construct a saddle point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The variable z, however, could be either a convex or concave variable, and so must appear in one of the lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The role of a variable can also be inferred from the constraints: Any variable that appears in a constraint with convex (concave) variables must also be convex (concave).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' With the objective above, the constraint z + v <= 1 would serve to classify z as a concave variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' With this constraint, we could pass empty variable lists to the saddle point constructor, since the roles of all variables can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When the roles of all variables are unambiguous, the lists are optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The roles of the variables in a saddle point problem prob can be found by calling prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() and prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables(), which return lists of vari- ables, and is a partition of all the variables appearing in the objective or constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is useful for debugging, to be sure that DSP agrees with you about the roles of all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A DSP-compliant saddle point problem must have an empty list of affine variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (If it did not, the problem would be ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') Solving a saddle point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The solve() method of a SaddlePointProblem object solves the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The solve method returns the optimal saddle value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', the value of the objective at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' As in CVXPY, the solve method has the side effect of writing all variables’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Here we create and solve a matrix game, continuing the example above where f was defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We do not need to pass in lists of variables since their roles can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Creating and solving a matrix game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 obj = MinimizeMaximize(f) 2 constraints = [x >= 0, cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(x) == 1, y >= 0, cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(y) == 1] 3 prob = SaddlePointProblem(obj, constraints) 4 5 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='is_dsp() # True 6 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() # [x] 7 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables() # [y] 8 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='affine_variables() # [] 9 10 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='solve() # solves the problem 20 11 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='6666666666666667 12 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value # array([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='66666667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='33333333]) 13 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value # array([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='33333333, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='66666667]) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4 Saddle extremum functions Local variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' An SE function has one of the forms G(x) = sup y∈Y f(x, y) or H(y) = inf x∈X f(x, y), where f is saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Note that y in the definition of G, and x in the definition of H, are local or dummy variables, understood to have no connection to any other variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Their scope extends only to the definition, and not beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' To express this subtlety in DSP, we use the class LocalVariable to represent these dummy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The variables that are maximized over (in a saddle max function) or minimized over (in a saddle min function) must be declared using the LocalVariable() constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Any LocalVariable in an SE function cannot appear in any other SE function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Constructing SE functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We construct SE functions in DSP using saddle_max(f, constraints) or saddle_min(f, constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Here, f is a CVXPY scalar expression, and constraints is a list of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We now describe the rules for constructing a DSP-compliant SE function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' If a saddle_max is being constructed, f must be DSP-compliant, and the function’s con- cave variables, and all variables appearing in the list of constraints, must be LocalVariables, while the function’s convex variables must all be regular Variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A similar rule applies for saddle_min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The list of constraints is used to specify the set over which the sup or inf is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Each constraint must be DCP-compliant, and can only contain LocalVariables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' With x a Variable, y_loc a LocalVariable, z_loc a LocalVariable, and z a Variable, consider the following two SE functions: 1 f_1 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1]) 2 f_2 = saddle_max(inner(x, y_loc) + z_loc, [y_loc <= 1, z_loc <= 1]) Both are DSP-compliant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For the first, calling f_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() would return [x, z], and calling f_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables() would return [y_loc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' For the second, calling f_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='convex_variables() would return [x], and f_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='concave_variables() return [y_loc, z_loc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Let y be a Variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Both of the following are not DSP-compliant: 1 f_3 = saddle_max(inner(x, y_loc) + z, [y_loc <= 1, z <= 1]) 2 f_4 = saddle_max(inner(x, y) + z_loc, [y_loc <= 1, z_loc <= 1]) 21 The first is not DSP-compliant because z is not a LocalVariable, but appears in the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The second is not DSP-compliant because y is not a LocalVariable, but appears as a concave variable in the saddle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' SE functions are DCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When they are DSP-compliant, a saddle_max is a convex func- tion, and a saddle_min is a concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' They can be used anywhere in CVXPY that a convex or concave function is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' You can add them, compose them (in appropriate ways), use them in the objective or either side of constraints (in appropriate ways).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Now we provide full examples demonstrating construction of a saddle_max, which we can use to solve the matrix game described in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 as a saddle problem involving an SE function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Creating a saddle max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 # Creating variables 2 x = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(2) 3 4 # Creating local variables 5 y_loc = LocalVariable(2) 6 7 # Convex in x, concave in y_loc 8 f = saddle_inner(C @ x, y_loc) 9 10 # maximizes over y_loc 11 G = saddle_max(f, [y_loc >= 0, cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(y_loc) == 1]) Note that G is a CVXPY expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Constructing a saddle_min works exactly the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='5 Saddle problems A saddle problem is a convex problem that uses SE functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' To be DSP-compliant, the problem must be DCP (which implies all SE functions are DSP-compliant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' When you call the solve method on a saddle problem involving SE functions, and the solve is suc- cessful, then all variables’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value fields are overwritten with optimal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This includes LocalVariables that the SE functions maximized or minimized over;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' they are assigned to the value of a particular maximizer or minimizer of the SE function at the value of the non-local variables, with no further guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We continue our example from §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4 and solve the matrix game using either a saddle max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 22 Creating and solving a saddle problem using a saddle max to solve the matrix game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 prob = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Problem(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Minimize(G), [x >= 0, cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(x) == 1]) 2 3 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='is_dsp() # True 4 5 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='solve() # solving the problem 6 prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value # 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='6666666666666667 7 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='value # array([0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='66666667, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='33333333]) 6 Examples In this section we give numerical examples, taken from §3, showing how to create DSP- compliant problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The specific problem instances we take are small, since our main point is to show how easily the problems can be specified in DSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' But DSP will scale to far larger problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1 Robust bond portfolio construction Our first example is the robust bond portfolio construction problem described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We consider portfolios of n = 20 bonds, over a period T = 60 half-years, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The bonds are taken as representative ones in a global investment grade bond portfolio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' for more detail, see [LSB22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The payments from the bonds are given by C ∈ R20×60, with cash flow of bond i in period t denoted ci,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The portfolio constraint set H is given by H = {h | h ≥ 0, pTh = B}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', the investments must be nonnegative and have a total value (budget) B, which we take to be $100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Here p ∈ R20 + denotes the price of the bonds on September 12, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The portfolio objective is φ(h) = 1 2∥(h − hmkt) ◦ p∥1, where hmkt is the market portfolio scaled to a value of $100, and ◦ denotes Hadamard or elementwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is called the turn-over distance, since it tells us how much we would need to buy and sell to convert our portfolio to the market portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The yield curve set Y is described in terms of perturbations to the nominal or current yield curve ynom, which is the yield curve on September 12, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We take Y = � ynom + δ ����� ∥δ∥∞ ≤ δmax, ∥δ∥1 ≤ κ, T−1 � t=1 (δt+1 − δt)2 ≤ ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We interpret δ as a shock to the yield curve, which we limit elementwise, in absolute sum, and in smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The specific parameter values are given by δmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='02, κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='9, ω = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 23 In the robust bond portfolio problem (10) we take V lim = 90, that is, the worst case value of the portfolio cannot drop below $90 for any y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We solve the problem using the following code, where we assume the cash flow matrix C, the price vector p, the nominal yield curve y_nom, and the market portfolio h_mkt are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Robust bond portfolio construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 # Constants and parameters 2 n, T = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='shape 3 delta_max, kappa, omega = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='9, 1e-6 4 B = 100 5 V_lim = 90 6 7 # Creating variables 8 h = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(n, nonneg=True) 9 10 delta = LocalVariable(T) 11 y = y_nom + delta 12 13 # Objective 14 phi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='5 * cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='norm1(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='multiply(h, p) - cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='multiply(h_mkt, p)) 15 16 # Creating saddle min function 17 V = 0 18 for i in range(n): 19 t_plus_1 = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='arange(T) + 1 # Account for zero-indexing 20 V += saddle_inner(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='exp(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='multiply(-t_plus_1, y)), h[i] * C[i]) 21 22 Y = [ 23 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='norm_inf(delta) <= delta_max, 24 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='norm1(delta) <= kappa, 25 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum_squares(delta[1:] - delta[:-1]) <= omega, 26 ] 27 28 V_wc = saddle_min(V, Y) 29 30 # Creating and solving the problem 31 problem = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Problem(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Minimize(phi), [h @ p == B, V_wc >= V_lim]) 32 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='solve() # 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='32 We first define the constants and parameters in lines 2–5, before creating the variable for the holdings h in line 8, and the LocalVariable delta, which gives the yield curve 24 Nominal portfolio Robust portfolio Turn-over distance $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='00 $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='32 Worst-case value $86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='99 $90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='00 Table 1: Turn-over distance and worst-case value for the nominal (market) portfolio and the robust portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The nominal portfolio does not meet our requirement that the worst-case value be at least $90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' perturbation, in line 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In line 11 we define y as the sum of the current yield curve y_nom and the perturbation delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The objective function is defined in line 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Lines 17–20 define the saddle function V via the saddle_inner atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The yield uncertainty set Y is defined in lines 22–26, and the worst case portfolio value is defined in line 25 using saddle_min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We use the concave expression saddle_min to create and solve a CVXPY problem in lines 31–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Table 1 summarizes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The nominal portfolio is the market portfolio, which has zero turn-over distance to the market portfolio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=', zero objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This nominal portfolio, however, does not satisfy the worst-case portfolio value constraint, since there are yield curves in Y that cause the portfolio value to drop to around $87, less than our limit of $90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The solution of the robust problem has turn-over distance $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='32, and satisfies the constraint that the worst-case value be at least $90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2 Model fitting robust to data weights We consider an instance of the model fitting problem described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We use the well known Titanic data set [HC17], which gives several attributes for each passenger on the ill- fated Titanic voyage, including whether they survived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' A classifier is fit to predict survival based on the features sex, age (binned into three groups, 0–26, 26–53, and 53–80), and class (1, 2, or 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' These features are encoded as a Boolean vector ai ∈ R7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The label yi = 1 means passenger i survived, and yi = −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' There are 1046 examples, but we fit our model using only the m = 50 passengers who embarked from Queenstown, one of three ports of embarkation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This is a somewhat non-representative sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' for example, the survival rate among Queenstown departures is 26%, whereas the overall survival rate is 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We seek a linear classifier ˆyi = sign(aT i θ + β0), where θ ∈ R7 is the classifier parameter vector and β0 ∈ R is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The hinge loss and ℓ2 regularization are used, given by ℓi(θ) = max(0, 1 − yiaT i θ), r(θ) = η∥θ∥2 2, with η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The data is weighted to partially correct for the different survival rates for our training set (26%) and the whole data set (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' To do this we set wi = z1 when yi = 1 and wi = z2 when yi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We require w ≥ 0 and 1Tw = 1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='408 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05 ≤ � yi=1 wi ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='408 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 25 Thus W consists of weights on the Queenstown departure samples that correct the survival rate to within 5% of the overall survival rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The code shown below solves this problem, where we assume the data matrix is already defined as A_train (with rows aT i ), the survival label vector is defined as y_train, and the indicator of survival in the training set is defined as surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Model fitting robust to data weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 # Constants and parameters 2 m, n = A_train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='shape 3 inds_0 = surv == 0 4 inds_1 = surv == 1 5 eta = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05 6 7 # Creating variables 8 theta = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(n) 9 beta_0 = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable() 10 weights = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(m, nonneg=True) 11 surv_weight_0 = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable() 12 surv_weight_1 = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable() 13 14 # Defining the loss function and the weight constraints 15 y_hat = A_train @ theta + beta_0 16 loss = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='pos(1 - cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='multiply(y_train, y_hat)) 17 objective = MinimizeMaximize(saddle_inner(loss, weights) 18 + eta * cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum_squares(theta)) 19 20 constraints = [ 21 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(weights) == 1, 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='408 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05 <= weights @ surv, 23 weights @ surv <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='408 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='05, 24 weights[inds_0] == surv_weight_0, 25 weights[inds_1] == surv_weight_1, 26 ] 27 28 # Creating and solving the problem 29 problem = SaddlePointProblem(objective, constraints) 30 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='solve() After defining the constants and parameters in lines 2–5, we specify the variables for the model coefficient and the weights in lines 8–9 and 10–12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The loss function and regularizer which make up the objective are defined next in lines 15–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The weight constraints are defined in lines 20–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The saddle point problem is created and solved in 26 Nominal classifier Robust classifier Train accuracy 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='0% Test accuracy 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='0% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='6% Table 2: Nominal and worst-case objective values for classification and robust classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' lines 29 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The results are shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We report the test accuracy on all samples in the dataset with a different port of embarkation than Queenstown (996 samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We see that while the robust classification model has slightly lower training accuracy than the nominal model, it achieves a higher test accuracy, generalizing from the non-representative training data better than the nominal classifier, which uses uniform weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='3 Robust Markowitz portfolio construction We consider the robust Markowitz portfolio construction problem described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' We take n = 6 assets, which are the (five) Fama-French factors [FF15] plus a risk-free asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The data is obtained from the Kenneth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' French data library [Fre22], with monthly return data available from July 1963 to October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The nominal return and risk are the empirical mean and covariance of the returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' (These obviously involve look-ahead, but the point of the example is how to specify and solve the problem with DSP, not the construction of a real portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=') We take parameters ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='02, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2, and risk aversion parameter γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In the code, we use mu and Sigma for the mean and covariance estimates, respectively, and the parameters are denoted rho, eta, and gamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Robust Markowitz portfolio construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 1 # Constants and parameters 2 n = len(mu) 3 rho, eta, gamma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='2, 1 4 5 # Creating variables 6 w = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Variable(n, nonneg=True) 7 8 delta_loc = LocalVariable(n) 9 Sigma_perturbed = LocalVariable((n, n), PSD=True) 10 Delta_loc = LocalVariable((n, n)) 11 12 # Creating saddle min function 13 f = w @ mu + saddle_inner(delta_loc, w) \\ 14 gamma * saddle_quad_form(w, Sigma_perturbed) 15 16 Sigma_diag = Sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='diagonal() 27 Nominal portfolio Robust portfolio Nominal objective .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='295 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='291 Robust objective .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='065 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='076 Table 3: Nominal and worst-case objective for the nominal and robust portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 17 local_constraints = [ 18 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='abs(delta_loc) <= rho, Sigma_perturbed == Sigma + Delta_loc, 19 cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='abs(Delta_loc) <= eta * np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sqrt(np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='outer(Sigma_diag, Sigma_diag)) 20 ] 21 22 G = saddle_min(f, local_constraints) 23 24 # Creating and solving the problem 25 problem = cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Problem(cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='Maximize(G), [cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='sum(w) == 1]) 26 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='solve() # 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='076 We first define the constants and parameters, before creating the weights variable in line 6, and the local variables for the perturbations in lines 8–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The saddle function for the objective is defined in line 13, followed by the constraints on the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Both are combined into the concave saddle min function, which is maximized over the portfolio constraints in lines 25–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The results are shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' The robust portfolio yields a slightly lower risk adjusted return of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='291 compared to the nominal optimal portfolio with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' But the robust portfolio attains a higher worst-case risk adjusted return of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='076, compared to the nominal optimal portfolio which attains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content='065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Acknowledgements P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Schiele is supported by a fellowship within the IFI program of the German Academic Exchange Service (DAAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' This research was partially supported by ACCESS (AI Chip Center for Emerging Smart Systems), sponsored by InnoHK funding, Hong Kong SAR, and by ONR N000142212121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 28 References [Mar52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Markowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' “Portfolio Selection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In: Journal of Finance 7 (1 1952), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 77– 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' [MVN53] O.' metadata={'source': 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+page_content=' French Data Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' [JN22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Juditsky and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Nemirovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' “On well-structured convex–concave saddle point problems and variational inequalities with monotone operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' In: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Robust Bond Portfolio Construction via Convex-Concave Saddle Point Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' [RY22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Ryu and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Large-Scale Convex Optimization: Algorithms & Analyses via Monotone Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' Cambridge University Press, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} +page_content=' 32' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQf3TZi/content/2301.13427v1.pdf'} diff --git a/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/2301.05286v1.pdf.txt b/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/2301.05286v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84f981b80db4dd5490510a66e0d81ea2270c8406 --- /dev/null +++ b/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/2301.05286v1.pdf.txt @@ -0,0 +1,810 @@ +GOVERNMENT LICENSE +The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne +National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science lab- +oratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for +itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in +said article to reproduce, prepare derivative works, distribute copies to the public, and perform +publicly and display publicly, by or on behalf of the Government. The Department of Energy will +provide public access to these results of federally sponsored research in accordance with the DOE +Public Access Plan. http://energy.gov/downloads/doe-public-access-plan. +1 +arXiv:2301.05286v1 [physics.app-ph] 12 Jan 2023 + +Demonstration of an AI-driven workflow for autonomous high-resolution scanning +microscopy +Saugat Kandel,1 Tao Zhou,2 Anakha V Babu,1 Zichao Di,3 Xinxin Li,2, 4 Xuedan Ma,2, 4 +Martin Holt,2 Antonino Miceli,1 Charudatta Phatak,5 and Mathew Cherukara1, a) +1)Advanced Photon Source, Argonne National Laboratory, Lemont, +IL 60439. +2)Nanoscience and Technology Division, Argonne National Laboratory, Lemont, +IL 60439. +3)Mathematics and Computer Science, Argonne National Laboratory, Lemont, +IL 60439. +4)Consortium for Advanced Science and Engineering, University of Chicago, Chicago, +Illinois 60637, USA +5)Materials Science Division, Argonne National Laboratory, Lemont, +IL 60439. +(Dated: 16 January 2023) +2 + +With the continuing advances in scientific instrumentation, scanning microscopes are now +able to image physical systems with up to sub-atomic-level spatial resolutions and sub- +picosecond time resolutions. Commensurately, they are generating ever-increasing vol- +umes of data, storing and analysis of which is becoming an increasingly difficult prospect. +One approach to address this challenge is through self-driving experimentation techniques +that can actively analyze the data being collected and use this information to make on- +the-fly measurement choices, such that the data collected is sparse but representative of +the sample and sufficiently informative. Here, we report the Fast Autonomous Scanning +Toolkit (FAST) that combines a trained neural network, a route optimization technique, +and efficient hardware control methods to enable a self-driving scanning microscopy ex- +periment. The key features of our method are that: it does not require any prior information +about the sample, it has a very low computational cost, and that it uses generic hardware +controls with minimal experiment-specific wrapping. We test this toolkit in numerical ex- +periments and a scanning dark-field x-ray microscopy experiment of a WSe2 thin film, +where our experiments show that a FAST scan of <25% of the sample is sufficient to pro- +duce both a high-fidelity image and a quantitative analysis of the surface distortions in the +sample. We show that FAST can autonomously identify all features of interest in the sam- +ple while significantly reducing the scan time, the volume of data acquired, and dose on +the sample. The FAST toolkit is easy to apply for any scanning microscopy modalities and +we anticipate adoption of this technique will empower broader multi-level studies of the +evolution of physical phenomena with respect to time, temperature, or other experimental +parameters. +a)Electronic mail: mcherukara@anl.gov, skandel@anl.gov +3 + +I. +INTRODUCTION +Scanning microscopes are versatile instruments that use photons, electrons, ions, neutrons, or +mechanical probes to interrogate atomic-scale composition, topography, and functionality of ma- +terials, with up to sub-atomic spatial resolution and sub-picosecond time resolution1–3. Notwith- +standing the variation in the probe modalities, these instruments all rely on a scan of the sample +to generate spatially resolved signals that are then collected to form an image of the sample. On- +going advances in instrumentation, such as the development of next-generation x-ray and electron +detectors4,5, has meant that scanning microscopes can now image faster, and at higher resolutions, +than ever before. We can now envision a broad use of these instruments to study not only static +systems, but also multi-level studies of dynamic evolution of materials with time, temperature, or +other parameters, even in situ or operando6. Fine-resolution large-field-of-view scanning exper- +iments, however, come with some significant drawbacks: the volume of data generated and the +probe-induced damage to the sample can be prohibitively large. For example, it is now routinely +possible to perform x-ray imaging of 1 mm3 volumes at ≈10 nm resolution, but this generates +≈ 1015 voxels of data7,8 and requires a commensurately high probe dose9. Meanwhile, the in- +formation of interest in these experiments is often concentrated in sparse regions that contain +interfaces, defects, or other specific structural elements. Directing the scan to only these locations +could greatly reduce the scan time and data volume, but it is difficult to obtain this information a +priori. Addressing this challenge with a human-in-the-loop protocol, where an experienced user +examines the data acquired to identify trends and guide the scan, can be tedious and prohibitively +time consuming (in comparison to the experimental acquisition time). Given these factors, the +development of autonomous acquisition techniques that can continuously analyze acquired data +and drive the sampling specifically towards regions of interest is imperative so as to make full use +of the potential of these scientific instruments. +In parallel to the advances in scientific instrumentation, the last decade has also seen the rapid +development of deep learning (DL) techniques and their applications in all domains of science +and technology, including for the acceleration and enhancement of advanced microscopy meth- +4 + +ods10–13. These DL-based inversion methods are enabling real-time data analysis, which is in turn +opening the door to self-driving techniques that make real-time acquisition decisions based on the +real-time data streams. Such self-driving or autonomous experimentation methods14 are methods +that combine automated experimental control with on-the-fly data-driven decision making so that +an algorithm adaptively explores parameter spaces of interest and conducts new experiments until +it achieves a pre-defined completion criterion15. These methods therefore have the potential to +not only remove the need for constant human supervision and intervention in experiments, but +also make optimal choices in parameter spaces that are too large for humans to easily contextual- +ize. As such, they have the potential to revolutionize experimental design in many scientific fields +including the field of imaging and materials characterization. +In general, the use of data-driven priors to direct future experiments is a Bayesian search prob- +lem, for which the use of off-the-shelf deep learning methods usually do not suffice16. Specific +to microscopy, a popular Bayesian search approach is to use unsupervised (without pre-training) +Gaussian Processes (GPs) that could continuously determine the spatial locations that we are most +uncertain about, then direct the scanning to these locations17–22. While GPs are powerful tech- +niques, their computational cost tends to scale cubically with the number of points acquired. The +decision making time increases during the experiment and quickly exceeds the acquisition time +for the measurement itself. The development of scalable GPs is a significant area of research, but +these methods are not yet ready for application in large-scale imaging problems23. General super- +vised alternatives such as reinforcement learning can be powerful and fast, but they often require +costly pre-training and tend to ignore the global state of the parameter space in exchange for a +local search; as such they have only found limited traction for scanning imaging modalities24. +Specifically for scanning microscopy applications, Godaliyadda et al.25 have proposed to +achieve computationally efficient autonomous sampling with the Supervised Learning Approach +for Dynamic Sampling (SLADS) technique. The SLADS technique uses curated feature maps +to quantify the current measurement state and predict the total image quality improvement ob- +tained by measuring a given point, thereby informing the choice of which point to measure next. +Variations of this technique have found applications in live steering for dose-efficient crystal posi- +5 + +tioning for crystallography26, and for imaging with transmission electron microscopy 27 and mass +spectrometry28 methods. These works, however, either involve training with and reconstruction +of binary images only26,27, or, require extensive training with images closely related to the sample +under study28. As such, they are difficult to translate to imaging settings with more complex im- +ages, particularly for imaging without any prior assumptions about the sample. Meanwhile, Zhang +et al.29 have incorporated a neural network (NN) within the SLADS method (for the SLADS-Net +method) and shown in numerical experiments that it is sufficient to train the method on only a +generic image, eschewing any prior knowledge about the sample, to produce high-fidelity image +with sparse sampling. However, this has not yet been demonstrated in experiment. +In this work, we report the Fast Autonomous Scanning Toolkit (FAST) that combines the +SLADS-Net method, a route optimization technique, and efficient and modular hardware controls +to make on-the-fly sampling and scan path choices for synchrotron-based scanning microscopy. +This method relies on sample-agnostic training to dynamically measure and reconstruct a com- +plicated (non-binary) sample, distinguishing this toolkit from existing SLADS-based workflows. +Moreover, its computational cost is negligible compared to the acquisition time even when run on +a low-power edge computing device placed at a synchrotron beamline, which presents a signifi- +cant advantage over more generic autonomous experimentation techniques. These characteristics +enable the application of our workflow in the high-precision nanoscale scanning x-ray microscopy +instrument present at the hard x-ray nanoprobe beamline at the Advanced Photon Source. +We validate the FAST scheme through real time demonstration at the hard x-ray nanoprobe +beamline at the APS30. A few-layer exfoliated two-dimensional WSe2 thin film was chosen as a +representative example; the preparation process for the thin film often leaves microscopic air bub- +bles trapped underneath the thin film, deforming the 2D material. We show that an adaptive scan +of < 25% of the sample is sufficient to produce a high-fidelity reconstruction that identifies all the +bubbles within the field of view, and even to acquire quantitative information about the film curva- +ture induced by these bubbles. The scheme quickly identifies the deformed part of the 2D material +and focuses its attention there, while ignoring regions of the film that are flat and homogeneous. +Film curvature reconstructed from the adaptive scan (< 25% coverage) is consistent with that re- +6 + +FIG. 1. (Artist’s representation) The APS synchrotron produces a coherent x-ray beam that is focused using +a zone plate setup. It strikes a WSe2 film (green) exfoliated onto a Si substrate (blue), which generates +diffraction patterns that are collected by a two-dimensional detector. Above the bubbles, the lattice of the +film rotates, shifting the diffracted intensities away from its nominal positions. The beam position as well +as the detector acquisition are autonomously controlled by the FAST AI-based workflow. +constructed from full-grid scan (100% coverage). Given these characteristics, the FAST scheme +can be directly applied in other scanning techniques and instruments at the APS and elsewhere, +and may underpin the development of many multi-level experimental studies. +7 + +II. +RESULTS +Figure 1 shows the experimental setup that scans a focused x-ray beam on a sample while ac- +quiring a two-dimensional diffraction image at each point. The live demonstration was performed +on a few-layer WSe2 sample with the detector placed along the 008 Bragg peak, with 2θ = 43.1° +at 10.4 keV. The diffraction patterns were processed on the detector computer (see Methods) to +generate the integrated intensities for use in the FAST workflow. The final output of the workflow +is a dark-field image of the WSe2 sample. +A. +Self-driving scanning microscopy workflow +Figure 2A broadly illustrates the FAST workflow for the experiments reported here. To ini- +tiate the workflow, a low-discrepancy quasi-random selection (generated using the Hammersely +sequence31) of sample position is measured corresponding to 1% of the total area of interest. The +integrated intensities of the measurements are transferred to the edge device, an NVIDIA Jetson +Xavier AGX32 located adjacent to the detector, which used Inverse Distance Weighted (IDW) in- +terpolation to estimate the dark-field image. The estimated image serves as input for the decision- +making step whereby the prospective measurement points are identified. +This self-driving workflow adopts the Supervised Learning Approach for Dynamic Sampling +using Deep Neural Networks (SLADS-Net) algorithm29 to find the prospective measurement +points. In effect, the SLADS-Net algorithm uses the current measurements to identify the best +unmeasured points that, when added to the existing dataset, would have the greatest effect on the +quality of the reconstructed image. As illustrated in Figure 2B, this is accomplished by, first, +representing each unmeasured point as a feature vector with elements that depend on the mea- +surement state in the neighborhood of the point. These feature vectors are used as input for a +pre-trained multi-layer perceptron. The neural network then predicts the expected reduction in +distortion (ERD), a metric (loosely speaking) for the expected improvement in the reconstruction +quality obtained from measuring this unmeasured point, individually for each unmeasured point. +The original SLADS-Net algorithm simply uses the unmeasured point with the highest ERD for +8 + +COMPUTING CANDIDATES +OPTIMIZING PATH +INITIAL ESTIMATION +INITIAL MEASUREMENTS +AI @ EDGE +FINAL RESULT +NEW ESTIMATION +NEW MEASUREMENT +𝒓 +Generate +Features +A +B +FIG. 2. The FAST workflow: (A) A set of random initial measurements are transferred to the edge device +which sequentially generates an initial sample estimate, computes the candidate points to be measured +next, and calculates the travel path for the measurement. The new measurements are combined with the +existing measurements and used to calculate a new estimate, and the process is repeated until it achieves +a completion criterion. (B) The candidate computation starts by examining the local neighborhood (with +radius r) of each unmeasured point P, with the highlighted points indicating points already measured, to +generate a 6-dimensional feature vector. The feature vector is transformed to a 50-dimensional vector using +the Radial Basis Function (RBF) kernel and used as input to a multi-layer NN. The NN then predicts the +expected improvement in the image (ERD) from measuring the point P. A set of unmeasured pixels with +the highest ERD are selected as candidates for the next measurement. +9 + +5 hidden layers +inputs +50 nodes per layer +V1 +output +ERD +02 +V50nVIDIAthe next measurement, and repeats this procedure pointwise. In practice, if the measurement pro- +cedure and the motor movements are fast, then the ERD calculation also has to be commensurately +fast to reduce the dead-time in the experiment. In this work, we mitigate this requirement by in- +stead selecting a batch of points that have the highest ERD, sorted in descending order—we found +that a batch of 50 points adequately minimized the experimental dead-time while still ensuring +that the overall measurement was adequately sparse. +The coordinates of these 50 points are passed on to a route optimization algorithm, based +on Google’s OR-Tools33, to generate the shortest path for the motors to visit all of the them. +This path is appended to the look-up table in the EPICS34 scan record, which then kicks off the +data acquisition. Henceforth, the scan is automatically paused after every 50 points, raising a +flag which event triggers a callback function on the edge device. There, a new estimated dark +field image of the sample is generated, and the coordinates for the next 50 prospective points are +computed. The scan is resumed after the EPICS scan record receives the new coordinates for the +optimized scanning path. The actual scanning of the focused x-ray beam is achieved by moving +two piezoelectric linear translation motors in step mode. The detector exposure time is set to 0.5 s +and comes with an overhead of 0.2 s. +For the 200×40 pixels object described in Section II C, the workflow required ≈0.15 s to +compute the new positions, ≈42 s to scan the set of 50 positions, and a total of ≈0.37 s to process +the diffraction patterns and communicate the measurements. This represents an overhead of ⪅ 2%. +The workflow is currently entirely CPU-bound, relying on the on-board 8-core ARM CPUs, and +does not take advantage of the GPU bundled into the NVIDIA AGX device. In the future, we +expect to perform the computation in a parallelized and asynchronous fashion, which would further +reduce this overhead. These timing results showcase the rapid data-driven decision-making ability +that is characteristic of the FAST workflow. +We also note that, for all the results reported in this work, the underlying NN was trained on a +single generic image with no relation to microscopy. For details about the SLADS-Net algorithm +and the sample-agnostic training procedure, the reader is referred to the Methods section. +10 + +B. +Numerical demonstration for scanning dark-field microscopy +Reconstructions (10% scan ) +True +Measured points +Raster grid (RG) +LD random (LDR) +FAST +A +B +C +D +E +G +H +I +F +FIG. 3. Numerical comparison of sampling methods: (A) shows the ground truth with the color scale +representing the normalized intensity, (B-D) show respectively the RG, LDR, and FAST reconstructions at +10% scan coverage, and (G-I) show the actual scan points that produce these reconstructions. (E-F) show +the evolution of the NRMSE (lower is better) and SSIM (higher is better) as a function of the scan coverage. +The FAST reconstruction stabilizes at 27% coverage while the other techniques take significantly longer to +reach the same quality. +We first validated the performance of the proposed workflow through a numerical experiment +on a set of pre-acquired dark-field microscopy data. Here, we compared the FAST sampling with +three static sampling techniques: +1. Raster grid (RG) For a test sampling percentage, we generated a equally spaced raster grid +that provides a uniform coverage of the sample. +2. Uniform random (UR) sampling The measurement pixels were drawn from a uniform +11 + +LDR +UR +RG +FAST150 μm. +· +.. +:random distribution. +3. Low-discrepancy (LDR) random sampling For each measurement percentage, we gener- +ated a low-discrepancy sampling grid using the quasi-random Hammersly sequence. +The test dataset is a dark field image of size 600×400 pixels which represents 240,000 possible +measurement positions. This covers a physical area of 900 µm×600 µm and encloses multiple +flakes of WSe2 with various thicknesses, with the thicker regions associated with regions of higher +brightness in the image (Figure 3). At this spatial resolution, only medium and large sized bubbles +(with diameter > 2 um) can be observed. As explained previously, the bubbles deform the surface +and shift the Bragg peak of the 2D materials away from their theoretical (flat region) positions, +resulting in regions of darker contrast. Finally, the image also contains flake-free regions that have +zero integrated intensities. +For this comparison, we first initialized the FAST sampling with a 1% measurement coverage +(as described above), then successively measured 50 additional points at iteration. For each FAST +measurement, we also generate RG, UR, and LDR measurement masks with the same number +of scan points. In this fashion, we generate a sequence of sampling masks and the associated +reconstructions until we achieve 100% sampling. +We present the numerical results in Figure 3, where we show a comparison of the various meth- +ods at 10% sampling. Note that while the proposed method internally uses the fast IDW algorithm +for the inpainting, the final images presented here are calculated using the higher quality bihar- +monic inpainting technique35. The uniform random scheme performs worse than the LD-random +and raster grid schemes and is not shown in the figure. In Figure 3A-D, we can see that the FAST +sampling is able to reproduce with high fidelity the flake boundaries, the bubbles, and the regions +of transition between the varying levels of thicknesses. In contrast, the LDR and raster schemes +produce much lower quality reconstructions of these features. Figure 3E shows an evolution of the +normalized root mean squared error (NRMSE) and fig. 3F the structural similarity metric (SSIM) +(which measures multiscale perceptual similarity) for the different sampling techniques. It is ev- +ident that FAST produces high quality reconstructions at much lower measurement percentages +12 + +than the examined static sampling techniques. We note that the result could be further improved +in the future by using a more sophisticated inpainting technique within the FAST method. To un- +derstand how FAST outperforms the other methods under the same sampling condition, we show +the actual measured positions of the various schemes at 10% coverage (Figure 3G-I). FAST pref- +erentially samples the regions with significant heterogeneity over the homogeneous regions. This +is particularly useful for sparse samples, where the time spent sampling from empty regions adds +little additional information. +C. +Experimental demonstration +Full scan image +FAST reconstructions +Measured points +A +C +E +G +B +D +F +Measurements between 15-20% +H +FIG. 4. Evolution of the FAST scan: (A, C, E) show the reconstruction at 5%, 15%, and 20% reconstructions +respectively, (B, D, F) show the corresponding actual measurement points. (G) shows the image obtained +through a full-grid pointwise scan. The color scale in (A-G) show the normalized intensities. (H) shows +only the points sampled between 15% and 20% coverage. +We next demonstrate the application of the FAST workflow in a live experiment at a syn- +chrotron beamline. A video showing the sampling, recorded live during the actual experiment, is +available here36. Other than starting the workflow scripts at the beginning, the entire experiment +was unmanned and fully automated. In order to measure the deformed WSe2 flakes in details, a +higher spatial resolution of 100 nm was chosen. This limits the field of view to 20 µm×4 µm for +13 + +5%:15%+++20%2 μma scan point density of 200×40 points. +In Figure 4 we show the reconstructed dark field image (subplots A,C,E) and the measurement +points (subplots B,D,F) from 5 % to 20 % coverage and compare them to that obtained from raster +scanning the sample with 100% coverage(subplot G). We see that the FAST method identifies +some of the regions of hetereogeneity — the edges of the bubbles — and starts to preferentially +sample these regions within 5 % coverage of the sample. At 15 % coverage, these regions are +extensively sampled. The reconstruction does not change significantly between 15 % to 20 %, +indicating that the reconstruction has stabilized. Moreover, the 20 % reconstruction also contains +sharp and accurate reproductions of all the major features present in the full scan image. +A point of interest is that the partially scanned bubble at the bottom right corners of Figure 4E- +G shows up only in the 20% scan, and not in the 15% scan. To explain this, we note that the +5% scan, and therefore the initial 1% random sampling, does not contain any measurements in +the neighborhood of this bubble. The FAST scheme favors exploitation of regions it knows to +be heterogeneous over exploration of this fully unknown region, and therefore only explores this +region much later in the measurement process (Figure 4H). This is, in fact, an instance of the +general exploration-exploitation tradeoff that exists in all Bayesian search procedures37. Potential +mitigation steps could be to sample more initially (say 5% random points), or to deliberately +introduce diversity into each batch of measurement points. +So far we have reduced the diffraction image measured at each point to one single quantity +(integrated intensity) in order to guide the automated experiment. These images often need to +be reprocessed after the experiment to extract additional physically relevant results. Notably, the +intensity distribution in the diffraction patterns contains information about the strain as well as +the rotation of the crystal lattice, and in this case, the curvature of the 2D materials due to the +bubbles underneath. A simple center of mass calculation in the X direction (CoMx) would yield +the magnitude of the film curved in the XZ plane. The curvature (deviation of the CoMx from its +nominal value) is the smallest around the center of the bubble and the largest at the edge. It also +changes sign going from the left side to the right side. Center of mass calculation in the Y direction +yields the magnitude of the film curved in the YZ plane. The results look slightly different from +14 + +the CoMx calculations due to the way the shifted Bragg peak intersects with the Ewald’s sphere. +Figure 5A and B shows respectively the CoMx and CoMy obtained from raster scan with 100% +coverage on the area of interest. The unit is the number of pixel shift, relative to the center of the +nominal diffraction pattern. Figure 5C and B shows respectively the CoMx and CoMy obtained +with FAST. The curvature information of the film were faithfully reproduced despite scanning just +20% of the entire area. For more information on the reconstruction of the CoM maps, he reader is +referred to the Methods section. +CoMx +CoMy +FAST +Full +FIG. 5. Comparison of the per measured point center of mass of the diffraction patterns between the FAST +scan at 20% coverage and full-grid scan. Subplots (A) and (B) show the inpainted COMx and COMy, +respectively, for the full-grid raster FAST scan, and subplots (C) and (D) for the FAST scan. +III. +DISCUSSION +In this work, we have showcased the FAST workflow that combines a sparse sampling algo- +rithm with route planning to drive a scanning diffraction microscopy experiment at a synchrotron +beamline. In addition to being an effective alternative to a full pointwise scan to acquire a dark- +field image of the sample, FAST also produces accurate quantitative measurements of its phys- +ical properties. For our live demonstration of a 200 points×40 points with a measurement time +of 0.5 s/point, the FAST decision-making time was negligible, leading to an overall saving of +≈80 min (about ≈65 %) of the experiment time. This saving was facilitated by our choice to ac- +quire a batch of 50 measurements between the selection of the prospective measurement points. +This ensured that the communication time stayed negligible with no noticeable loss in the quality +of points acquired when compared to a pointwise candidate selection scheme (see Supplemental +15 + +CDB +2 μmMaterials, Fig. S1). +The generalizability of the FAST method comes from the fact that the key NN-based compo- +nent of this workflow is trained on just the standard cameraman image38, not on close analogues of +a sample of interest. While this generalizability results in a slight loss of performance of the tech- +nique , it still shows excellent sparsity performance for cases tested in previous research29,39 and +in the current work. This has the benefit that we do not need a priori knowledge of the sample. As +such, while general pre-training would be difficult to satisfy for new and expensive experiments, +the FAST approach can be used directly. Furthermore, the batch prediction and route optimization +approach we implement can also be directly applied in any application of choice. Moreover, the +experimental application of our work uses an extensible edge device and the widely used EPICS +platform for hardware control, both of which can be incorporated into any instrument even with +the SLADS-Net replaced by any other sampling strategies. For example, we could just replace the +dark-field detection procedure described here with a fluorescence counting setup and use exactly +the FAST scheme for a fluorescence-based imaging of the sample. Alternatively, since all the in- +struments at the APS rely on EPICS controls, one can perform transmission, surface scattering, +or any other 2D scanning experiment in any applicable beamline with only minor changes to the +FAST routine. +The computations in the current workflow have a time complexity of O(2N logN +kM logN), +where N is the number of measured points, M the number of unmeasured points, and k the num- +ber of nearest neighboring measurements (k = 10 in our case) that we use for the feature vector +calculations. Here, the first term accounts for the creation of the nearest neighbor K-d tree and +the second term for the nearest neighbor calculation. The remainder of the algorithm has a linear +time complexity and could be performed in parallel for the unmeasured points. We expect that it is +possible to reduce this complexity using an approximate nearest neighbor search method instead +of the K-d tree approach. As such, a GPU-based implementation that takes advantage of the par- +allelization and the approximation would likely significantly reduce the computation time. This +stands in stark contrast with the time complexity of O +� +N3� +(for N measured points) for Gaussian +Processes, a similarly training-free method that is widely used for autonomous experimentation. +16 + +For an illustrative example, Vasudevan et al20 report a GP-based scanning microscopy experiment +where the calculation of each set of measurement candidates takes ≈6 s on an NVIDIA DGX-2 +GPU for a 50×50 image; our workflow performs an equivalent calculation for a larger 200×40 +image within ≈1.5 s in a low-power CPU. We note, however, that GPs remain a very powerful and +generalizable approach with a bevy of applications beyond only scanning microscopy. +We caution that our workflow suffers from three important challenges. First, it depends heav- +ily on the initial 1% random sampling to discover regions of heterogeneity. If an isolated feature +present in an otherwise homogeneous region is not partially sampled during this random sam- +pling step, then such a feature can be missed until much later in the scanning experiment (see +Figure 4H). A related second limitation is that this method produces sub-optimal reconstructions +if the sample is sufficiently heterogeneous that the data in each pixel changes significantly from +pixel to pixel throughout the image (Supplemental Material in Hujsak et al27). The third limi- +tation, more practical in nature, is that the scan paths require significant motor movement, often +including a retracing over points already measured. As such, there could exist scenarios in which +the time required for the motor movement eclipses the time required for a single measurement. +We expect to address these limitations by explicitly including a measurement-density-based term +39 or a movement-time-based term in the candidate selection procedure40, or by using a line-based +sampling technique41. +Despite these challenges, we believe that the proposed FAST technique has great potential. It +is an ideal tool for use cases with limited sampling or dosage budgets. It can be used to isolate +regions of interest in sparse settings, to prepare for pointwise scanning in these regions. More +generally, it can be used to guide any scanning microscopy experiment where we do not need +full pointwise information. In the future, we expect to extend this method for 3D imaging, fly +scans, ptychography, and other imaging applications. We expect that these developments will +significantly enhance the efficacy of scanning microscopy experiments, bolstering their use for the +study of dynamic physical phenomena. +17 + +IV. +METHODS +A. +The SLADS-Net algorithm +The SLADS-Net algorithm29 used within the FAST workflow is an adaptation of the Super- +vised Learning Approach for Dynamic Sampling (SLADS) algorithm originally developed by +Godaliyadda et al25, and the algorithms differ only in their training approaches ( Section IV B). To +explain the SLADS algorithm, we first denote the object we want to measure as A ∈ RN, where N +is the total number of pixels in the image. Further, we can denote the pixel at location 1 ≤ s ≤ N +as as so that a measurement at the location s extracts the value as; each measurement is thus +characterized by the pair (s, as). After k measurements, then, we get the k×2 measurement vector +Y k = +� +������� +s1 +as1 +s2 +as2 +... +sk +ask +� +������� +(1) +Using these k measurements, then, we can reconstruct (e.g. via interpolation) an estimate ˆAkof the +true object A. The difference between A and ˆAk is denoted as the distortion D(A, ˆAk) and can be +calculated using any chosen metric. In the current work, we define D(A, ˆAk) to be the L2 norm: +D(A, ˆAk) = ||A− ˆAk||2. +Given the measurement Y k and the reconstruction ˆAk, a new measurement at any location s will +presumably reduce the distortion in the reconstruction. We can denote this reduction in distortion +(RD) as +Rk,s = D(A, ˆAk)−D(A, ˆAk,s) +(2) +where ˆAk,s is the reconstruction that includes the newly added measurement at s. The goal of +the SLADS algorithm is then to identify the pixel location that would maximize this reduction in +distortion: +sk+1 = argmax +s +Rk,s +(3) +18 + +Of course, since we cannot know the value of the measurement as or the ground truth A, SLADS +bases its selection on the conditional expectation of reduction in distortion (ERD), which is defined +as: +Rk,s = E +� +Rk,s��Y k� +so that +sk+1 = argmax +s +Rk,s. +(4) +The algorithm assumes that we can compute the ERD at s based on just the measurement state Yk +as +Rk,s = g(vk,s) +(5) +where vk,s is a location-dependent feature vector calculated using the measurement state Yk. The +goal of the SLADS training procedure is to estimate the function g. +B. +Training +The training procedure for the SLADS/SLADS-Net algorithm is a supervised procedure in +which we generate a large number of (vk,s,Rk,s) pairs and use these to estimate g. Note that this +is a pixelwise computation that is performed independently for each measurement location s; for +each measurement s we have to calculate a reconstruction ˆAk,s before we can calculate the RD Rk,s. +To make this computationally tractable, the Godaliyadda et al25 use approximations that ensure +that the RD of each pixel only depends on its local neighborhood. Correspondingly, instead of +working with the full measurement state Y k, the training procedure uses carefully designed feature +vectors that capture the local neighborhood of the pixel at location s. As shown in Figure 2B, the +feature vector for the pixel P consists of six features: (i) ∇x and ∇y are the spatial gradients at +P, (ii) σ1,r and σ2,r measure the deviation of the estimated value for P from the nearby measured +values (highlighted in red), and (iii) L (which is the distance of P from the closest measured point) +and ρr measure the density of measurements around P. +The original SLADS algorithm assumes that this feature vector is linearly related to the RD, +and the training therefore is a linear regression procedure. The SLADS-Net adaptation first uses +an radial basis function (RBF) kernelization to transform the 6-dimensional feature vector to a +19 + +50-dimensional vector, then replaces the linear predictor with a nonlinear fully-connected neural +network that contains 5 hidden layers with 50 nodes each. +In this work, we train the SLADS-Net neural network on only the standard cameraman image, +without using any a priori information about the sample. For the training, we generate a mea- +surement state Y k by randomly choosing a fixed number number of measurement locations, then +calculate the feature vector vk,s and the RD Rk,s for each unmeasured pixel. We generate such +sets of training pairs for 10 different sample coverage percentages between 1% and 80%. This +overall comprises our training dataset. We use this data to train the neural network for 100 epochs +using the Adam optimizer with the learning rate 0.001. We use this trained model for all the simu- +lated and experimental measurements. We provide an example of a training measurement set—the +measured points, the interpolated reconstruction, and the corresponding RD for the unmeasured +points—in the supplemental materials (Fig. S2) +C. +Experimental measurements +At each point of the measurement, a tight region of interest (RoI) around the expected position +of the thin film Bragg peak was extracted from the corresponding diffraction image. Integrated +intensities of the RoI were used to guide the NN prediction. For the flat region, the integrated +intensity is high, showing up as brighter contrast on the dark field image. For the deformed region, +the integrated intensity is low (darker contrast on the dark field image) as the illuminated film +diffraction partially exits the selected RoI (see Supplemental Materials, Fig. S3). +For the FAST experiment, the predicted ERD and the dark-field reconstruction served as visual +guides to inform when to stop the experiment.. During the experiment, we noted that the ERD and +the reconstruction had stabilized by ≈20 % scan coverage, but we let the experiment run to ≈35 % +coverage to ensure that this behavior persisted (see Supplemental Materials, Fig. S4). While we +used this visual criterion for our exploratory experiment, it is straightforward to design a numerical +stopping criterion based on the absolute or relative convergence of the ERD, or on the per-iteration +change in the reconstructed image. +20 + +DATA AND CODE AVAILABILITY +The data and code will be made available at https://github.com/saugatkandel/fast_ +smart_scanning +ACKNOWLEDGMENTS +Work performed at the Center for Nanoscale Materials and Advanced Photon Source, both +U.S. Department of Energy Office of Science User Facilities, was supported by the U.S. DOE, +Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We also acknowl- +edge support from Argonne LDRD 2021-0090 – AutoPtycho: Autonomous, Sparse-sampled Pty- +chographic Imaging. We gratefully acknowledge the computing resources provided on Bebop, a +high-performance computing cluster operated by the Laboratory Computing Resource Center at +Argonne National Laboratory. 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In 2020 IEEE International Conference on Robotics and Automation +(ICRA), 328–334 (2020). ISSN: 2577-087X. +41Helminiak, D., Hu, H., Laskin, J. & Hye Ye, D. Deep Learning Approach for Dynamic Sparse +Sampling for High-Throughput Mass Spectrometry Imaging. Electronic Imaging 33, 290–1– +290–7 (2021). URL https://library.imaging.org/ei/articles/33/15/art00007. +25 + +Supplemental material +January 16, 2023 +1 +arXiv:2301.05286v1 [physics.app-ph] 12 Jan 2023 + +A +B +Figure S1: Comparison of the FAST reconstructions for scan batch size of 1 +(FAST-1) and 50 (FAST-50) as a function of the scan coverage for the numerical +simulation described in Section II.B. We observe that FAST-1 initially performs +better, with lower NRMSE and higher SSIM, than FAST-50, but this advantage +erodes quickly. We ended the FAST-1 experiment at ≈8.2 % sampling.due due to +simulation time limitations. +2 + +A +B +C +Figure S2: Example of training data. (A) shows a set of randomly selected mea- +surement points. (B)shows the reconstruction calculated by interpolating these +measurements. (c) shows the ERDs calculated for the unmeasured points, with the +ERD highest at regions of hetereogeneity. The location of the measured points, +the measured values, and the reconstruction are used to generate feature vectors +for the training, and the ERDs are used as the training labels. +3 + +CoMx +CoMy +Figure S3: Example of ROI selection and change in diffraction patterns around +the bubbles. (A) and (B) respectively show the CoMx and CoMy calcualted from +the FAST scan with 20% covergae, as discussed in Section II.C. The diffraction +patterns for the points marked with the ×, �, and + are shown in the bottom row. +The × point is in a region without a bubble and has the diffraction pattern at the +Bragg angle. The points marked with � and + are located at the top and bottom +edges of the bubble, and therefore show additional anomalous diffraction spots. +The dashed square boxes in the diffraction pattern figures indicate the ROI used +for the dark-field image reconstructions (shown in Figure 4 in the main paper). +The CoM calculations use the regions outside the dashed square boxes as the RoI. +4 + +AB +XOX0+0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +ERD +1e6 +0 +10 +20 +30 +40 +50 +60 +Scan iteration +0 +5 +10 +15 +20 +25 +30 +35 +Scan coverage (%) +20% coverage +Figure S4: Evolution in the ERD for the experimental demonstration. The ERD +initially decreases rapidly, during which point the each batch of 50 points signifi- +cantly improves the sample reconstruction. At per-iteration change in the ERD is +much smaller at 20% coverage. +5 + diff --git a/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/load_file.txt b/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8888552adc28ffa5eb3bd02b3cb2aee592f1505 --- /dev/null +++ b/4tE4T4oBgHgl3EQf1A0X/content/tmp_files/load_file.txt @@ -0,0 +1,682 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf,len=681 +page_content='GOVERNMENT LICENSE The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Argonne, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Department of Energy Office of Science lab- oratory, is operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' http://energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='gov/downloads/doe-public-access-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='05286v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='app-ph] 12 Jan 2023 Demonstration of an AI-driven workflow for autonomous high-resolution scanning microscopy Saugat Kandel,1 Tao Zhou,2 Anakha V Babu,1 Zichao Di,3 Xinxin Li,2, 4 Xuedan Ma,2, 4 Martin Holt,2 Antonino Miceli,1 Charudatta Phatak,5 and Mathew Cherukara1, a) 1)Advanced Photon Source, Argonne National Laboratory, Lemont, IL 60439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 2)Nanoscience and Technology Division, Argonne National Laboratory, Lemont, IL 60439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 3)Mathematics and Computer Science, Argonne National Laboratory, Lemont, IL 60439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 4)Consortium for Advanced Science and Engineering, University of Chicago, Chicago, Illinois 60637, USA 5)Materials Science Division, Argonne National Laboratory, Lemont, IL 60439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (Dated: 16 January 2023) 2 With the continuing advances in scientific instrumentation, scanning microscopes are now able to image physical systems with up to sub-atomic-level spatial resolutions and sub- picosecond time resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Commensurately, they are generating ever-increasing vol- umes of data, storing and analysis of which is becoming an increasingly difficult prospect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' One approach to address this challenge is through self-driving experimentation techniques that can actively analyze the data being collected and use this information to make on- the-fly measurement choices, such that the data collected is sparse but representative of the sample and sufficiently informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Here, we report the Fast Autonomous Scanning Toolkit (FAST) that combines a trained neural network, a route optimization technique, and efficient hardware control methods to enable a self-driving scanning microscopy ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The key features of our method are that: it does not require any prior information about the sample, it has a very low computational cost, and that it uses generic hardware controls with minimal experiment-specific wrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We test this toolkit in numerical ex- periments and a scanning dark-field x-ray microscopy experiment of a WSe2 thin film, where our experiments show that a FAST scan of <25% of the sample is sufficient to pro- duce both a high-fidelity image and a quantitative analysis of the surface distortions in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We show that FAST can autonomously identify all features of interest in the sam- ple while significantly reducing the scan time, the volume of data acquired, and dose on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The FAST toolkit is easy to apply for any scanning microscopy modalities and we anticipate adoption of this technique will empower broader multi-level studies of the evolution of physical phenomena with respect to time, temperature, or other experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' a)Electronic mail: mcherukara@anl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='gov, skandel@anl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='gov 3 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' INTRODUCTION Scanning microscopes are versatile instruments that use photons, electrons, ions, neutrons, or mechanical probes to interrogate atomic-scale composition, topography, and functionality of ma- terials, with up to sub-atomic spatial resolution and sub-picosecond time resolution1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Notwith- standing the variation in the probe modalities, these instruments all rely on a scan of the sample to generate spatially resolved signals that are then collected to form an image of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' On- going advances in instrumentation, such as the development of next-generation x-ray and electron detectors4,5, has meant that scanning microscopes can now image faster, and at higher resolutions, than ever before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We can now envision a broad use of these instruments to study not only static systems, but also multi-level studies of dynamic evolution of materials with time, temperature, or other parameters, even in situ or operando6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Fine-resolution large-field-of-view scanning exper- iments, however, come with some significant drawbacks: the volume of data generated and the probe-induced damage to the sample can be prohibitively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For example, it is now routinely possible to perform x-ray imaging of 1 mm3 volumes at ≈10 nm resolution, but this generates ≈ 1015 voxels of data7,8 and requires a commensurately high probe dose9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Meanwhile, the in- formation of interest in these experiments is often concentrated in sparse regions that contain interfaces, defects, or other specific structural elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Directing the scan to only these locations could greatly reduce the scan time and data volume, but it is difficult to obtain this information a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Addressing this challenge with a human-in-the-loop protocol, where an experienced user examines the data acquired to identify trends and guide the scan, can be tedious and prohibitively time consuming (in comparison to the experimental acquisition time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Given these factors, the development of autonomous acquisition techniques that can continuously analyze acquired data and drive the sampling specifically towards regions of interest is imperative so as to make full use of the potential of these scientific instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In parallel to the advances in scientific instrumentation, the last decade has also seen the rapid development of deep learning (DL) techniques and their applications in all domains of science and technology, including for the acceleration and enhancement of advanced microscopy meth- 4 ods10–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These DL-based inversion methods are enabling real-time data analysis, which is in turn opening the door to self-driving techniques that make real-time acquisition decisions based on the real-time data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Such self-driving or autonomous experimentation methods14 are methods that combine automated experimental control with on-the-fly data-driven decision making so that an algorithm adaptively explores parameter spaces of interest and conducts new experiments until it achieves a pre-defined completion criterion15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These methods therefore have the potential to not only remove the need for constant human supervision and intervention in experiments, but also make optimal choices in parameter spaces that are too large for humans to easily contextual- ize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As such, they have the potential to revolutionize experimental design in many scientific fields including the field of imaging and materials characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In general, the use of data-driven priors to direct future experiments is a Bayesian search prob- lem, for which the use of off-the-shelf deep learning methods usually do not suffice16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Specific to microscopy, a popular Bayesian search approach is to use unsupervised (without pre-training) Gaussian Processes (GPs) that could continuously determine the spatial locations that we are most uncertain about, then direct the scanning to these locations17–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' While GPs are powerful tech- niques, their computational cost tends to scale cubically with the number of points acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The decision making time increases during the experiment and quickly exceeds the acquisition time for the measurement itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The development of scalable GPs is a significant area of research, but these methods are not yet ready for application in large-scale imaging problems23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' General super- vised alternatives such as reinforcement learning can be powerful and fast, but they often require costly pre-training and tend to ignore the global state of the parameter space in exchange for a local search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' as such they have only found limited traction for scanning imaging modalities24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Specifically for scanning microscopy applications, Godaliyadda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='25 have proposed to achieve computationally efficient autonomous sampling with the Supervised Learning Approach for Dynamic Sampling (SLADS) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The SLADS technique uses curated feature maps to quantify the current measurement state and predict the total image quality improvement ob- tained by measuring a given point, thereby informing the choice of which point to measure next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Variations of this technique have found applications in live steering for dose-efficient crystal posi- 5 tioning for crystallography26, and for imaging with transmission electron microscopy 27 and mass spectrometry28 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These works, however, either involve training with and reconstruction of binary images only26,27, or, require extensive training with images closely related to the sample under study28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As such, they are difficult to translate to imaging settings with more complex im- ages, particularly for imaging without any prior assumptions about the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Meanwhile, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='29 have incorporated a neural network (NN) within the SLADS method (for the SLADS-Net method) and shown in numerical experiments that it is sufficient to train the method on only a generic image, eschewing any prior knowledge about the sample, to produce high-fidelity image with sparse sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' However, this has not yet been demonstrated in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In this work, we report the Fast Autonomous Scanning Toolkit (FAST) that combines the SLADS-Net method, a route optimization technique, and efficient and modular hardware controls to make on-the-fly sampling and scan path choices for synchrotron-based scanning microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This method relies on sample-agnostic training to dynamically measure and reconstruct a com- plicated (non-binary) sample, distinguishing this toolkit from existing SLADS-based workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Moreover, its computational cost is negligible compared to the acquisition time even when run on a low-power edge computing device placed at a synchrotron beamline, which presents a signifi- cant advantage over more generic autonomous experimentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These characteristics enable the application of our workflow in the high-precision nanoscale scanning x-ray microscopy instrument present at the hard x-ray nanoprobe beamline at the Advanced Photon Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We validate the FAST scheme through real time demonstration at the hard x-ray nanoprobe beamline at the APS30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A few-layer exfoliated two-dimensional WSe2 thin film was chosen as a representative example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' the preparation process for the thin film often leaves microscopic air bub- bles trapped underneath the thin film, deforming the 2D material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We show that an adaptive scan of < 25% of the sample is sufficient to produce a high-fidelity reconstruction that identifies all the bubbles within the field of view, and even to acquire quantitative information about the film curva- ture induced by these bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The scheme quickly identifies the deformed part of the 2D material and focuses its attention there, while ignoring regions of the film that are flat and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Film curvature reconstructed from the adaptive scan (< 25% coverage) is consistent with that re- 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (Artist’s representation) The APS synchrotron produces a coherent x-ray beam that is focused using a zone plate setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' It strikes a WSe2 film (green) exfoliated onto a Si substrate (blue), which generates diffraction patterns that are collected by a two-dimensional detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Above the bubbles, the lattice of the film rotates, shifting the diffracted intensities away from its nominal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The beam position as well as the detector acquisition are autonomously controlled by the FAST AI-based workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' constructed from full-grid scan (100% coverage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Given these characteristics, the FAST scheme can be directly applied in other scanning techniques and instruments at the APS and elsewhere, and may underpin the development of many multi-level experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 7 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' RESULTS Figure 1 shows the experimental setup that scans a focused x-ray beam on a sample while ac- quiring a two-dimensional diffraction image at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The live demonstration was performed on a few-layer WSe2 sample with the detector placed along the 008 Bragg peak, with 2θ = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='1° at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The diffraction patterns were processed on the detector computer (see Methods) to generate the integrated intensities for use in the FAST workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The final output of the workflow is a dark-field image of the WSe2 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Self-driving scanning microscopy workflow Figure 2A broadly illustrates the FAST workflow for the experiments reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' To ini- tiate the workflow, a low-discrepancy quasi-random selection (generated using the Hammersely sequence31) of sample position is measured corresponding to 1% of the total area of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The integrated intensities of the measurements are transferred to the edge device, an NVIDIA Jetson Xavier AGX32 located adjacent to the detector, which used Inverse Distance Weighted (IDW) in- terpolation to estimate the dark-field image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The estimated image serves as input for the decision- making step whereby the prospective measurement points are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This self-driving workflow adopts the Supervised Learning Approach for Dynamic Sampling using Deep Neural Networks (SLADS-Net) algorithm29 to find the prospective measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In effect, the SLADS-Net algorithm uses the current measurements to identify the best unmeasured points that, when added to the existing dataset, would have the greatest effect on the quality of the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As illustrated in Figure 2B, this is accomplished by, first, representing each unmeasured point as a feature vector with elements that depend on the mea- surement state in the neighborhood of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These feature vectors are used as input for a pre-trained multi-layer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The neural network then predicts the expected reduction in distortion (ERD), a metric (loosely speaking) for the expected improvement in the reconstruction quality obtained from measuring this unmeasured point, individually for each unmeasured point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The original SLADS-Net algorithm simply uses the unmeasured point with the highest ERD for 8 COMPUTING CANDIDATES OPTIMIZING PATH INITIAL ESTIMATION INITIAL MEASUREMENTS AI @ EDGE FINAL RESULT NEW ESTIMATION NEW MEASUREMENT 𝒓 Generate Features A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The FAST workflow: (A) A set of random initial measurements are transferred to the edge device which sequentially generates an initial sample estimate, computes the candidate points to be measured next, and calculates the travel path for the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The new measurements are combined with the existing measurements and used to calculate a new estimate, and the process is repeated until it achieves a completion criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (B) The candidate computation starts by examining the local neighborhood (with radius r) of each unmeasured point P, with the highlighted points indicating points already measured, to generate a 6-dimensional feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The feature vector is transformed to a 50-dimensional vector using the Radial Basis Function (RBF) kernel and used as input to a multi-layer NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The NN then predicts the expected improvement in the image (ERD) from measuring the point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A set of unmeasured pixels with the highest ERD are selected as candidates for the next measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 9 5 hidden layers inputs 50 nodes per layer V1 output ERD 02 V50nVIDIAthe next measurement, and repeats this procedure pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In practice, if the measurement pro- cedure and the motor movements are fast, then the ERD calculation also has to be commensurately fast to reduce the dead-time in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In this work, we mitigate this requirement by in- stead selecting a batch of points that have the highest ERD, sorted in descending order—we found that a batch of 50 points adequately minimized the experimental dead-time while still ensuring that the overall measurement was adequately sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The coordinates of these 50 points are passed on to a route optimization algorithm, based on Google’s OR-Tools33, to generate the shortest path for the motors to visit all of the them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This path is appended to the look-up table in the EPICS34 scan record, which then kicks off the data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Henceforth, the scan is automatically paused after every 50 points, raising a flag which event triggers a callback function on the edge device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' There, a new estimated dark field image of the sample is generated, and the coordinates for the next 50 prospective points are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The scan is resumed after the EPICS scan record receives the new coordinates for the optimized scanning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The actual scanning of the focused x-ray beam is achieved by moving two piezoelectric linear translation motors in step mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The detector exposure time is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 s and comes with an overhead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For the 200×40 pixels object described in Section II C, the workflow required ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='15 s to compute the new positions, ≈42 s to scan the set of 50 positions, and a total of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='37 s to process the diffraction patterns and communicate the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This represents an overhead of ⪅ 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The workflow is currently entirely CPU-bound, relying on the on-board 8-core ARM CPUs, and does not take advantage of the GPU bundled into the NVIDIA AGX device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In the future, we expect to perform the computation in a parallelized and asynchronous fashion, which would further reduce this overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These timing results showcase the rapid data-driven decision-making ability that is characteristic of the FAST workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We also note that, for all the results reported in this work, the underlying NN was trained on a single generic image with no relation to microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For details about the SLADS-Net algorithm and the sample-agnostic training procedure, the reader is referred to the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Numerical demonstration for scanning dark-field microscopy Reconstructions (10% scan ) True Measured points Raster grid (RG) LD random (LDR) FAST A B C D E G H I F FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Numerical comparison of sampling methods: (A) shows the ground truth with the color scale representing the normalized intensity, (B-D) show respectively the RG, LDR, and FAST reconstructions at 10% scan coverage, and (G-I) show the actual scan points that produce these reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (E-F) show the evolution of the NRMSE (lower is better) and SSIM (higher is better) as a function of the scan coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The FAST reconstruction stabilizes at 27% coverage while the other techniques take significantly longer to reach the same quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We first validated the performance of the proposed workflow through a numerical experiment on a set of pre-acquired dark-field microscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Here, we compared the FAST sampling with three static sampling techniques: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Raster grid (RG) For a test sampling percentage, we generated a equally spaced raster grid that provides a uniform coverage of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Uniform random (UR) sampling The measurement pixels were drawn from a uniform 11 LDR UR RG FAST150 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='. :random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Low-discrepancy (LDR) random sampling For each measurement percentage, we gener- ated a low-discrepancy sampling grid using the quasi-random Hammersly sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The test dataset is a dark field image of size 600×400 pixels which represents 240,000 possible measurement positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This covers a physical area of 900 µm×600 µm and encloses multiple flakes of WSe2 with various thicknesses, with the thicker regions associated with regions of higher brightness in the image (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' At this spatial resolution, only medium and large sized bubbles (with diameter > 2 um) can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As explained previously, the bubbles deform the surface and shift the Bragg peak of the 2D materials away from their theoretical (flat region) positions, resulting in regions of darker contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Finally, the image also contains flake-free regions that have zero integrated intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For this comparison, we first initialized the FAST sampling with a 1% measurement coverage (as described above), then successively measured 50 additional points at iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For each FAST measurement, we also generate RG, UR, and LDR measurement masks with the same number of scan points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In this fashion, we generate a sequence of sampling masks and the associated reconstructions until we achieve 100% sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We present the numerical results in Figure 3, where we show a comparison of the various meth- ods at 10% sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Note that while the proposed method internally uses the fast IDW algorithm for the inpainting, the final images presented here are calculated using the higher quality bihar- monic inpainting technique35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The uniform random scheme performs worse than the LD-random and raster grid schemes and is not shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In Figure 3A-D, we can see that the FAST sampling is able to reproduce with high fidelity the flake boundaries, the bubbles, and the regions of transition between the varying levels of thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In contrast, the LDR and raster schemes produce much lower quality reconstructions of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Figure 3E shows an evolution of the normalized root mean squared error (NRMSE) and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 3F the structural similarity metric (SSIM) (which measures multiscale perceptual similarity) for the different sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' It is ev- ident that FAST produces high quality reconstructions at much lower measurement percentages 12 than the examined static sampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We note that the result could be further improved in the future by using a more sophisticated inpainting technique within the FAST method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' To un- derstand how FAST outperforms the other methods under the same sampling condition, we show the actual measured positions of the various schemes at 10% coverage (Figure 3G-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' FAST pref- erentially samples the regions with significant heterogeneity over the homogeneous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This is particularly useful for sparse samples, where the time spent sampling from empty regions adds little additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Experimental demonstration Full scan image FAST reconstructions Measured points A C E G B D F Measurements between 15-20% H FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Evolution of the FAST scan: (A, C, E) show the reconstruction at 5%, 15%, and 20% reconstructions respectively, (B, D, F) show the corresponding actual measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (G) shows the image obtained through a full-grid pointwise scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The color scale in (A-G) show the normalized intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (H) shows only the points sampled between 15% and 20% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We next demonstrate the application of the FAST workflow in a live experiment at a syn- chrotron beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A video showing the sampling, recorded live during the actual experiment, is available here36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Other than starting the workflow scripts at the beginning, the entire experiment was unmanned and fully automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In order to measure the deformed WSe2 flakes in details, a higher spatial resolution of 100 nm was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This limits the field of view to 20 µm×4 µm for 13 5%:15%+++20%2 μma scan point density of 200×40 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In Figure 4 we show the reconstructed dark field image (subplots A,C,E) and the measurement points (subplots B,D,F) from 5 % to 20 % coverage and compare them to that obtained from raster scanning the sample with 100% coverage(subplot G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We see that the FAST method identifies some of the regions of hetereogeneity — the edges of the bubbles — and starts to preferentially sample these regions within 5 % coverage of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' At 15 % coverage, these regions are extensively sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The reconstruction does not change significantly between 15 % to 20 %, indicating that the reconstruction has stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Moreover, the 20 % reconstruction also contains sharp and accurate reproductions of all the major features present in the full scan image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A point of interest is that the partially scanned bubble at the bottom right corners of Figure 4E- G shows up only in the 20% scan, and not in the 15% scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' To explain this, we note that the 5% scan, and therefore the initial 1% random sampling, does not contain any measurements in the neighborhood of this bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The FAST scheme favors exploitation of regions it knows to be heterogeneous over exploration of this fully unknown region, and therefore only explores this region much later in the measurement process (Figure 4H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This is, in fact, an instance of the general exploration-exploitation tradeoff that exists in all Bayesian search procedures37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Potential mitigation steps could be to sample more initially (say 5% random points), or to deliberately introduce diversity into each batch of measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' So far we have reduced the diffraction image measured at each point to one single quantity (integrated intensity) in order to guide the automated experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' These images often need to be reprocessed after the experiment to extract additional physically relevant results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Notably, the intensity distribution in the diffraction patterns contains information about the strain as well as the rotation of the crystal lattice, and in this case, the curvature of the 2D materials due to the bubbles underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A simple center of mass calculation in the X direction (CoMx) would yield the magnitude of the film curved in the XZ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The curvature (deviation of the CoMx from its nominal value) is the smallest around the center of the bubble and the largest at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' It also changes sign going from the left side to the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Center of mass calculation in the Y direction yields the magnitude of the film curved in the YZ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The results look slightly different from 14 the CoMx calculations due to the way the shifted Bragg peak intersects with the Ewald’s sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Figure 5A and B shows respectively the CoMx and CoMy obtained from raster scan with 100% coverage on the area of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The unit is the number of pixel shift, relative to the center of the nominal diffraction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Figure 5C and B shows respectively the CoMx and CoMy obtained with FAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The curvature information of the film were faithfully reproduced despite scanning just 20% of the entire area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For more information on the reconstruction of the CoM maps, he reader is referred to the Methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' CoMx CoMy FAST Full FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Comparison of the per measured point center of mass of the diffraction patterns between the FAST scan at 20% coverage and full-grid scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Subplots (A) and (B) show the inpainted COMx and COMy, respectively, for the full-grid raster FAST scan, and subplots (C) and (D) for the FAST scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' DISCUSSION In this work, we have showcased the FAST workflow that combines a sparse sampling algo- rithm with route planning to drive a scanning diffraction microscopy experiment at a synchrotron beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In addition to being an effective alternative to a full pointwise scan to acquire a dark- field image of the sample, FAST also produces accurate quantitative measurements of its phys- ical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For our live demonstration of a 200 points×40 points with a measurement time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 s/point, the FAST decision-making time was negligible, leading to an overall saving of ≈80 min (about ≈65 %) of the experiment time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This saving was facilitated by our choice to ac- quire a batch of 50 measurements between the selection of the prospective measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This ensured that the communication time stayed negligible with no noticeable loss in the quality of points acquired when compared to a pointwise candidate selection scheme (see Supplemental 15 CDB 2 μmMaterials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The generalizability of the FAST method comes from the fact that the key NN-based compo- nent of this workflow is trained on just the standard cameraman image38, not on close analogues of a sample of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' While this generalizability results in a slight loss of performance of the tech- nique , it still shows excellent sparsity performance for cases tested in previous research29,39 and in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This has the benefit that we do not need a priori knowledge of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As such, while general pre-training would be difficult to satisfy for new and expensive experiments, the FAST approach can be used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Furthermore, the batch prediction and route optimization approach we implement can also be directly applied in any application of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Moreover, the experimental application of our work uses an extensible edge device and the widely used EPICS platform for hardware control, both of which can be incorporated into any instrument even with the SLADS-Net replaced by any other sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For example, we could just replace the dark-field detection procedure described here with a fluorescence counting setup and use exactly the FAST scheme for a fluorescence-based imaging of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Alternatively, since all the in- struments at the APS rely on EPICS controls, one can perform transmission, surface scattering, or any other 2D scanning experiment in any applicable beamline with only minor changes to the FAST routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The computations in the current workflow have a time complexity of O(2N logN +kM logN), where N is the number of measured points, M the number of unmeasured points, and k the num- ber of nearest neighboring measurements (k = 10 in our case) that we use for the feature vector calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Here, the first term accounts for the creation of the nearest neighbor K-d tree and the second term for the nearest neighbor calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The remainder of the algorithm has a linear time complexity and could be performed in parallel for the unmeasured points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We expect that it is possible to reduce this complexity using an approximate nearest neighbor search method instead of the K-d tree approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As such, a GPU-based implementation that takes advantage of the par- allelization and the approximation would likely significantly reduce the computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This stands in stark contrast with the time complexity of O � N3� (for N measured points) for Gaussian Processes, a similarly training-free method that is widely used for autonomous experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 16 For an illustrative example, Vasudevan et al20 report a GP-based scanning microscopy experiment where the calculation of each set of measurement candidates takes ≈6 s on an NVIDIA DGX-2 GPU for a 50×50 image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' our workflow performs an equivalent calculation for a larger 200×40 image within ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 s in a low-power CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We note, however, that GPs remain a very powerful and generalizable approach with a bevy of applications beyond only scanning microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We caution that our workflow suffers from three important challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' First, it depends heav- ily on the initial 1% random sampling to discover regions of heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' If an isolated feature present in an otherwise homogeneous region is not partially sampled during this random sam- pling step, then such a feature can be missed until much later in the scanning experiment (see Figure 4H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' A related second limitation is that this method produces sub-optimal reconstructions if the sample is sufficiently heterogeneous that the data in each pixel changes significantly from pixel to pixel throughout the image (Supplemental Material in Hujsak et al27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The third limi- tation, more practical in nature, is that the scan paths require significant motor movement, often including a retracing over points already measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As such, there could exist scenarios in which the time required for the motor movement eclipses the time required for a single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We expect to address these limitations by explicitly including a measurement-density-based term 39 or a movement-time-based term in the candidate selection procedure40, or by using a line-based sampling technique41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Despite these challenges, we believe that the proposed FAST technique has great potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' It is an ideal tool for use cases with limited sampling or dosage budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' It can be used to isolate regions of interest in sparse settings, to prepare for pointwise scanning in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' More generally, it can be used to guide any scanning microscopy experiment where we do not need full pointwise information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In the future, we expect to extend this method for 3D imaging, fly scans, ptychography, and other imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We expect that these developments will significantly enhance the efficacy of scanning microscopy experiments, bolstering their use for the study of dynamic physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 17 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The SLADS-Net algorithm The SLADS-Net algorithm29 used within the FAST workflow is an adaptation of the Super- vised Learning Approach for Dynamic Sampling (SLADS) algorithm originally developed by Godaliyadda et al25, and the algorithms differ only in their training approaches ( Section IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' To explain the SLADS algorithm, we first denote the object we want to measure as A ∈ RN, where N is the total number of pixels in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Further, we can denote the pixel at location 1 ≤ s ≤ N as as so that a measurement at the location s extracts the value as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' each measurement is thus characterized by the pair (s, as).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' After k measurements, then, we get the k×2 measurement vector Y k = � ������� s1 as1 s2 as2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' sk ask � ������� (1) Using these k measurements, then, we can reconstruct (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' via interpolation) an estimate ˆAkof the true object A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The difference between A and ˆAk is denoted as the distortion D(A, ˆAk) and can be calculated using any chosen metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In the current work, we define D(A, ˆAk) to be the L2 norm: D(A, ˆAk) = ||A− ˆAk||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Given the measurement Y k and the reconstruction ˆAk, a new measurement at any location s will presumably reduce the distortion in the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We can denote this reduction in distortion (RD) as Rk,s = D(A, ˆAk)−D(A, ˆAk,s) (2) where ˆAk,s is the reconstruction that includes the newly added measurement at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The goal of the SLADS algorithm is then to identify the pixel location that would maximize this reduction in distortion: sk+1 = argmax s Rk,s (3) 18 Of course, since we cannot know the value of the measurement as or the ground truth A, SLADS bases its selection on the conditional expectation of reduction in distortion (ERD), which is defined as: Rk,s = E � Rk,s��Y k� so that sk+1 = argmax s Rk,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (4) The algorithm assumes that we can compute the ERD at s based on just the measurement state Yk as Rk,s = g(vk,s) (5) where vk,s is a location-dependent feature vector calculated using the measurement state Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The goal of the SLADS training procedure is to estimate the function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Training The training procedure for the SLADS/SLADS-Net algorithm is a supervised procedure in which we generate a large number of (vk,s,Rk,s) pairs and use these to estimate g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Note that this is a pixelwise computation that is performed independently for each measurement location s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' for each measurement s we have to calculate a reconstruction ˆAk,s before we can calculate the RD Rk,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' To make this computationally tractable, the Godaliyadda et al25 use approximations that ensure that the RD of each pixel only depends on its local neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Correspondingly, instead of working with the full measurement state Y k, the training procedure uses carefully designed feature vectors that capture the local neighborhood of the pixel at location s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' As shown in Figure 2B, the feature vector for the pixel P consists of six features: (i) ∇x and ∇y are the spatial gradients at P, (ii) σ1,r and σ2,r measure the deviation of the estimated value for P from the nearby measured values (highlighted in red), and (iii) L (which is the distance of P from the closest measured point) and ρr measure the density of measurements around P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The original SLADS algorithm assumes that this feature vector is linearly related to the RD, and the training therefore is a linear regression procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The SLADS-Net adaptation first uses an radial basis function (RBF) kernelization to transform the 6-dimensional feature vector to a 19 50-dimensional vector, then replaces the linear predictor with a nonlinear fully-connected neural network that contains 5 hidden layers with 50 nodes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' In this work, we train the SLADS-Net neural network on only the standard cameraman image, without using any a priori information about the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For the training, we generate a mea- surement state Y k by randomly choosing a fixed number number of measurement locations, then calculate the feature vector vk,s and the RD Rk,s for each unmeasured pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We generate such sets of training pairs for 10 different sample coverage percentages between 1% and 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' This overall comprises our training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We use this data to train the neural network for 100 epochs using the Adam optimizer with the learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We use this trained model for all the simu- lated and experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We provide an example of a training measurement set—the measured points, the interpolated reconstruction, and the corresponding RD for the unmeasured points—in the supplemental materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' S2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Experimental measurements At each point of the measurement, a tight region of interest (RoI) around the expected position of the thin film Bragg peak was extracted from the corresponding diffraction image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Integrated intensities of the RoI were used to guide the NN prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For the flat region, the integrated intensity is high, showing up as brighter contrast on the dark field image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For the deformed region, the integrated intensity is low (darker contrast on the dark field image) as the illuminated film diffraction partially exits the selected RoI (see Supplemental Materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' For the FAST experiment, the predicted ERD and the dark-field reconstruction served as visual guides to inform when to stop the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='. During the experiment, we noted that the ERD and the reconstruction had stabilized by ≈20 % scan coverage, but we let the experiment run to ≈35 % coverage to ensure that this behavior persisted (see Supplemental Materials, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' While we used this visual criterion for our exploratory experiment, it is straightforward to design a numerical stopping criterion based on the absolute or relative convergence of the ERD, or on the per-iteration change in the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 20 DATA AND CODE AVAILABILITY The data and code will be made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='com/saugatkandel/fast_ smart_scanning ACKNOWLEDGMENTS Work performed at the Center for Nanoscale Materials and Advanced Photon Source, both U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Department of Energy Office of Science User Facilities, was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' DOE, Office of Basic Energy Sciences, under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We also acknowl- edge support from Argonne LDRD 2021-0090 – AutoPtycho: Autonomous, Sparse-sampled Pty- chographic Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We gratefully acknowledge the computing resources provided on Bebop, a high-performance computing cluster operated by the Laboratory Computing Resource Center at Argonne National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='L.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=', Laskin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' & Hye Ye, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Deep Learning Approach for Dynamic Sparse Sampling for High-Throughput Mass Spectrometry Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' Electronic Imaging 33, 290–1– 290–7 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' URL https://library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='org/ei/articles/33/15/art00007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 25 Supplemental material January 16, 2023 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='05286v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='app-ph] 12 Jan 2023 A B Figure S1: Comparison of the FAST reconstructions for scan batch size of 1 (FAST-1) and 50 (FAST-50) as a function of the scan coverage for the numerical simulation described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We observe that FAST-1 initially performs better, with lower NRMSE and higher SSIM, than FAST-50, but this advantage erodes quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' We ended the FAST-1 experiment at ≈8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='2 % sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='due due to simulation time limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 2 A B C Figure S2: Example of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (A) shows a set of randomly selected mea- surement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (B)shows the reconstruction calculated by interpolating these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (c) shows the ERDs calculated for the unmeasured points, with the ERD highest at regions of hetereogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The location of the measured points, the measured values, and the reconstruction are used to generate feature vectors for the training, and the ERDs are used as the training labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 3 CoMx CoMy Figure S3: Example of ROI selection and change in diffraction patterns around the bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' (A) and (B) respectively show the CoMx and CoMy calcualted from the FAST scan with 20% covergae, as discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The diffraction patterns for the points marked with the ×, �, and + are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The × point is in a region without a bubble and has the diffraction pattern at the Bragg angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The points marked with � and + are located at the top and bottom edges of the bubble, and therefore show additional anomalous diffraction spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The dashed square boxes in the diffraction pattern figures indicate the ROI used for the dark-field image reconstructions (shown in Figure 4 in the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The CoM calculations use the regions outside the dashed square boxes as the RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 4 AB XOX0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content='5 ERD 1e6 0 10 20 30 40 50 60 Scan iteration 0 5 10 15 20 25 30 35 Scan coverage (%) 20% coverage Figure S4: Evolution in the ERD for the experimental demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' The ERD initially decreases rapidly, during which point the each batch of 50 points signifi- cantly improves the sample reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' At per-iteration change in the ERD is much smaller at 20% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE4T4oBgHgl3EQf1A0X/content/2301.05286v1.pdf'} diff --git a/5tAzT4oBgHgl3EQfEfox/content/2301.00993v1.pdf b/5tAzT4oBgHgl3EQfEfox/content/2301.00993v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8f4c54e3425e37ca6aa4f418e3ead5318284b5ba --- /dev/null +++ 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sha256:7a9539a31c671df33902b24d26ef1bb80a10a8e53d00f301184dea9d324077ef +size 91877 diff --git a/79A0T4oBgHgl3EQfOf-b/content/tmp_files/2301.02162v1.pdf.txt b/79A0T4oBgHgl3EQfOf-b/content/tmp_files/2301.02162v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a77f5de1395679e7570b69a67c67ef1bb7490de --- /dev/null +++ b/79A0T4oBgHgl3EQfOf-b/content/tmp_files/2301.02162v1.pdf.txt @@ -0,0 +1,1504 @@ +arXiv:2301.02162v1 [stat.ME] 5 Jan 2023 +Improve Efficiency of Doubly Robust Estimator when +Propensity Score is Misspecified +Liangbo Lv∗ and Molei Liu† +Abstract +Doubly robust (DR) estimation is a crucial technique in causal inference and miss- +ing data problems. We propose a novel Propensity score Augmentved Doubly robust +(PAD) estimator to enhance the commonly used DR estimator for average treatment +effect on the treated (ATT), or equivalently, the mean of the outcome under covariate +shift. Our proposed estimator attains a lower asymptotic variance than the conven- +tional DR estimator when the propensity score (PS) model is misspecified and the +outcome regression (OR) model is correct while maintaining the double robustness +property that it is valid when either the PS or OR model is correct. These are realized +by introducing some properly calibrated adjustment covariates to linearly augment the +PS model and solving a restricted weighted least square (RWLS) problem to minimize +the variance of the augmented estimator. Both the asymptotic analysis and simula- +tion studies demonstrate that PAD can significantly reduce the estimation variance +compared to the standard DR estimator when the PS model is wrong and the OR is +correct, and maintain close performance to DR when the PS model is correct. We +further applied our method to study the effects of eligibility for 401(k) plan on the +improvement of net total financial assets using data from the Survey of Income and +Program Participation of 1991. +Keywords: Causal inference; Covariate shift correction; Propensity score; Outcome regres- +sion; Double robustness; Intrinsic efficiency. +∗Liangbo Lv is an undergraduate student from the School of Statistics, Renmin University of China. +†Molei Liu is an assistant professor at Columbia University Mailman School of Public Health. +1 + +1 +Introduction +1.1 +Background +Doubly robust (DR) estimation has attracted extensive interest in the literature on semipara- +metric theory and causal inference and is frequently used in biomedical science, economics, +and policy science studies. It incorporates two nuisance models, a propensity score (PS) +model, and an outcome regression (OR) model to characterize distributions of the expo- +sure and outcome against the adjustment covariates respectively, and draws valid inferences +when either one of them is correctly specified. It has been well-established that when both +the PS and OR models are correct, the DR estimator is semiparametric efficient and its +asymptotic variance does not really depend on the estimating equations for the nuisance +models (Tsiatis, 2006, e.g.). Nevertheless, there still remains an intriguing question on how +to improve the asymptotic efficiency of the DR estimator when one nuisance model is mis- +specified. For the scenario with correct PS and wrong OR models, there is a track of work +(Cao et al., 2009; Tan, 2010, e.g.) proposing the so-called intrinsic efficient estimator that +will be reviewed in Section 1.3. This type of estimator preserves the double robustness prop- +erty and achieves improved efficiency over the standard DR estimator when the PS model +is correct and the OR is wrong. Interestingly, we notice that the dual problem of this, i.e., +improving the (intrinsic) efficiency of the DR estimator under wrong PS and correct OR, +is supposed to be equally important but has not been handled yet due to certain technical +reasons that will be discussed later. Aimed in this paper, filling this methodological blank +can effectively complement the existing tools for DR and semiparametric inference. +1.2 +Problem Setup +To make our idea easier to understand, we focus on a specific missing data problem: trans- +fer estimation of the outcome’s mean in the presence of covariate shift (Huang et al., 2007, +e.g.). This is also equivalent to estimating the average treatment effect on the treated (ATT) +(Hahn, 2004, e.g.) in the context of causal inference and matching-adjusted indirect com- +parison frequently conducted in biomedical studies (Signorovitch et al., 2010). Our method +could be generalized to other settings such as estimating the average treatment effect (ATE) +and transfer learning of regression models (Liu et al., 2020). +Suppose there are n labeled samples with observed outcome Y and covariates X ∈ +Rd, and N unlabeled samples only observed on X. Let ∆ = 1 indicate that the sample +is labeled and ∆ = 0 otherwise. +The labeled observations (Yi, Xi) are collected from a +source population S with ∆i = 1 for i = 1, 2, . . . , n. Assume (Yi, Xi) ∼ pS(x)q(y|x) for +2 + +i = 1, 2, . . . , n where pS(x) and q(y|x) represent the density of X on S and the conditional +density of Y given X = x respectively. Meanwhile, there are unlabeled samples from a target +population T indicated by ∆i = 0 and only observed on covariates Xi for i = n+1, . . . , N+n. +Assume that on T , (Yi, Xi) ∼ pT (x)q(y|x) with pT (x) representing the density of X on +T and the distribution of Y | X remaining to be the same as that on S. Our goal is to +estimate µ0 = ET Y , the marginal mean of Y on T . In the absence of observed Y on the +target samples, two simple strategies to estimate µ0 are introduced below. +(PS) Define the propensity score (PS) or density ratio between the two populations as +r0(x) = pT (x)/pS(x). Estimate r0(x) with some �r(x) and average the observed Yi +weighted by �r(Xi) over i = 1, 2, . . . , n from S. +(OR) Define the outcome regression (OR) or imputation model for Y as m0(x) = E[Y | X = +x]. Estimate m0(x) with some �m(x) obtained using the labeled samples and average +�m(Xi) over i = n + 1, . . . , n + N from T . +Both the PS and OR strategies are built upon the assumption that the distribution of Y | X +is the same between S and T so the knowledge of Y on S is transferable to T . This is in the +same spirit as the no unmeasured confounding assumption in the context of causal inference. +1.3 +Related literature +Our work is based on the doubly robust (DR) inference framework that has been fre- +quently studied and applied in the past years (Robins et al., 1994; Bang and Robins, 2005; +Kang and Schafer, 2007; Tan, 2010; Vermeulen and Vansteelandt, 2015, e.g.). It combines +the PS and OR models introduced in Section 1.2 to construct an estimator that is valid +when at least one of the two nuisance models are correct and, thus, regarded as a more ro- +bust statistical inference procedure than the simple PS and OR strategies. Early work in DR +inference (Bang and Robins, 2005; Kang and Schafer, 2007, e.g.) mainly used working low- +dimensional parametric regression to construct the PS and OR models. Recent progress has +been made to accommodate the use of high-dimensional regression or complex machine learn- +ing methods in estimating the nuisance models (Chernozhukov et al., 2018; Tan, 2020, e.g.), +which is less prone to model misspecification. We focus the scope of this paper on the low- +dimensional parametric setting that is technically less involved but more user-friendly and +less sensitive to over-fitting in practice. It is also possible and valuable to generalize our work +to the settings of high-dimensional parametric (Tan, 2020; Dukes and Vansteelandt, 2020, +e.g.) or semi-non-parametric (Liu et al., 2020) nuisance models, in which model misspecifi- +cation is still an important concern. +3 + +There has risen great interest in studying and improving the asymptotic efficiency of the +DR estimator. One track of literature studied the local efficiency of the DR estimator, i.e., if +it is semiparametric efficient when both the PS and OR models are known or correctly spec- +ified. While it was shown that the standard DR estimator for the ATE (Robins et al., 1994) +achieves such local efficiency (Hahn, 1998; Tsiatis, 2006). This result cannot be directly ap- +plied to the ATT estimator because unlike ATE, the PS model of ATT is informative (or +non-ancillary) (Hahn, 1998, 2004). Shu and Tan (2018) further studied this subtle issue and +proposed locally efficient DR estimators for ATT based on its influence function. +Meanwhile, another track of literature focuses on improving the efficiency of the DR es- +timator in the presence of correct PS and potentially wrong OR models and, thus, is more +relevant to our work that also aims at automatic variance reduction under model misspec- +ification. +A class of intrinsic efficient DR estimator has been proposed for the efficient +estimation of ATE (Cao et al., 2009; Tan, 2010), ATT (Shu and Tan, 2018), casual regres- +sion model (Rotnitzky et al., 2012), longitudinal data (Han, 2016), individual treatment rule +(Pan and Zhao, 2021), etc. This type of estimator is (i) valid when either nuisance model is +correct; (ii) equivalent with the standard DR estimator when both models are correct; and +(iii) of the minimum variance under correct PS and wrong OR, among all the DR estimators +with the same parametric specification of the OR model, and, consequently, more efficient +than the standard DR estimator. In addition, it was shown that including more prognostic +covariates or auxiliary basis in the PS model can always help to reduce the variance of the +ATE estimator (Hahn, 2004; Tsiatis, 2006). Motivated by this, Cheng et al. (2020) proposed +a double-index PS estimator for ATE that smooths the treatment over the parametric PS +and OR models to achieve the DR property as well as variance reduction under correct PS +and wrong OR. Nevertheless, such a strategy may also incur over-fitting issues and cause +poor performance in finite or small sample studies (Gronsbell et al., 2022). +Although the correct PS and wrong OR setting has been frequently studied, there is still a +paucity of solutions to its dual problem, i.e., enhancing the DR estimator under the wrong PS +and correct OR. Some early work like Kang and Schafer (2007) and Cao et al. (2009) argued +that the simple OR strategy is an ideal choice when one knows the PS model is wrong since +it is free of PS weighting that may decrease the effective sample size. However, since there +are no perfect ways to examine model correctness without any additional assumptions, this +strategy can never be as robust as the DR estimator to misspecification of the OR model. +We also notice a large body of work in statistical learning and causal inference that aims +at leveraging some auxiliary data or information to boost the asymptotic efficiency of certain +estimators using the idea of augmentation. For example, Kawakita and Kanamori (2013), +Chakrabortty et al. (2018) and Azriel et al. (2021) proposed different semi-supervised learn- +4 + +ing methods that improve estimation efficiency of the linear model leveraging large unlabeled +data drawn from the same distribution as the labeled samples. Methods like Chen and Chen (2000) +and Yang and Ding (2019) utilized external data with error-prone outcomes or covariates to +construct control variate for variance reduction. These methods, as well as other examples, +rely on some auxiliary data to construct estimators that always converge to zero and are +asymptotically correlated with the target estimator. These zero estimators are then used +to augment the target estimator properly for variance reduction. Our work also adapts the +high-level idea of augmentation. But different from these methods, ours does not leverage +any auxiliary samples or knowledge and additionally cares about the need of prioritizing va- +lidity (double robustness) over statistical power. Consequently, the asymptotic behavior of +our augmented estimator actually varies according to the correctness of the nuisance models +and is more technically involved in to study. +1.4 +Our contribution +To estimate µ0 introduced in Section 1.2 efficiently, we propose a novel Propensity score +Augmented Doubly robust (PAD) estimation method that enhances the standard DR es- +timator of µ0 by linearly augmenting the PS model with some functions of X. Both the +augmentation functions and their linear coefficients are wisely and carefully constructed such +that the augmentation term always reduces the variance of the DR estimator if the PS is +wrong and the OR is correct while it automatically converges to zero if the PS is correct, +in order to avoid bias and ensure double robustness. Also, when both models are correct, +our PAD estimator becomes asymptotically equivalent to the standard DR estimator. To +our best knowledge, the proposed estimator is the first one to simultaneously have the DR +property and a smaller variance than the standard DR estimator under wrong PS and correct +OR models. Thus, our work serves as an important complement to existing DR inference +approaches, especially to the intrinsically efficient DR estimators proposed to work for the +setting with correct PS and wrong OR (Cao et al., 2009; Tan, 2010, e.g.). +2 +Method +2.1 +Doubly robust estimator +As a prerequisite of our proposal, we first introduce the standard DR estimator for µ0 un- +der the setup described in Section 1.2, which has been studied for years (Hahn, 1998, 2004; +Shu and Tan, 2018, e.g.). Following a common strategy (Bang and Robins, 2005; Shu and Tan, 2018; +Liu et al., 2020, e.g.), we form the PS and OR models as r(x) = exp(xTγ) and m(x) = +5 + +g(xTα) where γ and α are model coefficients and g(·) is a known and differentiable link +function. We say that the PS (or OR) model is correct if there exists γ0 (or α0) such that +the true r0(x) = exp(xTγ0) (or m0(x) = g(xTα0)). Denote the empirical mean operator on +S and T as �ES and �ET such that +�ESa(X, Y ) = n−1 +n +� +i=1 +a(Xi, Yi), +�ET a(X, Y ) = N−1 +n+N +� +i=n+1 +a(Xi, Yi) +for any function a(·). Suppose the two nuisance estimators �γ and �α are obtained respectively +by solving the estimating equations: +�ESX exp(X +Tγ) = �ET X, +�ESX{Y − g(X +Tα)} = 0. +(1) +The estimating equations for γ in (1) is usually referred as covariate balancing (Imai and Ratkovic, 2014; +Zhao and Percival, 2017), and those for α correspond to the ordinary least square regression +when g(a) = a and the logistic regression when Y is binary and g(a) = expit(a) = ea/(1+ea). +Note that one can use alternative estimation procedures to obtain γ and α, e.g., running a +logistic regression on ∆ against X to estimate γ, and our proposed method could naturally +adapt to different choices on this. +Based on �γ and �α, the PS and OR estimators introduced in Section 1.2 can be specified as +�µPS = �ESY exp(X +T�γ) and �µOR = �ET g(X +T�α) respectively. Then the standard DR estimator +is constructed by augmenting one of them with another nuisance model: +�µDR = �ES{Y − g(X +T�α)} exp(X +T�γ) + �ET g(X +T�α). +(2) +When the PS model is correct and �γ converges to γ0, �ET g(X +T�α) − �ESg(X +T�α) exp(X +T�γ) +converges to zero and the remainder term �ESY exp(X +T�γ) is exactly the PS estimator con- +verging to µ0. Similarly, when OR is correct, we can show that �ES{Y − g(X +T�α)} exp(X +T�γ) +converges to zero and �ET g(X +T�α) converges to µ0. Thus �µDR is doubly robust in the sense +that it is consistent when either the PS or OR model is correctly and consistently estimated. +2.2 +Expansion of DR estimator under correct OR model +To help the readers understand our method more intuitively, we now heuristically derive and +analyze the asymptotic expansion of �µDR when the OR model is correctly specified. Suppose +that �γ and �α converge to some ¯γ and ¯α defined as the solutions to the population-level +estimating equations ESX exp(X +Tγ) = ET X and ESX{Y − g(X +Tα)} = 0, respectively. +Let �r(x) = exp(X +T�γ), ¯r(x) = exp(X +T¯γ), and S(α) = S(Y, X, α) = X{Y − g(X +Tα)}. +Suppose that the OR model is correct, i.e., m0(x) = g(X +Tα0) and α0 = ¯α, and n1/2(�α − +6 + +¯α, �γ − ¯γ) is asymptotically normal with mean zero following the standard M-estimation +theory (Van der Vaart, 2000). Then we have +�ES{Y − g(X +T�α)}{�r(X) − ¯r(X)} = op(n−1/2) +due to Neyman orthogonality (Neyman, 1959), which, as will be strictly proved in Section +3, implies that �µDR defined in (2) is asymptotically equivalent with +�µDR =�ES{Y − g(X +T ¯α)}¯r(X) + �ET g(X +T ¯α) ++ +� +�ES{g(X +T ¯α) − g(X +T�α)}¯r(X) + �ET {g(X +T�α) − g(X +T ¯α)} +� +≈�ES{Y − g(X +T ¯α)}¯r(X) + �ET g(X +T ¯α) + L +T�ESX{Y − g(X +T ¯α)}, +where L = − ¯H−1 {ESX ˙g(X +T ¯α)¯r(X) − ET X ˙g(X +T ¯α)}, ¯H = ESXX +T ˙g(X +T ¯α), and ˙g(a) is +the derivative of g(a). To derive the above result, we use the standard asymptotic expansion +of �α given by our Lemma B3 in Appendix, and the symbol “≈” indicates that the difference +between the two lines is up to op(n−1/2) and, thus, asymptotically negligible. So when OR +is correct, the asymptotic variance of n1/2(�µDR − µ0) is equal to that of n1/2(�µDR − µ0), which +can be expressed as +aVar{n1/2(�µDR − µ0)} = ES{¯r(X)}2v(X) + 2L +TESX¯r(X)v(X) + C, +(3) +where v(x) = Var(Y | X) and C is some positive constant free of ¯r(·) and, thus, needs not +to be considered in the following derivation. Note that when the PS model also is correct, +i.e., ¯r(·) = r0(·), we further have L = 0. +Empirically, term L in (3) can be estimated by +�L = − �H−1 � +�ESX ˙g(X +T�α) exp(X +T�γ) − �ET X ˙g(X +T�α) +� +, +(4) +where �H = �ESXX +T ˙g(X +T�α). Estimation of v(x) relies on our working assumption on the +form of Var(Y | X). For example, one may assume Y = m0(X) + ǫ where ǫ ∼ N(0, σ2) +so v(x) is invariant of x and can be simply imputed with the moment estimator of σ2. +Also, for the common Poisson model Y ∼ Poisson{exp(X +Tα0)} and logistic model Y ∼ +Bernoulli{expit(X +Tα0)}, one can naturally estimate v(x) by exp(xT�α) and expit(xT�α){1 − +expit(xT�α)} respectively. To preserve generality, we introduce a working model vθ(x) for +v(x) with some nuisance parameter θ to be estimated as �θ that could be partially or fully +determined by �α. Suppose that �θ converges to some ¯θ. As will be shown in Section 3, +violation of this conditional variance model, i.e., v(x) ̸= v¯θ(x) does not impact the double +robustness of our proposed estimator but only affects its efficiency gain when PS is wrong +and OR is correct. +7 + +2.3 +PAD estimator +Now we formally introduce the propensity score augmented doubly robust (PAD) estimator. +Our central idea is to augment the PS model ¯r(X) = exp(X +T¯γ) as ¯raug(X; β) = exp(X +T¯γ)+ +Ψ +Tβ and use ¯raug(·) to replace ¯r(·) in the DR estimator. Here Ψ is some properly constructed +basis function of X and β is some loading coefficient vector to be estimated. We first describe +the empirical construction procedures for PAD in Algorithm 1 and then discuss the reason +and intuition of the key steps in this algorithm. +Algorithm 1 Propensity score Augmented Doubly robust (PAD) estimation +[Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional +variance estimator as �θ. +[Step 2] Specify Φ = φ(X) of larger dimensionality than X using any basis function φ(·), +and take �Ψ = Φ − �ET [Φv�θ(X)]/�ET v�θ(X). +[Step 3] Solve the restricted weighted least square (RWLS) problem: +�β = argminβ �Vµ(β), +s.t. +�ESX ˙g(X +T�α) �Ψ +Tβ = 0, +(5) +where +�Vµ(β) = �ES{exp(X +T�γ) + �Ψ +Tβ}2v�θ(X) + 2�L +T�ESX{exp(X +T�γ) + �Ψ +Tβ}v�θ(X), +(6) +and �L is as defined in equation (4). +[Step 4] Obtain the PAD estimator through +�µPAD = �ES{Y − g(X +T�α)}{exp(X +T�γ) + �Ψ +T�β} + �ET g(X +T�α). +For heuristic analysis, suppose that all estimators used in (6) converge to their limiting +values. Then let Ψ = Φ −ET [Φv¯θ(X)]/ET vθ(X) be the limits of �Ψ, ¯β the limits of �β, with +its specific form given by Lemma B1 in Appendix, and +Vµ(β) = ES{exp(X +T¯γ) + Ψ +Tβ}2v¯θ(X) + 2L +TESX{exp(X +T¯γ) + Ψ +Tβ}v¯θ(X) +the limiting function of �Vµ(β) specified in Algorithm 1. We shall consider two scenarios +separately to demonstrate that our proposed PAD estimator not only maintains double +robustness property but also has a lower asymptotic variance than �µDR when the OR model +is correctly specified and PS is wrong. Rigorous justification for these results will be provided +in Section 3. +8 + +Correct PS model. +When the PS model is correct, we easily have L = 0 as stated in +Section 2.2 so Vµ(β) = ES{exp(X +T¯γ) + Ψ +Tβ}2v¯θ(X), and +∂Vµ(β) +∂β += ESΨ exp(X +T¯γ)v¯θ(X) = ET Ψv¯θ(X). +By definition of Ψ, we have ET Ψv¯θ(X) = 0, as ensured by the mean shift of Φ in Step +2 of Algorithm 1. Thus, β = 0 minimizes Vµ(β) and consequently, is the solution of the +population-level version of the RWLS problem (5) since the linear constraints in (5) is trivially +satisfied by β = 0. This implies that as long as the PS model is correct, �β converges to 0 +so the augmented PS estimator exp(X +T�γ) + �Ψ +T�β converges to the correct PS model, which +ensures �µPAD to converge to the true µ0. Meanwhile, it is clear that the augmentation of PS +does not change the OR model at all. Therefore, �µPAD preserves the same DR property as +�µDR, i.e., being (root-n) consistent whenever the PS or the OR model is correctly specified. +Correct OR and wrong PS. +Note that �µPAD = �µDR + �ES �Ψ +T�β{Y − g(X +T�α)} and when +the OR model is correct, +�ES �Ψ +T�β{Y − g(X +T�α)} =�ES �Ψ +T�β{Y − g(X +Tα0)} + �ES �Ψ +T�β{g(X +Tα0) − g(X +T�α)} +≈�ESΨ +T¯β{Y − g(X +Tα0)} + �ES(α0 − �α) +TX ˙g(X +T�α) �Ψ +T�β, +(7) +in which we use the orthogonality between �Ψ +T�β − Ψ +T¯β and Y − g(X +Tα0) on the first term, +as well as expansion on g(X +Tα0) − g(X +T�α) in the second term of the first line, to derive +the “≈” relation shown in the second line. Here, “≈” in (7) again means that the difference +between the first and second line is up to op(n−1/2) and, thus, becomes asymptotically +negligible. +In addition, according to the moment constraint in the RWLS problem (5), +�ESX ˙g(X +T�α) �Ψ +T�β converges to 0. +So the second term in the second line of (7) is also +negligible and �µPAD ≈ �µDR + �ESΨ +T¯β{Y − g(X +Tα0)}. Combining this with equation (3) as +well as the asymptotic equivalence between �µDR and �µDR discussed in Section 2.2, we have +aVar{n1/2(�µPAD − µ0)} = ES{¯r(X) + Ψ +T¯β}2v(X) + 2L +TESX{¯r(X) + Ψ +T¯β}v(X) + C, (8) +which, after dropping the invariant C, is equal to Vµ(¯β), the limiting value of the minimized +objective function �Vµ(�β) in the RWLS problem (5). Note that β = 0 is always feasible to +the linear constraint in (5) and if we simply replace ¯β with 0 in the right-hand side of (8), +it reduces to the asymptotic variance of n1/2(�µDR − µ0) derived in (3). Meanwhile, when the +PS model is wrong, ∂Vµ(β)/∂β is typically not 0 at β = 0 so the population-level minimizer +¯β ̸= 0. Thus, aVar{n1/2(�µPAD − µ0)} ≤ aVar{n1/2(�µDR − µ0)} when the OR model is correct +and the strict “<” will hold in general when the PS model is wrong. +9 + +3 +Asymptotic analysis +In this section, we rigorously present the asymptotic properties of the proposed PAD estima- +tor and compare PAD with the standard DR estimator. We first introduce some mild and +common regularity assumptions. Without loss of generality, we assume that n/N = O(1) so +the desirable parametric rate of the DR estimators will be O(n−1/2). +Assumption 1. The supports of X and Φ are compact and EY 4 < ∞. +Assumption 2. The link function g(·) is differentiable with derivative ˙g(·) and there exists +a constant L such that |˙g(x1) − ˙g(x2)| < L|x1 − x2| for all x1, x2 ∈ R. +Assumption 3. The dimension of Ψ is larger than that of X. Matrices ES{ΨΨ +Tv¯θ(X)}, +ES{XX +T exp(X +T¯γ)}, ES{XX +T ˙g(X +T ¯α)} and ES{ΨX +T ˙g(X +T ¯α)} have all their eigenvalues +bounded and staying away from zero. +Assumption 4. The conditional variance function vθ(x) is differentiable on θ with a bounded +partial derivative ∂θvθ(x). The estimator �θ converges to some ¯θ in probability and satisfies +that n1/2(�θ − ¯θ) is asymptotic normal with mean zero. +Remark 1. Assumptions 1–3 are all mild, standard, and commonly used to justify the +asymptotic properties of M-estimation (Van der Vaart, 2000). Note that in Assumption 3, +we take Ψ to have larger dimension than X and make regularity conditions on ES{XX +T ˙g(X +T ¯α)} +and ES{ΨX +T ˙g(X +T ¯α)}. These are to ensure that �β is not zero and properly converges to ¯β. +Assumption 4 constrains the way of specifying vθ(x) and estimating θ. Under Assumptions +1–3, this assumption is satisfied when either θ is fully determined by α, e.g., in a Pois- +son or logistic model for Y against X, or when θ is estimated by additionally fitting some +parametric model of Var(Y | X) against X. +Now we present the main results about the robustness and efficiency of our proposed PAD +estimator in Theorem 1 with its proof given in Section B of the Appendix. Some important +heuristics of this theorem has already been discussed in Section 2.3. +Theorem 1. Under Assumptions 1–4, it holds that +(i) Double robustness. When either the PS or the OR model is correctly specified, i.e., +r0(x) = exp(xTγ0) for some γ0 or m0(x) = g(xTα0) for some α0, �µPAD +p−→ µ0 and +n1/2(�µPAD − µ0) weakly converges to some normal distribution with mean zero. +(ii) Variance reduction under wrong PS. When the OR model is correct while the +PS model may be misspecified, the asymptotic variance of n1/2(�µPAD − µ0) is always +10 + +not larger than that of n1/2(�µDR − µ0). Further when ¯β ̸= 0 (the explicit form of ¯β is +given in Lemma B1), n1/2(�µPAD − µ0) has a strictly smaller asymptotic variance than +n1/2(�µDR − µ0). +(iii) Equivalence under correct PS and OR. When both the PS and OR models are +correct, n1/2(�µPAD − µ0) and n1/2(�µDR − µ0) are asymptotically equivalent and have the +same asymptotic variance. +4 +Simulation study +We conducted simulation studies to evaluate our proposed estimator and compare it with +the standard DR estimator. +In our studies, we generate covariates X = (X1, X2, X3)T +from N(0, Σ) with Σ = (σij) ∈ R3×3 and σij = 0.3|i−j|. For generation of the population +assignment ∆ and outcome Y , we consider six settings, namely: +(G1) Gassuian Y , Correct PS, Correct OR. Pr(∆ = 1 | X)} = expit(X1 − 2X2 + X3) +and Y = 0.5X1 + 0.5X2 + X3 + ǫ where ǫ | X ∼ N(0, 1). +(G2) Gassuian Y , Correct PS, Wrong OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) +and Y = 0.5X1 + 0.5X2 + sin(X2 + 0.5X3) + ǫ. +(G3) Gassuian Y , Wrong PS, Correct OR. Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 − +1.5|X1| − 1.5|X2| − |X3|) and Y = 0.5X1 + 0.5X2 + X3 + ǫ. +(L1) Binary Y , Correct PS, Correct OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and +Pr(Y = 1 | X) = expit(0.5X1 + 0.5X2 + X3). +(L2) Binary Y , Correct PS, Wrong OR. Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and +Pr(Y = 1 | X)} = expit(0.5X1 + 0.5X2 + sin(X2 + 0.5X3)) +(L3) Binary Y , Wrong PS, Correct OR. Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 − +1.5|X1| − 1.5|X2| − |X3|) and Pr(Y = 1 | X) = expit(0.5X1 + 0.5X2 + X3). +In Settings (G1)–(G3), Y is a gaussian variable and we fit linear models for Y ∼ X with +vθ(x) = 1. While in Settings (L1)–(L3), we fit logistic models for the binary Y against +X with vθ(x) = expit(X +Tα){1 − expit(X +Tα)}. We consider different scenarios about the +correctness of the PS and OR models to examine the robustness and efficiency of PAD. +Bootstrap is used for estimating the asymptotic variance and constructing the confidence +interval (CI). For effective variance reduction on PAD when PS is wrong, i.e. under Settings +11 + +(G3) and (L3), we include in the augmentation covariates Φ a decent amount of X’s basis +functions including Xj, exp(Xj), |Xj|, exp(−Xj1 − Xj2), and exp(−X1 − X2 − X3) for all j +and j1 ̸= j2 ∈ {1, 2, 3}. We set N = n = 500 or N = n = 1000 separately and generate 1000 +realizations for each setting. +Table 1 reports the absolute average bias (Bias), standard error (SE), and coverage +probability (CP) of the 95% CI of the DR and PAD estimators. When at least one nuisance +models are correct, DR and PAD attain very close bias, which is much smaller compared to +their SE and, thus, grants their CPs to be close to the nominal level. This indicates that +PAD achieves the double robustness property just like the standard DR estimator under +finite samples. To compare PAD and DR in terms of their estimation variance and efficiency, +we present in Table 2 their relative efficiency (RE) defined as Var(�µDR)/ Var(�µPAD). Under +Settings (G1), (G2), (L1), and (L2) where the PS model is correct, the two estimators show +nearly identical variance, with their REs located between 1 ±0.04. Under Settings (G3) and +(L3) with misspecified PS and correct OR models, our proposed PAD estimator shows 20% +to 40% smaller variance than the standard DR estimator. All these results demonstrate that +conclusions in Theorem 1 also apply well for finite samples. In specific, PAD performs very +closely to the standard DR when the PS model is correct and is potentially better than DR +in the presence of wrong PS models. +5 +Real example +The effects of the 401(k) program have been investigated for a long time (Abadie, 2003; +Chernozhukov et al., 2018, e.g.). Different from other plans like Individual Retirement Ac- +counts (IRAs), eligibility for 401(k) is completely decided by employers. Therefore, unob- +served personal preferences for savings may make little difference in 401(k) eligibility. How- +ever, there may be some other confounders affecting the causal studies of 401(k), such as job +choice, income, and age. To address this problem, (Abadie, 2003) and (Chernozhukov et al., 2018) +proposed to adjust for certain covariates related to job choice so that 401(k) eligibility can +be regarded exogenous. +Whether 401(k) eligibility contributes to the improvement of people’s net total financial +assets is an important topic studied in existing literature like Abadie (2003) and Chernozhukov et al. (2018). +However, whether 401(k) can improve the financial assets of those actually not eligible for +401(k) is still an open and interesting problem. To investigate this problem, we analyze the +data from the Survey of Income and Program Participation of 1991. The data set consists of +n + N = 9275 observations. The outcome of our interests, Y is defined as the indication of +having positive net total financial assets. There are 9 adjustment covariates in X, including +12 + +Table 1: +The absolute average bias (Bias), standard error (SE), and coverage probability (CP) of the 95% +confidence intervals of the DR and PAD estimators under the settings described in Section 4. All results are +produced based on 1000 repetitions. +n = N = 500 +n = N = 1000 +Setting +Method +Bias +SE +CP +bias +SE +CP +(G1) +DR +0.006 +0.145 +0.94 +0.005 +0.106 +0.92 +PAD +0.005 +0.142 +0.93 +0.004 +0.105 +0.92 +(G2) +DR +0.007 +0.152 +0.92 +0.008 +0.111 +0.92 +PAD +0.005 +0.149 +0.92 +0.007 +0.112 +0.92 +(G3) +DR +0.010 +0.162 +0.93 +0.001 +0.121 +0.92 +PAD +0.005 +0.136 +0.93 +0.001 +0.105 +0.93 +(L1) +DR +0.000 +0.055 +0.92 +0.001 +0.040 +0.92 +PAD +0.001 +0.054 +0.93 +0.001 +0.040 +0.93 +(L2) +DR +0.001 +0.054 +0.92 +0.004 +0.040 +0.92 +PAD +0.001 +0.053 +0.92 +0.004 +0.040 +0.92 +(L3) +DR +0.005 +0.057 +0.91 +0.003 +0.038 +0.92 +PAD +0.005 +0.052 +0.93 +0.002 +0.035 +0.93 +Table 2: +Relative efficiency (RE) between DR and PAD, i.e., Var(�µDR)/ Var(�µPAD), under the settings de- +scribed in Section 4. +n, N +(G1) +(G2) +(G3) +(L1) +(L2) +(L3) +500 +1.04 +1.04 +1.42 +1.04 +1.04 +1.20 +1000 +1.02 +0.98 +1.33 +1.00 +1.00 +1.18 +age, income, family size, years of education, benefit pension status, marriage, two-earner +household status, individual participation in IRA plan, and home ownership status. The +source (treated) samples S with ∆ = 1 are taken as those eligible for 401(k) and the target +(untreated) samples T are those without 401(k) eligibility. We applied PAD and standard +DR to estimate µ, the effect of 401(k) eligibility on improving the positive rate of net to- +tal financial assets among people without 401(k) eligibility. The PS model is specified as +exp(X +Tγ) and the OR model is expit(X +Tα). In our method, the augmentation covariates +vector Φ consists of X, exp(−0.3Xj), |Xj|, and X2 +j for all Xj’s that are not binary. We +again use bootstrap to estimate SEs and construct CIs. +In Table 3, we report the point estimation, their estimated standard errors (ESE), and +95% CIs for the treatment effect µ, obtained using the standard DR and our proposed PAD +13 + +methods. Outputs of both methods indicate that 401(k) eligibility has a significant effect on +improving the rate of having positive net total financial assets among people who are actually +not eligible for 401(k). The estimated treatment effect is 0.169 (95% CI: 0.142, 0.196) by the +standard DR and 0.150 (95% CI: 0.126, 0.175) by PAD. Moreover, the ESE of our proposed +PAD estimator is remarkably smaller than that of the standard DR estimator, with their +estimated RE, i.e., Var(�µDR)/ Var(�µPAD) being around 1.25. This means our proposed PAD +method can characterize the treatment effect µ more precisely than DR in this example. +Table 3: The point estimation (PE), its estimated standard error (ESE), and 95% confidence interval (CI) +for µ, the effect of 401(k) eligibility on improving the positive rate of net total financial assets among people +without 401(k) eligibility, derived using the standard DR and the PAD methods. +Method +PE +ESE +CI +DR +0.169 +0.0140 +(0.142, 0.196) +PAD +0.150 +0.0125 +(0.126, 0.175) +6 +Discussion +In analogy to our PS model augmentation strategy, we also propose an OR model augmen- +tation strategy (OAD) that augments the OR model with some bases of X satisfying certain +moment conditions like Ψ in Algorithm 1. Description and discussion of this method are +presented in Section A of the Appendix. Similar to Theorem 1, we are able to show that +this OAD estimator is doubly robust, of a smaller variance than the standard DR estimator +when the PS model is correct but the OR model is wrong, and equivalent with DR when +both nuisance models are correct. Just like PAD, this OAD method is easy to implement +and only requires convex optimization. We notice that some existing methods in intrinsic +efficient DR estimation like Rotnitzky et al. (2012) and Gronsbell et al. (2022) rely on non- +convex training to construct the OR model when it is not linear. This OAD strategy could +mitigate this practical problem and still achieves the purpose of variance reduction in the +presence of misspecified OR models. +For ease of demonstration, we focus on covariate shift correction, or equivalently ATT +estimation in this paper. 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Journal of Causal +Inference, 5(1). +17 + +A +Dual construction to augment OR +In analogy to our PAD estimator, to improve the efficiency our the DR estimator under the +correct PS and wrong OR models, we propose the Outcome regression Augmented Doubly +robust (OAD) estimator in the following algorithm. +Algorithm A1 Outcome regression Augmented Doubly robust (OAD) estimation +[Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional +variance estimator as �θ. +[Step 2] Let Φ = φ(X) with function φ(·), �g(X +T�α) = g(X +T�α) − �ET g(X +T�α) and +�Ψ = Φ − +�ET Φ�g(X +T�α) +�ET �g2(X +T�α) +�g(X +T�α). +[Step 3] Solve the restricted weighted least square (RWLS) problem: +�β = argminβ �Vµ,OAD(β), +s.t. +�ESX �Ψ +Tβ exp(X +T�γ) = 0, +(A1) +where +�Vµ,OAD(β) =n−1� +VarS[{Y − g(X +T�α) − �Ψ +T�β} exp(X +T�γ)] + N−1� +VarT {g(X +T�α) + �Ψ +T�β} ++ 2� +L∗ +T[N−1 � +CovT (X, �Ψ +T�β) + n−1 � +CovS{X exp(X +T�γ), �Ψ +T�β exp(X +T�γ)}], +(A2) +and � +L∗ = {�ESX exp(X +T�γ)X +T}−1�ES{Y − g(X +T�α)} exp(X +T�γ)X. +[Step 4] Obtain the OAD estimator: +�µOAD = �ES{Y − g(X +T�α) − �Ψ +T�β} exp(X +T�γ) + �ET {g(X +T�α) + �Ψ +T�β}. +To demonstrate how Algorithm A1 works, we define that +�µOAD =�ES{Y − g(X +T ¯α) − Ψ +T¯β} exp(X +T¯γ) + �ET {g(X +T ¯α) + Ψ +T¯β} ++ ES{Y − g(X +T ¯α)} exp(X +T¯γ)X +T{ESX exp(X +T¯γ)X +T}−1{�ET X − �ESX exp(X +T¯γ)}. +Then similar to our analysis in Section 2.2, when the PS model is correct, �µOAD is asymptot- +ically equivalent to �µOAD, with its variance being: +Vµ,OAD(β) =n−1 VarS[{Y − g(X +T ¯α) − Ψ +T¯β} exp(X +T¯γ)] + N−1 VarT {g(X +T ¯α) + Ψ +T¯β} ++ 2L∗T[N−1CovT (X, Ψ +T¯β) + n−1CovS{X exp(X +T¯γ), Ψ +T¯β exp(X +T¯γ)}], +1 + +the limiting function of �Vµ,OAD(β) specified in Algorithm A1, where +L∗ = {ESX exp(X +T¯γ)X +T}−1ES{Y − g(X +T ¯α)} exp(X +T¯γ)X. +This corresponds to the objective function in equation (A2). Similar to the PAD construc- +tion, when β = 0, Vµ,OAD(β) reduces to the asymptotic variance of the standard DR estimator. +Thus, �µOAD has a smaller variance than the standard DR estimator when the PS model is +correct and the OR model is wrong, under which we typically have ¯β ̸= 0. +On the other hand, when OR is correctly specified, we have ¯α = α0, L∗ = 0, and thus +∂Vµ,OAD(β) +∂β +|β=0 = CovT (g(X +T ¯α), Ψ). +By definition of Ψ, we have CovT (g(X +T ¯α), Ψ) = 0. Hence, similar to the analysis in Section +2.2, �µOAD preserved the same DR property as �µDR, i.e., being root-n consistent whenever the +PS or the OR model is correctly specified. +B +Asymptotic justification +B.1 +Technical lemma +Lemma B1. Define a := ESΨ ∂g(XTα) +∂αT +|¯α, b := ESΨ exp(X +T¯γ)v¯θ(X) + ESΨX +Tv¯θ(X)LT, +and Σ := ESΨΨ +Tv¯θ(X), under Condition 2-5, the solution of the RWLS problem (5) is +¯β = Σ−1a(a +TΣ−1a)−1a +TΣ−1b − Σ−1b. +Proof. First we introduce Lagrange multiplier λ and write (5) as the Lagrange form: +¯β = argminβ ES{exp(X +T¯γ)+Ψ +Tβ}2v¯θ(X)+2LESX{exp(X +T¯γ)+Ψ +Tβ}v¯θ(X)−λ +TES +∂g(X +Tα) +∂α +|¯αΨ +Tβ. +Then we have the partial derivative of λ and β: +ES +∂g(X +Tα) +∂α +|¯αΨ +Tβ = 0, +(B1) +and +2ESΨ{exp(X +T¯γ) + Ψ +Tβ}v¯θ(X) + 2ESΨX +Tv¯θ(X)L +T − ESΨ∂g(X +Tα) +∂αT +|¯αλ = 0. +(B2) +From (B2) we have +β = {2ESΨΨ +Tv¯θ(X)}−1{ESΨ∂g(X +Tα) +∂αT +|¯αλ−2ESΨ exp(X +T¯γ)v¯θ(X)−2ESΨX +Tv¯θ(X)L +T}, +2 + +together with (B1), we have +ES +∂g(X +Tα) +∂α +|¯αΨ +T{ESΨΨ +Tv¯θ(X)}−1 +∗ {ESΨ∂g(X +Tα) +∂αT +|¯αλ − 2ESΨ exp(X +T¯γ)v¯θ(X) − 2ESΨX +Tv¯θ(X)L +T} = 0. +this function can be simplified as +a +TΣ−1(aλ − 2b) = 0, +and we further have +λ = 2(a +TΣ−1a)−1a +TΣ−1b. +Hence, we have +¯β = Σ−1a(a +TΣ−1a)−1a +TΣ−1b − Σ−1b, +and we can estimate it by +�β = �Σ +−1�a(�a +T �Σ +−1�a)−1�a +T �Σ +−1�b − �Σ +−1�b +for �a := �ES �Ψ∂g(XTα) +∂αT +|�α, �b := �ES �Ψ exp(X +T�γ)v�θ(X)+�ES �ΨX +Tv�θ(X)�LT, and �Σ := �ES �Ψ �Ψ +Tv�θ(X) +Lemma B2. Under Condition 3, 4 and 6, we have that �Ψ − Ψ = Op(n−1/2). +Proof. By definition, we would have that +�Ψ − Ψ = +�ET {Φv�θ(X)} +�ET v�θ(X) +− ET {Φv¯θ(X)} +ET v¯θ(X) +. +Under Condition 4 and 6, we have that +�ET v�θ(X) − ET v¯θ(X) = �ET v¯θ(X) + �ET +∂vθ(X) +∂θ +|�θ(�θ − ¯θ) − ET v¯θ(X) = Op(n−1/2) +(B3) +for �θ between �θ and ¯θ. +By using the same techniques, we have that �ET {Φv�θ(X)} − +ET {Φv¯θ(X)} = Op(n−1/2). And we have +�Ψ − Ψ = +�ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}�ET v�θ(X) +�ET v�θ(X)ET v¯θ(X) += +�ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}ET v¯θ(X) − [ET {Φv¯θ(X)}�ET v�θ(X) − ET {Φv¯θ(X)}ET v¯θ(X)] +{ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X) += Op(n−1/2)ET v¯θ(X) − ET {Φv¯θ(X)}Op(n−1/2) +{ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X) += Op(n−1/2). +3 + +Lemma B3. Under Condition 1, 2 and 4, 5, we have that �γ − ¯γ = Op(n−1/2) and �α − ¯α = +Op(n−1/2). +Proof. The estimation of γ has been given as +�ESX exp(X +T�γ) = �ET X, +by applying Taylor series expansion, we have +n−1 +n +� +i=1 +Xi exp(X +T +i ¯γ) + n−1 +n +� +i=1 +Xi exp(X +T +i �γ)X +T +i (�γ − ¯γ) = N−1 +n+N +� +i=n+1 +Xi, +where �γ is some vector between �γ and ¯γ. According to (Van der Vaart, 2000), we have +�γ − ¯γ = op(1). Let J represent matrix n−1 �n +i=1 Xi exp(X +T +i �γ)X +T +i , and we have that +J = n−1 +n +� +i=1 +Xi exp(X +T +i ¯γ)X +T +i +n−1 +n +� +i=1 +Xi exp(X +T +i γ∗)X +T +i Xi(�γ−¯γ) = ESX exp(X +T¯γ)X +T+op(1) +for γ∗ between �γ and ¯γ. Hence, by central limit theorem and Slutsky theorem, we have +that, under Condition 1 and 4 +�γ − ¯γ = J−1 +� +N−1 +n+N +� +i=n+1 +Xi − n−1 +n +� +i=1 +Xi exp(X +T +i ¯γ) +� +=J−1 +� +N−1 +n+N +� +i=n+1 +Xi − ET X + ESX exp(X +T¯γ) − n−1 +n +� +i=1 +Xi exp(X +T +i ¯γ) +� += Op(n−1/2). +Furthermore, The estimation equation of �α is given by +�ESS(�α) = �ESX{Y − g(X +T�α)} = 0, +by using Taylor series expansion, we have that +�ESX{Y − g(X +T ¯α)} + �ES +∂S(α) +∂αT +���� +�α +(�α − ¯α) = 0 +for �α between �α and ¯α, and we have +�α − ¯α = −�ES +�∂S(α) +∂αT +���� +�α +�−1 +�ESX{Y − g(X +T ¯α)}. +By using the same techniques as those for obtaining the asymptotic properties of �γ, under +Condition 2, 4 and 5, we have �α − ¯α = Op(n−1/2). +4 + +Lemma B4. Under Condition 1-6 and Lemma A1-A3, we can obtain that �β−¯β = Op(n−1/2). +In addition, when the PS is correctly specified, we further have ¯β = 0 and �β = Op(n−1/2). +Proof. By using the same techniques as (B3), under Condition 2-4, we first have that +�a−a = �ES �Ψ∂g(X +Tα) +∂αT +|�α−ESΨ∂g(X +Tα) +∂αT +|¯α = �ESΨ∂g(X +Tα) +∂αT +|¯α−ESΨ∂g(X +Tα) +∂αT +|¯α+Op(n−1/2) = Op(n−1/2). +In addition, we can have that �b − b = Op(n−1/2) and �Σ − Σ = Op(n−1/2). Furthermore, we +can easily have that +�Σ +−1 − Σ−1 = Σ−1Σ{Σ + Op(n−1/2)}−1 − Σ−1 += Σ−1[Σ{Σ + Op(n−1/2)}−1 − {Σ + Op(n−1/2)}{Σ + Op(n−1/2)}−1] = Op(n−1/2), +based on which we can have (�aT �Σ +−1�a)−1−(aTΣ−1a)−1 = Op(n−1/2). Let �Ω denote �Σ +−1�a(�aT �Σ +−1�a)−1�aT �Σ +−1 +and Ω denote Σ−1a(aTΣ−1a)−1aTΣ−1. We can have that �Ω − Ω = Op(n−1/2), hence, we +have that �β − ¯β = �Ω�b − Ωb = Op(n−1/2). +On the other hand, when the PS is correctly specified, L = 0 and ESΨ exp(X +T¯γ)v¯θ(X) = +ET Ψv¯θ(X) = 0, which means +¯β = Ωb = Ω{ESΨ exp(X +T¯γ)v¯θ(X) + ESΨX +Tv¯θ(X)L +T} = Ω0 = 0. +And at the same time, we have �β = Op(n−1/2). +B.2 +Proof of Theorem 1 +Proof. Proof of Theorem 1 (i). +When the OR is correctly specified, ¯α = α0. Consider �µOR where +�µOR = �ES{Y − g(X +T ¯α)}{exp(X +T¯γ) + Ψ +T¯β} + �ET g(X +T ¯α) ++ +� +ES +∂g(X +Tα) +∂αT +���� +¯α +exp(X +T¯γ) − ET +∂g(X +Tα) +∂αT +���� +¯α +� +ES +�∂S(α) +∂αT +���� +¯α +�−1 +�ESX{Y − g(X +T ¯α)}. +It is obvious that E�µOR = ET g(X +T ¯α) = µ0. Hence, by using central limit theorem, we have +that �µOR − µ0 = Op(n−1/2), n1/2(�µOR − µ0) weakly converges to gaussian distribution with +mean 0. On the other hand, we have that +�µP AD − �µOR = �ES{Y − g(X +Tα0)}{exp(X +T¯γ)X +T(�γ − ¯γ) + Ψ +T(�β − ¯β) + ( �Ψ − Ψ) +T¯β} +− +� +�ES +∂g(X +Tα) +∂αT +���� +α0 +{exp(X +T¯γ) + Ψ +T¯β} − �ET +∂g(X +Tα) +∂αT +���� +α0 +� +(�α − ¯α) + op(n−1/2) +5 + +− +� +ES +∂g(X +Tα) +∂αT +���� +α0 +exp(X +T¯γ) − ET +∂g(X +Tα) +∂αT +���� +α0 +� +ES +�∂S(α) +∂αT +���� +α0 +�−1 +�ESX{Y − g(X +Tα0)}, +by using central limit theorem, along with Lemma A2-A4, we have that +�ES{Y − g(X +Tα0)}{exp(X +T¯γ)X +T(�γ − ¯γ) + Ψ +T(�β − ¯β) + ( �Ψ − Ψ) +T¯β} += [�ES{Y − g(X +Tα0)} exp(X +T¯γ)X +T](�γ − ¯γ) + [�ES{Y − g(X +Tα0)}Ψ +T](�β − ¯β) ++ [�ES{Y − g(X +Tα0)}¯β +T]( �Ψ − Ψ) = Op(n−1/2)op(1) + Op(n−1/2)op(1) + Op(n−1/2)op(1) = op(n−1/2). +On the other hand, +− +� +�ES +∂g(X +Tα) +∂αT +���� +α0 +{exp(X +T¯γ) + Ψ +T¯β} − �ET +∂g(X +Tα) +∂αT +���� +α0 +� +(�α − ¯α) += +� +�ES +∂g(X +Tα) +∂αT +���� +α0 +{exp(X +T¯γ) + Ψ +T¯β} − �ET +∂g(X +Tα) +∂αT +���� +α0 +� +�ES +�∂S(α) +∂αT +���� +¯α +�−1 +�ESX{Y − g(X +T ¯α)} += +� +ES +∂g(X +Tα) +∂αT +���� +α0 +exp(X +T¯γ) − ET +∂g(X +Tα) +∂αT +���� +α0 ++ Op(n−1/2) +� +∗ +� +ES +�∂S(α) +∂αT +���� +α0 +�−1 ++ Op(n−1/2) +� +�ESX{Y − g(X +Tα0)}. +(B4) +Hence, we have that +− +� +�ES +∂g(X +Tα) +∂αT +���� +α0 +{exp(X +T¯γ) + Ψ +T¯β} − �ET +∂g(X +Tα) +∂αT +���� +α0 +� +(�α − ¯α) +− +� +ES +∂g(X +Tα) +∂αT +���� +α0 +exp(X +T¯γ) − ET +∂g(X +Tα) +∂αT +���� +α0 +� +ES +�∂S(α) +∂αT +���� +α0 +�−1 +�ESX{Y − g(X +Tα0)} += �ESX{Y − g(X +Tα0)}Op(n−1/2) = op(n−1/2). +Thus, from previous results, we have that �µP AD − �µOR = op(n−1/2). Together with Slutsky +theorem, we futher have that �µP AD − µ0 = Op(n−1/2) and n1/2(�µP AD − µ0) weakly converges +to gaussian distribution with mean 0. +When the PS is correctly specified, ¯γ = γ0, we consider �µPS where +�µPS = �ES{Y − g(X +T ¯α)}{exp(X +Tγ0) + Ψ +T¯β} + �ET g(X +T ¯α) ++ ES{Y − g(X +T ¯α)} exp(X +T¯γ)X +T{ESX exp(X +Tγ0)X +T}−1{�ET X − �ESX exp(X +Tγ0)}. +Together with the results from Lemma A4, we have that E�µPS = ESY exp(X +T¯γ) = ET Y = +µ0. By using the central limit theorem, we have that �µPS − µ0 = Op(n−1/2), n1/2(�µPS − µ0) +weakly converges to gaussian distribution with mean 0. On the other hand, we have that +�µP AD − �µPS = �ES{Y − g(X +T ¯α)}{exp(X +T¯γ)X +T(�γ − ¯γ) + Ψ +T(�β − ¯β)} +6 + +− ES{Y − g(X +T ¯α)} exp(X +T¯γ)X +T{ESX exp(X +Tγ0)X +T}−1{�ET X − �ESX exp(X +Tγ0)} + op(n−1/2) +By using the techniques from (B4), we would have +�ES{Y − g(X +T ¯α)} exp(X +T¯γ)X +T(�γ − ¯γ) +− ES{Y − g(X +T ¯α)} exp(X +T¯γ)X +T{ESX exp(X +Tγ0)X +T}−1{�ET X − �ESX exp(X +Tγ0)} = op(n−1/2) +And from Lemma A4, we have �β = Op(n−1/2). Thus, we have �µP AD − µ0 = Op(n−1/2). On +the other hand, it is worth noticing that �β is the continuous function of �θ, �γ and �α, so +under central limit theorem and Slutsky theorem, we would have the asymptotic normality +of �β. Hence, we further have that n1/2(�µPS − µ0) weakly converges to gaussian distribution +with mean 0. +Proof. Proof of Theorem 1 (ii). +First we denote U as +U = VarT (E(Y |X)) + LESXX +T Var(Y |X)L +T +When the OR is correctly specified, the asymptotic variance of �µP AD, Var{n−1/2(�µP AD−µ0)} +is +ES{exp(X +T¯γ) + Ψ +T¯β}2v¯θ(X) + 2LESX{exp(X +T¯γ) + Ψ +T¯β}v¯θ(X) + U, +and ¯β contributes to minimizing this variance. When ¯β = 0, the function above is written +as +ES{exp(X +T¯γ)}2v¯θ(X) + 2LESX{exp(X +T¯γ)}v¯θ(X) + U, +which is the same as the asymptotic variance of �µDR, Var{n−1/2(�µDR − µ0)}. Hence, when +¯β ̸= 0, �µP AD has the smaller asymptotic variance than standard doubly robust estimator +�µDR. +Proof. Proof of Theorem 1 (iii). +When both the PS and OR is correctly specified, consider �µB, where +�µB = �ES{Y − g(X +Tα0)} exp(X +Tγ0) + �ET g(X +Tα0). +By using central limit theorem, n1/2(�µB −µ0) weakly converges to gaussian distribution with +mean 0. On the other hand, by using Taylor series expansion, we would have �µP AD − �µB = +op(n−1/2) and �µDR − �µB = op(n−1/2). Hence, they have the same asymptotic variance. +7 + +This figure "DMLmse.png" is available in "png"� format from: +http://arxiv.org/ps/2301.02162v1 + +This figure "biasdml.png" is available in "png"� format from: +http://arxiv.org/ps/2301.02162v1 + diff --git a/79A0T4oBgHgl3EQfOf-b/content/tmp_files/load_file.txt b/79A0T4oBgHgl3EQfOf-b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e79bb8fdaecc3f1371f4d1629cafc288103eba8e --- /dev/null +++ b/79A0T4oBgHgl3EQfOf-b/content/tmp_files/load_file.txt @@ -0,0 +1,769 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf,len=768 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='02162v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='ME] 5 Jan 2023 Improve Efficiency of Doubly Robust Estimator when Propensity Score is Misspecified Liangbo Lv∗ and Molei Liu† Abstract Doubly robust (DR) estimation is a crucial technique in causal inference and miss- ing data problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We propose a novel Propensity score Augmentved Doubly robust (PAD) estimator to enhance the commonly used DR estimator for average treatment effect on the treated (ATT), or equivalently, the mean of the outcome under covariate shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our proposed estimator attains a lower asymptotic variance than the conven- tional DR estimator when the propensity score (PS) model is misspecified and the outcome regression (OR) model is correct while maintaining the double robustness property that it is valid when either the PS or OR model is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' These are realized by introducing some properly calibrated adjustment covariates to linearly augment the PS model and solving a restricted weighted least square (RWLS) problem to minimize the variance of the augmented estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Both the asymptotic analysis and simula- tion studies demonstrate that PAD can significantly reduce the estimation variance compared to the standard DR estimator when the PS model is wrong and the OR is correct, and maintain close performance to DR when the PS model is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We further applied our method to study the effects of eligibility for 401(k) plan on the improvement of net total financial assets using data from the Survey of Income and Program Participation of 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Keywords: Causal inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Covariate shift correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Propensity score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Outcome regres- sion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Double robustness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Intrinsic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ∗Liangbo Lv is an undergraduate student from the School of Statistics, Renmin University of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' †Molei Liu is an assistant professor at Columbia University Mailman School of Public Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 1 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='1 Background Doubly robust (DR) estimation has attracted extensive interest in the literature on semipara- metric theory and causal inference and is frequently used in biomedical science, economics, and policy science studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' It incorporates two nuisance models, a propensity score (PS) model, and an outcome regression (OR) model to characterize distributions of the expo- sure and outcome against the adjustment covariates respectively, and draws valid inferences when either one of them is correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' It has been well-established that when both the PS and OR models are correct, the DR estimator is semiparametric efficient and its asymptotic variance does not really depend on the estimating equations for the nuisance models (Tsiatis, 2006, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Nevertheless, there still remains an intriguing question on how to improve the asymptotic efficiency of the DR estimator when one nuisance model is mis- specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For the scenario with correct PS and wrong OR models, there is a track of work (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tan, 2010, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=') proposing the so-called intrinsic efficient estimator that will be reviewed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This type of estimator preserves the double robustness prop- erty and achieves improved efficiency over the standard DR estimator when the PS model is correct and the OR is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Interestingly, we notice that the dual problem of this, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', improving the (intrinsic) efficiency of the DR estimator under wrong PS and correct OR, is supposed to be equally important but has not been handled yet due to certain technical reasons that will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Aimed in this paper, filling this methodological blank can effectively complement the existing tools for DR and semiparametric inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 Problem Setup To make our idea easier to understand, we focus on a specific missing data problem: trans- fer estimation of the outcome’s mean in the presence of covariate shift (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2007, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This is also equivalent to estimating the average treatment effect on the treated (ATT) (Hahn, 2004, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=') in the context of causal inference and matching-adjusted indirect com- parison frequently conducted in biomedical studies (Signorovitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our method could be generalized to other settings such as estimating the average treatment effect (ATE) and transfer learning of regression models (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Suppose there are n labeled samples with observed outcome Y and covariates X ∈ Rd, and N unlabeled samples only observed on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Let ∆ = 1 indicate that the sample is labeled and ∆ = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The labeled observations (Yi, Xi) are collected from a source population S with ∆i = 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assume (Yi, Xi) ∼ pS(x)q(y|x) for 2 i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' , n where pS(x) and q(y|x) represent the density of X on S and the conditional density of Y given X = x respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Meanwhile, there are unlabeled samples from a target population T indicated by ∆i = 0 and only observed on covariates Xi for i = n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' , N+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assume that on T , (Yi, Xi) ∼ pT (x)q(y|x) with pT (x) representing the density of X on T and the distribution of Y | X remaining to be the same as that on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our goal is to estimate µ0 = ET Y , the marginal mean of Y on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In the absence of observed Y on the target samples, two simple strategies to estimate µ0 are introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (PS) Define the propensity score (PS) or density ratio between the two populations as r0(x) = pT (x)/pS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Estimate r0(x) with some �r(x) and average the observed Yi weighted by �r(Xi) over i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' , n from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (OR) Define the outcome regression (OR) or imputation model for Y as m0(x) = E[Y | X = x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Estimate m0(x) with some �m(x) obtained using the labeled samples and average �m(Xi) over i = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' , n + N from T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Both the PS and OR strategies are built upon the assumption that the distribution of Y | X is the same between S and T so the knowledge of Y on S is transferable to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This is in the same spirit as the no unmeasured confounding assumption in the context of causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3 Related literature Our work is based on the doubly robust (DR) inference framework that has been fre- quently studied and applied in the past years (Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Bang and Robins, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Kang and Schafer, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tan, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Vermeulen and Vansteelandt, 2015, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' It combines the PS and OR models introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 to construct an estimator that is valid when at least one of the two nuisance models are correct and, thus, regarded as a more ro- bust statistical inference procedure than the simple PS and OR strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Early work in DR inference (Bang and Robins, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Kang and Schafer, 2007, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=') mainly used working low- dimensional parametric regression to construct the PS and OR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Recent progress has been made to accommodate the use of high-dimensional regression or complex machine learn- ing methods in estimating the nuisance models (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tan, 2020, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ), which is less prone to model misspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We focus the scope of this paper on the low- dimensional parametric setting that is technically less involved but more user-friendly and less sensitive to over-fitting in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' It is also possible and valuable to generalize our work to the settings of high-dimensional parametric (Tan, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Dukes and Vansteelandt, 2020, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=') or semi-non-parametric (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2020) nuisance models, in which model misspecifi- cation is still an important concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 3 There has risen great interest in studying and improving the asymptotic efficiency of the DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' One track of literature studied the local efficiency of the DR estimator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', if it is semiparametric efficient when both the PS and OR models are known or correctly spec- ified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' While it was shown that the standard DR estimator for the ATE (Robins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 1994) achieves such local efficiency (Hahn, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tsiatis, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This result cannot be directly ap- plied to the ATT estimator because unlike ATE, the PS model of ATT is informative (or non-ancillary) (Hahn, 1998, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Shu and Tan (2018) further studied this subtle issue and proposed locally efficient DR estimators for ATT based on its influence function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Meanwhile, another track of literature focuses on improving the efficiency of the DR es- timator in the presence of correct PS and potentially wrong OR models and, thus, is more relevant to our work that also aims at automatic variance reduction under model misspec- ification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' A class of intrinsic efficient DR estimator has been proposed for the efficient estimation of ATE (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tan, 2010), ATT (Shu and Tan, 2018), casual regres- sion model (Rotnitzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2012), longitudinal data (Han, 2016), individual treatment rule (Pan and Zhao, 2021), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This type of estimator is (i) valid when either nuisance model is correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (ii) equivalent with the standard DR estimator when both models are correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' and (iii) of the minimum variance under correct PS and wrong OR, among all the DR estimators with the same parametric specification of the OR model, and, consequently, more efficient than the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In addition, it was shown that including more prognostic covariates or auxiliary basis in the PS model can always help to reduce the variance of the ATE estimator (Hahn, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tsiatis, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Motivated by this, Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2020) proposed a double-index PS estimator for ATE that smooths the treatment over the parametric PS and OR models to achieve the DR property as well as variance reduction under correct PS and wrong OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Nevertheless, such a strategy may also incur over-fitting issues and cause poor performance in finite or small sample studies (Gronsbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Although the correct PS and wrong OR setting has been frequently studied, there is still a paucity of solutions to its dual problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', enhancing the DR estimator under the wrong PS and correct OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Some early work like Kang and Schafer (2007) and Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2009) argued that the simple OR strategy is an ideal choice when one knows the PS model is wrong since it is free of PS weighting that may decrease the effective sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' However, since there are no perfect ways to examine model correctness without any additional assumptions, this strategy can never be as robust as the DR estimator to misspecification of the OR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We also notice a large body of work in statistical learning and causal inference that aims at leveraging some auxiliary data or information to boost the asymptotic efficiency of certain estimators using the idea of augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For example, Kawakita and Kanamori (2013), Chakrabortty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2018) and Azriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2021) proposed different semi-supervised learn- 4 ing methods that improve estimation efficiency of the linear model leveraging large unlabeled data drawn from the same distribution as the labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Methods like Chen and Chen (2000) and Yang and Ding (2019) utilized external data with error-prone outcomes or covariates to construct control variate for variance reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' These methods, as well as other examples, rely on some auxiliary data to construct estimators that always converge to zero and are asymptotically correlated with the target estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' These zero estimators are then used to augment the target estimator properly for variance reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our work also adapts the high-level idea of augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' But different from these methods, ours does not leverage any auxiliary samples or knowledge and additionally cares about the need of prioritizing va- lidity (double robustness) over statistical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Consequently, the asymptotic behavior of our augmented estimator actually varies according to the correctness of the nuisance models and is more technically involved in to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='4 Our contribution To estimate µ0 introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 efficiently, we propose a novel Propensity score Augmented Doubly robust (PAD) estimation method that enhances the standard DR es- timator of µ0 by linearly augmenting the PS model with some functions of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Both the augmentation functions and their linear coefficients are wisely and carefully constructed such that the augmentation term always reduces the variance of the DR estimator if the PS is wrong and the OR is correct while it automatically converges to zero if the PS is correct, in order to avoid bias and ensure double robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Also, when both models are correct, our PAD estimator becomes asymptotically equivalent to the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To our best knowledge, the proposed estimator is the first one to simultaneously have the DR property and a smaller variance than the standard DR estimator under wrong PS and correct OR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, our work serves as an important complement to existing DR inference approaches, especially to the intrinsically efficient DR estimators proposed to work for the setting with correct PS and wrong OR (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Tan, 2010, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 2 Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='1 Doubly robust estimator As a prerequisite of our proposal, we first introduce the standard DR estimator for µ0 un- der the setup described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2, which has been studied for years (Hahn, 1998, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Shu and Tan, 2018, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Following a common strategy (Bang and Robins, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Shu and Tan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2020, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ), we form the PS and OR models as r(x) = exp(xTγ) and m(x) = 5 g(xTα) where γ and α are model coefficients and g(·) is a known and differentiable link function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We say that the PS (or OR) model is correct if there exists γ0 (or α0) such that the true r0(x) = exp(xTγ0) (or m0(x) = g(xTα0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Denote the empirical mean operator on S and T as �ES and �ET such that �ESa(X, Y ) = n−1 n � i=1 a(Xi, Yi), �ET a(X, Y ) = N−1 n+N � i=n+1 a(Xi, Yi) for any function a(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Suppose the two nuisance estimators �γ and �α are obtained respectively by solving the estimating equations: �ESX exp(X Tγ) = �ET X, �ESX{Y − g(X Tα)} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (1) The estimating equations for γ in (1) is usually referred as covariate balancing (Imai and Ratkovic, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Zhao and Percival, 2017), and those for α correspond to the ordinary least square regression when g(a) = a and the logistic regression when Y is binary and g(a) = expit(a) = ea/(1+ea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Note that one can use alternative estimation procedures to obtain γ and α, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', running a logistic regression on ∆ against X to estimate γ, and our proposed method could naturally adapt to different choices on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Based on �γ and �α, the PS and OR estimators introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 can be specified as �µPS = �ESY exp(X T�γ) and �µOR = �ET g(X T�α) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Then the standard DR estimator is constructed by augmenting one of them with another nuisance model: �µDR = �ES{Y − g(X T�α)} exp(X T�γ) + �ET g(X T�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2) When the PS model is correct and �γ converges to γ0, �ET g(X T�α) − �ESg(X T�α) exp(X T�γ) converges to zero and the remainder term �ESY exp(X T�γ) is exactly the PS estimator con- verging to µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Similarly, when OR is correct, we can show that �ES{Y − g(X T�α)} exp(X T�γ) converges to zero and �ET g(X T�α) converges to µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus �µDR is doubly robust in the sense that it is consistent when either the PS or OR model is correctly and consistently estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 Expansion of DR estimator under correct OR model To help the readers understand our method more intuitively, we now heuristically derive and analyze the asymptotic expansion of �µDR when the OR model is correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Suppose that �γ and �α converge to some ¯γ and ¯α defined as the solutions to the population-level estimating equations ESX exp(X Tγ) = ET X and ESX{Y − g(X Tα)} = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Let �r(x) = exp(X T�γ), ¯r(x) = exp(X T¯γ), and S(α) = S(Y, X, α) = X{Y − g(X Tα)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Suppose that the OR model is correct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', m0(x) = g(X Tα0) and α0 = ¯α, and n1/2(�α − 6 ¯α, �γ − ¯γ) is asymptotically normal with mean zero following the standard M-estimation theory (Van der Vaart, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Then we have �ES{Y − g(X T�α)}{�r(X) − ¯r(X)} = op(n−1/2) due to Neyman orthogonality (Neyman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 1959),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' as will be strictly proved in Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' implies that �µDR defined in (2) is asymptotically equivalent with �µDR =�ES{Y − g(X T ¯α)}¯r(X) + �ET g(X T ¯α) + � �ES{g(X T ¯α) − g(X T�α)}¯r(X) + �ET {g(X T�α) − g(X T ¯α)} � ≈�ES{Y − g(X T ¯α)}¯r(X) + �ET g(X T ¯α) + L T�ESX{Y − g(X T ¯α)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' where L = − ¯H−1 {ESX ˙g(X T ¯α)¯r(X) − ET X ˙g(X T ¯α)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ¯H = ESXX T ˙g(X T ¯α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' and ˙g(a) is the derivative of g(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To derive the above result, we use the standard asymptotic expansion of �α given by our Lemma B3 in Appendix, and the symbol “≈” indicates that the difference between the two lines is up to op(n−1/2) and, thus, asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' So when OR is correct, the asymptotic variance of n1/2(�µDR − µ0) is equal to that of n1/2(�µDR − µ0), which can be expressed as aVar{n1/2(�µDR − µ0)} = ES{¯r(X)}2v(X) + 2L TESX¯r(X)v(X) + C, (3) where v(x) = Var(Y | X) and C is some positive constant free of ¯r(·) and, thus, needs not to be considered in the following derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Note that when the PS model also is correct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', ¯r(·) = r0(·), we further have L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Empirically, term L in (3) can be estimated by �L = − �H−1 � �ESX ˙g(X T�α) exp(X T�γ) − �ET X ˙g(X T�α) � , (4) where �H = �ESXX T ˙g(X T�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Estimation of v(x) relies on our working assumption on the form of Var(Y | X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For example, one may assume Y = m0(X) + ǫ where ǫ ∼ N(0, σ2) so v(x) is invariant of x and can be simply imputed with the moment estimator of σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Also, for the common Poisson model Y ∼ Poisson{exp(X Tα0)} and logistic model Y ∼ Bernoulli{expit(X Tα0)}, one can naturally estimate v(x) by exp(xT�α) and expit(xT�α){1 − expit(xT�α)} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To preserve generality, we introduce a working model vθ(x) for v(x) with some nuisance parameter θ to be estimated as �θ that could be partially or fully determined by �α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Suppose that �θ converges to some ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' As will be shown in Section 3, violation of this conditional variance model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', v(x) ̸= v¯θ(x) does not impact the double robustness of our proposed estimator but only affects its efficiency gain when PS is wrong and OR is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3 PAD estimator Now we formally introduce the propensity score augmented doubly robust (PAD) estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our central idea is to augment the PS model ¯r(X) = exp(X T¯γ) as ¯raug(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' β) = exp(X T¯γ)+ Ψ Tβ and use ¯raug(·) to replace ¯r(·) in the DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Here Ψ is some properly constructed basis function of X and β is some loading coefficient vector to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We first describe the empirical construction procedures for PAD in Algorithm 1 and then discuss the reason and intuition of the key steps in this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Algorithm 1 Propensity score Augmented Doubly robust (PAD) estimation [Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional variance estimator as �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 2] Specify Φ = φ(X) of larger dimensionality than X using any basis function φ(·), and take �Ψ = Φ − �ET [Φv�θ(X)]/�ET v�θ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 3] Solve the restricted weighted least square (RWLS) problem: �β = argminβ �Vµ(β), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' �ESX ˙g(X T�α) �Ψ Tβ = 0, (5) where �Vµ(β) = �ES{exp(X T�γ) + �Ψ Tβ}2v�θ(X) + 2�L T�ESX{exp(X T�γ) + �Ψ Tβ}v�θ(X), (6) and �L is as defined in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 4] Obtain the PAD estimator through �µPAD = �ES{Y − g(X T�α)}{exp(X T�γ) + �Ψ T�β} + �ET g(X T�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For heuristic analysis, suppose that all estimators used in (6) converge to their limiting values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Then let Ψ = Φ −ET [Φv¯θ(X)]/ET vθ(X) be the limits of �Ψ, ¯β the limits of �β, with its specific form given by Lemma B1 in Appendix, and Vµ(β) = ES{exp(X T¯γ) + Ψ Tβ}2v¯θ(X) + 2L TESX{exp(X T¯γ) + Ψ Tβ}v¯θ(X) the limiting function of �Vµ(β) specified in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We shall consider two scenarios separately to demonstrate that our proposed PAD estimator not only maintains double robustness property but also has a lower asymptotic variance than �µDR when the OR model is correctly specified and PS is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Rigorous justification for these results will be provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 8 Correct PS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When the PS model is correct, we easily have L = 0 as stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 so Vµ(β) = ES{exp(X T¯γ) + Ψ Tβ}2v¯θ(X), and ∂Vµ(β) ∂β = ESΨ exp(X T¯γ)v¯θ(X) = ET Ψv¯θ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By definition of Ψ, we have ET Ψv¯θ(X) = 0, as ensured by the mean shift of Φ in Step 2 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, β = 0 minimizes Vµ(β) and consequently, is the solution of the population-level version of the RWLS problem (5) since the linear constraints in (5) is trivially satisfied by β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This implies that as long as the PS model is correct, �β converges to 0 so the augmented PS estimator exp(X T�γ) + �Ψ T�β converges to the correct PS model, which ensures �µPAD to converge to the true µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Meanwhile, it is clear that the augmentation of PS does not change the OR model at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Therefore, �µPAD preserves the same DR property as �µDR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', being (root-n) consistent whenever the PS or the OR model is correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Correct OR and wrong PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Note that �µPAD = �µDR + �ES �Ψ T�β{Y − g(X T�α)} and when the OR model is correct, �ES �Ψ T�β{Y − g(X T�α)} =�ES �Ψ T�β{Y − g(X Tα0)} + �ES �Ψ T�β{g(X Tα0) − g(X T�α)} ≈�ESΨ T¯β{Y − g(X Tα0)} + �ES(α0 − �α) TX ˙g(X T�α) �Ψ T�β, (7) in which we use the orthogonality between �Ψ T�β − Ψ T¯β and Y − g(X Tα0) on the first term, as well as expansion on g(X Tα0) − g(X T�α) in the second term of the first line, to derive the “≈” relation shown in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Here, “≈” in (7) again means that the difference between the first and second line is up to op(n−1/2) and, thus, becomes asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In addition, according to the moment constraint in the RWLS problem (5), �ESX ˙g(X T�α) �Ψ T�β converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' So the second term in the second line of (7) is also negligible and �µPAD ≈ �µDR + �ESΨ T¯β{Y − g(X Tα0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Combining this with equation (3) as well as the asymptotic equivalence between �µDR and �µDR discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2, we have aVar{n1/2(�µPAD − µ0)} = ES{¯r(X) + Ψ T¯β}2v(X) + 2L TESX{¯r(X) + Ψ T¯β}v(X) + C, (8) which, after dropping the invariant C, is equal to Vµ(¯β), the limiting value of the minimized objective function �Vµ(�β) in the RWLS problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Note that β = 0 is always feasible to the linear constraint in (5) and if we simply replace ¯β with 0 in the right-hand side of (8), it reduces to the asymptotic variance of n1/2(�µDR − µ0) derived in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Meanwhile, when the PS model is wrong, ∂Vµ(β)/∂β is typically not 0 at β = 0 so the population-level minimizer ¯β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, aVar{n1/2(�µPAD − µ0)} ≤ aVar{n1/2(�µDR − µ0)} when the OR model is correct and the strict “<” will hold in general when the PS model is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 9 3 Asymptotic analysis In this section, we rigorously present the asymptotic properties of the proposed PAD estima- tor and compare PAD with the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We first introduce some mild and common regularity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Without loss of generality, we assume that n/N = O(1) so the desirable parametric rate of the DR estimators will be O(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The supports of X and Φ are compact and EY 4 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The link function g(·) is differentiable with derivative ˙g(·) and there exists a constant L such that |˙g(x1) − ˙g(x2)| < L|x1 − x2| for all x1, x2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The dimension of Ψ is larger than that of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Matrices ES{ΨΨ Tv¯θ(X)}, ES{XX T exp(X T¯γ)}, ES{XX T ˙g(X T ¯α)} and ES{ΨX T ˙g(X T ¯α)} have all their eigenvalues bounded and staying away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The conditional variance function vθ(x) is differentiable on θ with a bounded partial derivative ∂θvθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The estimator �θ converges to some ¯θ in probability and satisfies that n1/2(�θ − ¯θ) is asymptotic normal with mean zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumptions 1–3 are all mild, standard, and commonly used to justify the asymptotic properties of M-estimation (Van der Vaart, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Note that in Assumption 3, we take Ψ to have larger dimension than X and make regularity conditions on ES{XX T ˙g(X T ¯α)} and ES{ΨX T ˙g(X T ¯α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' These are to ensure that �β is not zero and properly converges to ¯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Assumption 4 constrains the way of specifying vθ(x) and estimating θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Assumptions 1–3, this assumption is satisfied when either θ is fully determined by α, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', in a Pois- son or logistic model for Y against X, or when θ is estimated by additionally fitting some parametric model of Var(Y | X) against X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Now we present the main results about the robustness and efficiency of our proposed PAD estimator in Theorem 1 with its proof given in Section B of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Some important heuristics of this theorem has already been discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Assumptions 1–4, it holds that (i) Double robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When either the PS or the OR model is correctly specified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', r0(x) = exp(xTγ0) for some γ0 or m0(x) = g(xTα0) for some α0, �µPAD p−→ µ0 and n1/2(�µPAD − µ0) weakly converges to some normal distribution with mean zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (ii) Variance reduction under wrong PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When the OR model is correct while the PS model may be misspecified, the asymptotic variance of n1/2(�µPAD − µ0) is always 10 not larger than that of n1/2(�µDR − µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Further when ¯β ̸= 0 (the explicit form of ¯β is given in Lemma B1), n1/2(�µPAD − µ0) has a strictly smaller asymptotic variance than n1/2(�µDR − µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (iii) Equivalence under correct PS and OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When both the PS and OR models are correct, n1/2(�µPAD − µ0) and n1/2(�µDR − µ0) are asymptotically equivalent and have the same asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 4 Simulation study We conducted simulation studies to evaluate our proposed estimator and compare it with the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In our studies, we generate covariates X = (X1, X2, X3)T from N(0, Σ) with Σ = (σij) ∈ R3×3 and σij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3|i−j|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For generation of the population assignment ∆ and outcome Y , we consider six settings, namely: (G1) Gassuian Y , Correct PS, Correct OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X)} = expit(X1 − 2X2 + X3) and Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + X3 + ǫ where ǫ | X ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (G2) Gassuian Y , Correct PS, Wrong OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + sin(X2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X3) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (G3) Gassuian Y , Wrong PS, Correct OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5|X1| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5|X2| − |X3|) and Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + X3 + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (L1) Binary Y , Correct PS, Correct OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and Pr(Y = 1 | X) = expit(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (L2) Binary Y , Correct PS, Wrong OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X) = expit(X1 − 2X2 + X3) and Pr(Y = 1 | X)} = expit(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + sin(X2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X3)) (L3) Binary Y , Wrong PS, Correct OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Pr(∆ = 1 | X) = expit(4 + X1 + X2 + X3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5|X1| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5|X2| − |X3|) and Pr(Y = 1 | X) = expit(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='5X2 + X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In Settings (G1)–(G3), Y is a gaussian variable and we fit linear models for Y ∼ X with vθ(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' While in Settings (L1)–(L3), we fit logistic models for the binary Y against X with vθ(x) = expit(X Tα){1 − expit(X Tα)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We consider different scenarios about the correctness of the PS and OR models to examine the robustness and efficiency of PAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Bootstrap is used for estimating the asymptotic variance and constructing the confidence interval (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For effective variance reduction on PAD when PS is wrong, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' under Settings 11 (G3) and (L3), we include in the augmentation covariates Φ a decent amount of X’s basis functions including Xj, exp(Xj), |Xj|, exp(−Xj1 − Xj2), and exp(−X1 − X2 − X3) for all j and j1 ̸= j2 ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We set N = n = 500 or N = n = 1000 separately and generate 1000 realizations for each setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Table 1 reports the absolute average bias (Bias), standard error (SE), and coverage probability (CP) of the 95% CI of the DR and PAD estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When at least one nuisance models are correct, DR and PAD attain very close bias, which is much smaller compared to their SE and, thus, grants their CPs to be close to the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This indicates that PAD achieves the double robustness property just like the standard DR estimator under finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To compare PAD and DR in terms of their estimation variance and efficiency, we present in Table 2 their relative efficiency (RE) defined as Var(�µDR)/ Var(�µPAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Settings (G1), (G2), (L1), and (L2) where the PS model is correct, the two estimators show nearly identical variance, with their REs located between 1 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Settings (G3) and (L3) with misspecified PS and correct OR models, our proposed PAD estimator shows 20% to 40% smaller variance than the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' All these results demonstrate that conclusions in Theorem 1 also apply well for finite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In specific, PAD performs very closely to the standard DR when the PS model is correct and is potentially better than DR in the presence of wrong PS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 5 Real example The effects of the 401(k) program have been investigated for a long time (Abadie, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2018, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Different from other plans like Individual Retirement Ac- counts (IRAs), eligibility for 401(k) is completely decided by employers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Therefore, unob- served personal preferences for savings may make little difference in 401(k) eligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' How- ever, there may be some other confounders affecting the causal studies of 401(k), such as job choice, income, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To address this problem, (Abadie, 2003) and (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', 2018) proposed to adjust for certain covariates related to job choice so that 401(k) eligibility can be regarded exogenous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Whether 401(k) eligibility contributes to the improvement of people’s net total financial assets is an important topic studied in existing literature like Abadie (2003) and Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' However, whether 401(k) can improve the financial assets of those actually not eligible for 401(k) is still an open and interesting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To investigate this problem, we analyze the data from the Survey of Income and Program Participation of 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The data set consists of n + N = 9275 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The outcome of our interests, Y is defined as the indication of having positive net total financial assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' There are 9 adjustment covariates in X, including 12 Table 1: The absolute average bias (Bias), standard error (SE), and coverage probability (CP) of the 95% confidence intervals of the DR and PAD estimators under the settings described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' All results are produced based on 1000 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' n = N = 500 n = N = 1000 Setting Method Bias SE CP bias SE CP (G1) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 (G2) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 (G3) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 (L1) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 (L2) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 (L3) DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='92 PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='93 Table 2: Relative efficiency (RE) between DR and PAD, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', Var(�µDR)/ Var(�µPAD), under the settings de- scribed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' n, N (G1) (G2) (G3) (L1) (L2) (L3) 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='20 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='18 age, income, family size, years of education, benefit pension status, marriage, two-earner household status, individual participation in IRA plan, and home ownership status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The source (treated) samples S with ∆ = 1 are taken as those eligible for 401(k) and the target (untreated) samples T are those without 401(k) eligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We applied PAD and standard DR to estimate µ, the effect of 401(k) eligibility on improving the positive rate of net to- tal financial assets among people without 401(k) eligibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The PS model is specified as exp(X Tγ) and the OR model is expit(X Tα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In our method, the augmentation covariates vector Φ consists of X, exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='3Xj), |Xj|, and X2 j for all Xj’s that are not binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We again use bootstrap to estimate SEs and construct CIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In Table 3, we report the point estimation, their estimated standard errors (ESE), and 95% CIs for the treatment effect µ, obtained using the standard DR and our proposed PAD 13 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Outputs of both methods indicate that 401(k) eligibility has a significant effect on improving the rate of having positive net total financial assets among people who are actually not eligible for 401(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The estimated treatment effect is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='169 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='142, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='196) by the standard DR and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='150 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='126, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='175) by PAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Moreover, the ESE of our proposed PAD estimator is remarkably smaller than that of the standard DR estimator, with their estimated RE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', Var(�µDR)/ Var(�µPAD) being around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This means our proposed PAD method can characterize the treatment effect µ more precisely than DR in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Table 3: The point estimation (PE), its estimated standard error (ESE), and 95% confidence interval (CI) for µ, the effect of 401(k) eligibility on improving the positive rate of net total financial assets among people without 401(k) eligibility, derived using the standard DR and the PAD methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Method PE ESE CI DR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='0140 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='142, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='196) PAD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='0125 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='126, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='175) 6 Discussion In analogy to our PS model augmentation strategy, we also propose an OR model augmen- tation strategy (OAD) that augments the OR model with some bases of X satisfying certain moment conditions like Ψ in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Description and discussion of this method are presented in Section A of the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Similar to Theorem 1, we are able to show that this OAD estimator is doubly robust, of a smaller variance than the standard DR estimator when the PS model is correct but the OR model is wrong, and equivalent with DR when both nuisance models are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Just like PAD, this OAD method is easy to implement and only requires convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We notice that some existing methods in intrinsic efficient DR estimation like Rotnitzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2012) and Gronsbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2022) rely on non- convex training to construct the OR model when it is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This OAD strategy could mitigate this practical problem and still achieves the purpose of variance reduction in the presence of misspecified OR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' For ease of demonstration, we focus on covariate shift correction, or equivalently ATT estimation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Our proposed PAD estimation can be potentially generalized to address other causal or missing data problems like ATE estimation (Bang and Robins, 2005, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' ), casual model estimation Rotnitzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2012), transfer learning of a regression model Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Also, properly specifying the bases Φ is crucial for variance reduction in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The optimal choice of Φ for the most effective variance reduction is still 14 an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Related to this, it may be useful and interesting to extend our current framework for high-dimensional sparse or sieve construction of the augmentation term Ψ Tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Bias-reduced doubly robust estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Journal of the American Statistical Association, 110(511):1024–1036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' and Ding, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Combining multiple observational data sources to estimate causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Journal of the American Statistical Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' and Percival, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Entropy balancing is doubly robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Journal of Causal Inference, 5(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 17 A Dual construction to augment OR In analogy to our PAD estimator, to improve the efficiency our the DR estimator under the correct PS and wrong OR models, we propose the Outcome regression Augmented Doubly robust (OAD) estimator in the following algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Algorithm A1 Outcome regression Augmented Doubly robust (OAD) estimation [Step 1] Solve the estimating equations in (1) to obtain �γ and �α, and obtain the conditional variance estimator as �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 2] Let Φ = φ(X) with function φ(·), �g(X T�α) = g(X T�α) − �ET g(X T�α) and �Ψ = Φ − �ET Φ�g(X T�α) �ET �g2(X T�α) �g(X T�α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 3] Solve the restricted weighted least square (RWLS) problem: �β = argminβ �Vµ,OAD(β), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' �ESX �Ψ Tβ exp(X T�γ) = 0, (A1) where �Vµ,OAD(β) =n−1� VarS[{Y − g(X T�α) − �Ψ T�β} exp(X T�γ)] + N−1� VarT {g(X T�α) + �Ψ T�β} + 2� L∗ T[N−1 � CovT (X, �Ψ T�β) + n−1 � CovS{X exp(X T�γ), �Ψ T�β exp(X T�γ)}], (A2) and � L∗ = {�ESX exp(X T�γ)X T}−1�ES{Y − g(X T�α)} exp(X T�γ)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' [Step 4] Obtain the OAD estimator: �µOAD = �ES{Y − g(X T�α) − �Ψ T�β} exp(X T�γ) + �ET {g(X T�α) + �Ψ T�β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' To demonstrate how Algorithm A1 works, we define that �µOAD =�ES{Y − g(X T ¯α) − Ψ T¯β} exp(X T¯γ) + �ET {g(X T ¯α) + Ψ T¯β} + ES{Y − g(X T ¯α)} exp(X T¯γ)X T{ESX exp(X T¯γ)X T}−1{�ET X − �ESX exp(X T¯γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Then similar to our analysis in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2, when the PS model is correct, �µOAD is asymptot- ically equivalent to �µOAD, with its variance being: Vµ,OAD(β) =n−1 VarS[{Y − g(X T ¯α) − Ψ T¯β} exp(X T¯γ)] + N−1 VarT {g(X T ¯α) + Ψ T¯β} + 2L∗T[N−1CovT (X, Ψ T¯β) + n−1CovS{X exp(X T¯γ), Ψ T¯β exp(X T¯γ)}], 1 the limiting function of �Vµ,OAD(β) specified in Algorithm A1, where L∗ = {ESX exp(X T¯γ)X T}−1ES{Y − g(X T ¯α)} exp(X T¯γ)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' This corresponds to the objective function in equation (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Similar to the PAD construc- tion, when β = 0, Vµ,OAD(β) reduces to the asymptotic variance of the standard DR estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, �µOAD has a smaller variance than the standard DR estimator when the PS model is correct and the OR model is wrong, under which we typically have ¯β ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, when OR is correctly specified, we have ¯α = α0, L∗ = 0, and thus ∂Vµ,OAD(β) ∂β |β=0 = CovT (g(X T ¯α), Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By definition of Ψ, we have CovT (g(X T ¯α), Ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, similar to the analysis in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2, �µOAD preserved the same DR property as �µDR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=', being root-n consistent whenever the PS or the OR model is correctly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' B Asymptotic justification B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='1 Technical lemma Lemma B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Define a := ESΨ ∂g(XTα) ∂αT |¯α, b := ESΨ exp(X T¯γ)v¯θ(X) + ESΨX Tv¯θ(X)LT, and Σ := ESΨΨ Tv¯θ(X), under Condition 2-5, the solution of the RWLS problem (5) is ¯β = Σ−1a(a TΣ−1a)−1a TΣ−1b − Σ−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' First we introduce Lagrange multiplier λ and write (5) as the Lagrange form: ¯β = argminβ ES{exp(X T¯γ)+Ψ Tβ}2v¯θ(X)+2LESX{exp(X T¯γ)+Ψ Tβ}v¯θ(X)−λ TES ∂g(X Tα) ∂α |¯αΨ Tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Then we have the partial derivative of λ and β: ES ∂g(X Tα) ∂α |¯αΨ Tβ = 0, (B1) and 2ESΨ{exp(X T¯γ) + Ψ Tβ}v¯θ(X) + 2ESΨX Tv¯θ(X)L T − ESΨ∂g(X Tα) ∂αT |¯αλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (B2) From (B2) we have β = {2ESΨΨ Tv¯θ(X)}−1{ESΨ∂g(X Tα) ∂αT |¯αλ−2ESΨ exp(X T¯γ)v¯θ(X)−2ESΨX Tv¯θ(X)L T}, 2 together with (B1), we have ES ∂g(X Tα) ∂α |¯αΨ T{ESΨΨ Tv¯θ(X)}−1 ∗ {ESΨ∂g(X Tα) ∂αT |¯αλ − 2ESΨ exp(X T¯γ)v¯θ(X) − 2ESΨX Tv¯θ(X)L T} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' this function can be simplified as a TΣ−1(aλ − 2b) = 0, and we further have λ = 2(a TΣ−1a)−1a TΣ−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, we have ¯β = Σ−1a(a TΣ−1a)−1a TΣ−1b − Σ−1b, and we can estimate it by �β = �Σ −1�a(�a T �Σ −1�a)−1�a T �Σ −1�b − �Σ −1�b for �a := �ES �Ψ∂g(XTα) ∂αT |�α, �b := �ES �Ψ exp(X T�γ)v�θ(X)+�ES �ΨX Tv�θ(X)�LT, and �Σ := �ES �Ψ �Ψ Tv�θ(X) Lemma B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Condition 3, 4 and 6, we have that �Ψ − Ψ = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By definition, we would have that �Ψ − Ψ = �ET {Φv�θ(X)} �ET v�θ(X) − ET {Φv¯θ(X)} ET v¯θ(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Condition 4 and 6, we have that �ET v�θ(X) − ET v¯θ(X) = �ET v¯θ(X) + �ET ∂vθ(X) ∂θ |�θ(�θ − ¯θ) − ET v¯θ(X) = Op(n−1/2) (B3) for �θ between �θ and ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By using the same techniques, we have that �ET {Φv�θ(X)} − ET {Φv¯θ(X)} = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' And we have �Ψ − Ψ = �ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}�ET v�θ(X) �ET v�θ(X)ET v¯θ(X) = �ET {Φv�θ(X)}ET v¯θ(X) − ET {Φv¯θ(X)}ET v¯θ(X) − [ET {Φv¯θ(X)}�ET v�θ(X) − ET {Φv¯θ(X)}ET v¯θ(X)] {ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X) = Op(n−1/2)ET v¯θ(X) − ET {Φv¯θ(X)}Op(n−1/2) {ET v¯θ(X) + Op(n−1/2)}ET v¯θ(X) = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 3 Lemma B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Condition 1, 2 and 4, 5, we have that �γ − ¯γ = Op(n−1/2) and �α − ¯α = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' The estimation of γ has been given as �ESX exp(X T�γ) = �ET X, by applying Taylor series expansion, we have n−1 n � i=1 Xi exp(X T i ¯γ) + n−1 n � i=1 Xi exp(X T i �γ)X T i (�γ − ¯γ) = N−1 n+N � i=n+1 Xi, where �γ is some vector between �γ and ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' According to (Van der Vaart, 2000), we have �γ − ¯γ = op(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Let J represent matrix n−1 �n i=1 Xi exp(X T i �γ)X T i , and we have that J = n−1 n � i=1 Xi exp(X T i ¯γ)X T i +n−1 n � i=1 Xi exp(X T i γ∗)X T i Xi(�γ−¯γ) = ESX exp(X T¯γ)X T+op(1) for γ∗ between �γ and ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, by central limit theorem and Slutsky theorem, we have that, under Condition 1 and 4 �γ − ¯γ = J−1 � N−1 n+N � i=n+1 Xi − n−1 n � i=1 Xi exp(X T i ¯γ) � =J−1 � N−1 n+N � i=n+1 Xi − ET X + ESX exp(X T¯γ) − n−1 n � i=1 Xi exp(X T i ¯γ) � = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Furthermore, The estimation equation of �α is given by �ESS(�α) = �ESX{Y − g(X T�α)} = 0, by using Taylor series expansion, we have that �ESX{Y − g(X T ¯α)} + �ES ∂S(α) ∂αT ���� �α (�α − ¯α) = 0 for �α between �α and ¯α, and we have �α − ¯α = −�ES �∂S(α) ∂αT ���� �α �−1 �ESX{Y − g(X T ¯α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By using the same techniques as those for obtaining the asymptotic properties of �γ, under Condition 2, 4 and 5, we have �α − ¯α = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 4 Lemma B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Under Condition 1-6 and Lemma A1-A3, we can obtain that �β−¯β = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In addition, when the PS is correctly specified, we further have ¯β = 0 and �β = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By using the same techniques as (B3), under Condition 2-4, we first have that �a−a = �ES �Ψ∂g(X Tα) ∂αT |�α−ESΨ∂g(X Tα) ∂αT |¯α = �ESΨ∂g(X Tα) ∂αT |¯α−ESΨ∂g(X Tα) ∂αT |¯α+Op(n−1/2) = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' In addition, we can have that �b − b = Op(n−1/2) and �Σ − Σ = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Furthermore, we can easily have that �Σ −1 − Σ−1 = Σ−1Σ{Σ + Op(n−1/2)}−1 − Σ−1 = Σ−1[Σ{Σ + Op(n−1/2)}−1 − {Σ + Op(n−1/2)}{Σ + Op(n−1/2)}−1] = Op(n−1/2), based on which we can have (�aT �Σ −1�a)−1−(aTΣ−1a)−1 = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Let �Ω denote �Σ −1�a(�aT �Σ −1�a)−1�aT �Σ −1 and Ω denote Σ−1a(aTΣ−1a)−1aTΣ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' We can have that �Ω − Ω = Op(n−1/2), hence, we have that �β − ¯β = �Ω�b − Ωb = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, when the PS is correctly specified, L = 0 and ESΨ exp(X T¯γ)v¯θ(X) = ET Ψv¯θ(X) = 0, which means ¯β = Ωb = Ω{ESΨ exp(X T¯γ)v¯θ(X) + ESΨX Tv¯θ(X)L T} = Ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' And at the same time, we have �β = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='2 Proof of Theorem 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof of Theorem 1 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When the OR is correctly specified, ¯α = α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Consider �µOR where �µOR = �ES{Y − g(X T ¯α)}{exp(X T¯γ) + Ψ T¯β} + �ET g(X T ¯α) + � ES ∂g(X Tα) ∂αT ���� ¯α exp(X T¯γ) − ET ∂g(X Tα) ∂αT ���� ¯α � ES �∂S(α) ∂αT ���� ¯α �−1 �ESX{Y − g(X T ¯α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' It is obvious that E�µOR = ET g(X T ¯α) = µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, by using central limit theorem, we have that �µOR − µ0 = Op(n−1/2), n1/2(�µOR − µ0) weakly converges to gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' we have that �µP AD − �µOR = �ES{Y − g(X Tα0)}{exp(X T¯γ)X T(�γ − ¯γ) + Ψ T(�β − ¯β) + ( �Ψ − Ψ) T¯β} − � �ES ∂g(X Tα) ∂αT ���� α0 {exp(X T¯γ) + Ψ T¯β} − �ET ∂g(X Tα) ∂αT ���� α0 � (�α − ¯α) + op(n−1/2) 5 − � ES ∂g(X Tα) ∂αT ���� α0 exp(X T¯γ) − ET ∂g(X Tα) ∂αT ���� α0 � ES �∂S(α) ∂αT ���� α0 �−1 �ESX{Y − g(X Tα0)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' by using central limit theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' along with Lemma A2-A4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' we have that �ES{Y − g(X Tα0)}{exp(X T¯γ)X T(�γ − ¯γ) + Ψ T(�β − ¯β) + ( �Ψ − Ψ) T¯β} = [�ES{Y − g(X Tα0)} exp(X T¯γ)X T](�γ − ¯γ) + [�ES{Y − g(X Tα0)}Ψ T](�β − ¯β) + [�ES{Y − g(X Tα0)}¯β T]( �Ψ − Ψ) = Op(n−1/2)op(1) + Op(n−1/2)op(1) + Op(n−1/2)op(1) = op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, − � �ES ∂g(X Tα) ∂αT ���� α0 {exp(X T¯γ) + Ψ T¯β} − �ET ∂g(X Tα) ∂αT ���� α0 � (�α − ¯α) = � �ES ∂g(X Tα) ∂αT ���� α0 {exp(X T¯γ) + Ψ T¯β} − �ET ∂g(X Tα) ∂αT ���� α0 � �ES �∂S(α) ∂αT ���� ¯α �−1 �ESX{Y − g(X T ¯α)} = � ES ∂g(X Tα) ∂αT ���� α0 exp(X T¯γ) − ET ∂g(X Tα) ∂αT ���� α0 + Op(n−1/2) � ∗ � ES �∂S(α) ∂αT ���� α0 �−1 + Op(n−1/2) � �ESX{Y − g(X Tα0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' (B4) Hence, we have that − � �ES ∂g(X Tα) ∂αT ���� α0 {exp(X T¯γ) + Ψ T¯β} − �ET ∂g(X Tα) ∂αT ���� α0 � (�α − ¯α) − � ES ∂g(X Tα) ∂αT ���� α0 exp(X T¯γ) − ET ∂g(X Tα) ∂αT ���� α0 � ES �∂S(α) ∂αT ���� α0 �−1 �ESX{Y − g(X Tα0)} = �ESX{Y − g(X Tα0)}Op(n−1/2) = op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, from previous results, we have that �µP AD − �µOR = op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Together with Slutsky theorem, we futher have that �µP AD − µ0 = Op(n−1/2) and n1/2(�µP AD − µ0) weakly converges to gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When the PS is correctly specified, ¯γ = γ0, we consider �µPS where �µPS = �ES{Y − g(X T ¯α)}{exp(X Tγ0) + Ψ T¯β} + �ET g(X T ¯α) + ES{Y − g(X T ¯α)} exp(X T¯γ)X T{ESX exp(X Tγ0)X T}−1{�ET X − �ESX exp(X Tγ0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Together with the results from Lemma A4, we have that E�µPS = ESY exp(X T¯γ) = ET Y = µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By using the central limit theorem, we have that �µPS − µ0 = Op(n−1/2), n1/2(�µPS − µ0) weakly converges to gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, we have that �µP AD − �µPS = �ES{Y − g(X T ¯α)}{exp(X T¯γ)X T(�γ − ¯γ) + Ψ T(�β − ¯β)} 6 − ES{Y − g(X T ¯α)} exp(X T¯γ)X T{ESX exp(X Tγ0)X T}−1{�ET X − �ESX exp(X Tγ0)} + op(n−1/2) By using the techniques from (B4), we would have �ES{Y − g(X T ¯α)} exp(X T¯γ)X T(�γ − ¯γ) − ES{Y − g(X T ¯α)} exp(X T¯γ)X T{ESX exp(X Tγ0)X T}−1{�ET X − �ESX exp(X Tγ0)} = op(n−1/2) And from Lemma A4, we have �β = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Thus, we have �µP AD − µ0 = Op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, it is worth noticing that �β is the continuous function of �θ, �γ and �α, so under central limit theorem and Slutsky theorem, we would have the asymptotic normality of �β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, we further have that n1/2(�µPS − µ0) weakly converges to gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof of Theorem 1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' First we denote U as U = VarT (E(Y |X)) + LESXX T Var(Y |X)L T When the OR is correctly specified, the asymptotic variance of �µP AD, Var{n−1/2(�µP AD−µ0)} is ES{exp(X T¯γ) + Ψ T¯β}2v¯θ(X) + 2LESX{exp(X T¯γ) + Ψ T¯β}v¯θ(X) + U, and ¯β contributes to minimizing this variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When ¯β = 0, the function above is written as ES{exp(X T¯γ)}2v¯θ(X) + 2LESX{exp(X T¯γ)}v¯θ(X) + U, which is the same as the asymptotic variance of �µDR, Var{n−1/2(�µDR − µ0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, when ¯β ̸= 0, �µP AD has the smaller asymptotic variance than standard doubly robust estimator �µDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Proof of Theorem 1 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' When both the PS and OR is correctly specified, consider �µB, where �µB = �ES{Y − g(X Tα0)} exp(X Tγ0) + �ET g(X Tα0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' By using central limit theorem, n1/2(�µB −µ0) weakly converges to gaussian distribution with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' On the other hand, by using Taylor series expansion, we would have �µP AD − �µB = op(n−1/2) and �µDR − �µB = op(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' Hence, they have the same asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content=' 7 This figure "DMLmse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='02162v1 This figure "biasdml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} +page_content='02162v1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79A0T4oBgHgl3EQfOf-b/content/2301.02162v1.pdf'} diff --git a/79AyT4oBgHgl3EQf2_lP/content/2301.00760v1.pdf b/79AyT4oBgHgl3EQf2_lP/content/2301.00760v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e7e4aa409ae304fdf0af2e348ea23f60a7ce2c40 --- /dev/null +++ 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Alghunaim and Kun Yuan +Abstract—Gradient-tracking +(GT) +based +decentralized +methods have emerged as an effective and viable alterna- +tive method to decentralized (stochastic) gradient descent +(DSGD) when solving distributed online stochastic opti- +mization problems. Initial studies of GT methods implied +that GT methods have worse network dependent rate than +DSGD, contradicting experimental results. This dilemma +has recently been resolved, and tighter rates for GT methods +have been established, which improves upon DSGD. +In this work, we establish more enhanced rates for GT +methods under the online stochastic convex settings. We +present an alternative approach for analyzing GT methods +for convex problems and over static graphs. When compared +to previous analyses, this approach allows us to establish +enhanced network dependent rates. +Index Terms—Distributed stochastic optimization, decen- +tralized learning, gradient-tracking, adapt-then-combine. +I. Introduction +We consider the multi-agent consensus optimization prob- +lem, in which n agents work together to solve the following +stochastic optimization problem: +minimize +x∈Rd +f(x) = 1 +n +n +� +i=1 +fi(x) +fi(x) ≜ E[Fi(x; ξi)]. +(1) +Here, fi : Rd → R is the private cost function held by agent +i, which is defined as the expected value of some loss function +Fi(·, ξi) over local random variable ξi (e.g., data points). An +algorithm that solves (1) is said to be a decentralized method +if its implementation requires the agents to communicate only +with agents who are directly connected to them (i.e., neighbors) +based on the given network topology/graph. +One of the most popular decentralized methods to solve prob- +lem (1) is decentralized stochastic gradient descent (DSGD) +[1]–[3]. While DSGD is communication efficient and simple to +implement, it converges slowly when the local functions/data +are heterogeneous across nodes. Furthermore, because data +heterogeneity can be amplified by large and sparse network +topologies [4], DSGD performance is significantly degraded +with these topologies. +In this work, we analyze the performance of the gradient- +tracking method [5], [6], which is another well-known decentral- +ized method that solves problem (1). To describe the algorithm, +we let wij ≥ 0 denote the weight used by agent i to scale +information received from agent j with wij = 0 if j /∈ Ni where +Ni is the neighborhood of agent i. The adapt-then-combine +gradient-tracking (ATC-GT) method [5] is described as follows: +xk+1 +i += +� +j∈Ni +wij(xk +j − αgk +j ) +(2a) +S. A. Alghunaim (sulaiman.alghunaim@ku.edu.kw) is with the +Department of Electrical Engineering, Kuwait University, Kuwait. K. +Yuan (kunyuan@pku.edu.cn) is with the Center for Machine Learning +Research, Peking University, China. +gk+1 +i += +� +j∈Ni +wij +� +gk +j + ∇Fj(xk+1 +j +; ξk+1 +j +) − ∇Fj(xk +j ; ξk +j )� +(2b) +with initialization g0 +i = ∇Fi(x0 +i ; ξ0 +i ) and arbitrary x0 +i ∈ Rd. +Here, ∇Fi(xk +i ; ξk +i ) is the stochastic gradient and ξk +i is the data +sampled by agent i at iteration k. +Gradient-tracking can eliminate the impact of heterogeneity +between local functions [5]–[8]. In massive numerical experi- +ments reported in [9]–[12], GT can significantly outperform +DSGD in the online stochastic setting. Initial studies on the +convergence rate of GT methods are inadequate; they provide +loose convergence rates that are more sensitive to network +topology than vanilla DSGD. According to these findings, GT +will converge slower than DSGD on large and sparse networks, +which is counter-intuitive and contradicts numerical results +published in the literature. Recent works [13], [14] establish +the first convergence rates for GT that are faster than DSGD +and more robust to sparse topologies under stochastic and non- +convex settings. In this paper, we will provide additional en- +hancements for GT under convex and strongly convex settings. +A. Related works +Gradient-tracking (GT) methods, which utilize dynamic +tracking mechanisms [15] to approximate the globally averaged +gradient, have emerged as an alternative to decentralized gradi- +ent descent (DGD) [1]–[3], [16], [17] with exact convergence for +deterministic problems [5]–[8]. Since their inception, numerous +works have investigated GT methods in a variety of contexts [9], +[10], [18]–[28]. However, all of these works provide convergence +rates that can be worse than vanilla DSGD. In particular, these +results indicate that GT is less robust to sparse topologies even +if it can remove the influence of data heterogeneity. The work +[14] established refined bounds for various methods including +GT methods that improve upon DSGD under nonconvex set- +tings. Improved network dependent bounds for GT methods in +both convex and non-convex settings are also provided in [13]. +In this work, we provide additional improvements over previous +works in convex and strongly convex settings – see Table I. +It should be noted that there are other methods that are +different from GT methods but have been shown to have com- +parable or superior performance – see [14], [29] and references +therein. In contrast to these other methods, GT methods have +been shown to converge in a variety of scenarios, such as +directed graphs and time-varying graphs [18], [19], [22]. We +should also mention that there are modifications to GT ap- +proaches that can improve the rate at the price of knowing addi- +tional network information and/or more computation/memory +[21]. However, the focus of this study is on basic vanilla GT +methods. +B. Contributions +• We present an alternative approach for analyzing GT +methods in convex and static graph settings, which may +arXiv:2301.02855v1 [math.OC] 7 Jan 2023 + +2 +TABLE I: Convergence rate to reach ϵ accuracy. The strongly convex (SC) and PL condition rates ignores iteration logarithmic factors. +The quantity λ = ρ(W − 1 +n11T) ∈ (0, 1) is the mixing rate of the network where W is the network combination matrix. a0 = ∥¯x0 − x⋆∥2, +ς2 +⋆ = 1 +n +�n +i=1 ∥∇fi(x⋆)∥2, ς2 +0 = 1 +n +�n +i=1 ∥∇fi(x0) − ∇f(x0)∥2, x0 is the initialization for all nodes, and x⋆ is an optimal solution of (1). +Reference +Iterations to ϵ accuracy +Remark +Convex +[13] +1 +nϵ2 + +log( +1 +1−λ )1/2 +(1−λ)1/2 +1 +ϵ3/2 + +log( +1 +1−λ )(a0+ς2 +0 ) +1−λ +1 +ϵ +Rate holds only when iteration number K > +log( +1 +1−λ ) +1−λ +Convex +Our work +1 +nϵ2 + +1 +(1−λ)1/2 +1 +ϵ3/2 + (a0+ς2 +⋆) +(1−λ) +1 +ϵ +– +SC +[9] +1 +nϵ + +1 +(1−λ)3/2 +1 +√ϵ + C +√ϵ +C depends on 1/(1 − λ) +PL∗ +[10] +1 +nϵ + +1 +(1−λ)3/2 +1 +√ϵ + ˜C log 1 +ϵ +˜C depends on 1/(1 − λ) +SC +[13] +1 +nϵ + +log( +1 +1−λ )1/2 +(1−λ)1/2 +1 +√ϵ + +log( +1 +1−λ ) +(1−λ) +log +� +(a0+ς2 +0 ) +(1−λ)ϵ +� +Rate holds only when iteration number K > +log( +1 +1−λ ) +1−λ +PL∗ +[14] +1 +nϵ + +� +1 +(1−λ)1/2 + +1 +(1−λ)√n +� 1 +√ϵ + +1 +1−λ log +� +(a0+ς2 +⋆) +ϵ +� +Rate holds by tuning stepsize from [14, Theorem 2] +SC +Our work +1 +nϵ + +1 +(1−λ)1/2 +1 +√ϵ + +1 +1 − λ log +� +(a0+ς2 +⋆) +ϵ +� +– +∗ The PL condition is weaker than SC and can hold for nonconvex functions; any SC function satisfies the PL condition. +be useful for analyzing GT methods in other settings such +as variance-reduced gradients. +• In stochastic and convex environments, our convergence +rate improve and tighten existing GT bounds. We show, +in particular, that under convex settings, GT methods +have better dependence on network topologies than in +nonconvex settings [14]. Also, our bounds removes the +network dependent log factors in [13] – See Table I. +II. ATC-GT and Main Assumption +In this section, we describe the GT algorithm (2) in network +notation and list all necessary assumptions. We begin by defin- +ing some network quantities. +A. GT in network notation +We define xk +i ∈ Rd as the estimated value of x ∈ Rd at +agent i and iteration (time) k, and we introduce the augmented +network quantities: +xk ≜ col{xk +1, . . . , xk +n} ∈ Rdn +f(xk) ≜ +n +� +i=1 +fi(xk +i ) +∇f(xk) ≜ col{∇f1(xk +1), . . . , ∇fn(xk +n)} +∇F(xk) ≜ col{∇F1(xk +1; ξk +1), . . . , ∇Fn(xk +n; ξk +n)} +gk ≜ col{gk +1, . . . , gk +n} ∈ Rdn. +Here, col{·} is an operation to stack all vectors on top of each +other. In addition, we define +W ≜ [wij] ∈ Rn×n, +W ≜ W ⊗ Id, +(3) +where W is the network weight (or combination, mixing, gossip) +matrix with elements wij, and symbol ⊗ denotes the Kronecker +product operation. Using the above quantities, the ATC-GT +method (2) can be described as follows: +xk+1 = W[xk − αgk] +(4a) +gk+1 = W[gk + ∇F(xk+1) − ∇F(xk)], +(4b) +with initialization g0 = ∇F(x0) and arbitrary x0. +B. Assumptions +Here, we list the assumptions used in our analyses. Our first +assumption is on the network graph stated below. +Assumption 1 (Weight matrix). The network graph is as- +sumed to be static and, the weight matrix W to be doubly +stochastic and primitive. We further assume W to be symmetric +and positive semidefinite. +■ +It is important to note that assuming W +to be positive +semidefinite is not restrictive; given any doubly stochastic and +symmetric ˜ +W, we can easily construct a positive semidefinite +weight matrix by W = (I + ˜ +W)/2. We also remark that, under +Assumption 1, the mixing rate of the network is: +λ ≜ +��W − 1 +n11T�� = +max +i∈{2,...,n} |λi| < 1. +(5) +The next assumption is on the objective function. +Assumption 2 (Objective function). Each function fi : +Rd → R is L-smooth +∥∇fi(y) − ∇fi(z)∥ ≤ L∥y − z∥, +∀ y, z ∈ Rd +(6) +and (µ-strongly) convex for some L ≥ µ ≥ 0. As a result, the +aggregate function f(x) = +1 +n +�n +i=1 fi(x) is also L-smooth and +(µ-strongly) convex. (When µ = 0, then the objective functions +are simply convex.) +■ +We now state our final assumption related to the gradient +noise. +Assumption 3 (Gradient noise). For all {i}n +i=1 and k = +0, 1, . . ., we assume the following inequalities hold +E � +∇Fi(xk +i ; ξk +i ) − ∇fi(xk +i ) | F k� += 0, +(7a) +E � +∥∇Fi(xk +i ; ξk +i ) − ∇fi(xk +i )∥2 | F k� +≤ σ2, +(7b) +for some σ2 ≥ 0, where F k ≜ {x0, x2, . . . , xk} is the algorithm- +generated filtration. We further assume that conditioned on F k, +the random data {ξt +i} are independent of one another for any +{i}n +i=1 and {t}t≤k. +■ +III. Error Recursion +To establish the convergence of (4), we will first derive +an error recursion that will be key to our enhanced bounds. + +3 +Motivated by [14], the following result rewrites algorithm (4) +in an equivalent manner. +Lemma 1 (Equivalent GT form). Let x0 take any arbitrary +value and z0 = 0. Then for static graphs, the update for xk in +algorithm (4) is equivalent to following updates for k = 1, 2, . . . +xk+1 = (2W − I)xk − αW2∇F(xk) − Bzk +(8a) +zk+1 = zk + Bxk +(8b) +with initialization x1 = W(x0 − α∇F(x0)) and z1 = Bx0, and +B = I − W. +Proof. Clearly with the above initialization, both x1 are iden- +tical for the updates (4) and (8). Now, for k ≥ 1, it holds from +(8a) that +xk+1 − xk = (2W − I)(xk − xk−1) − B(zk − zk−1) +− αW2(∇F(xk) − ∇F(xk−1)). +Substituting zk − zk−1 = Bxk−1 ((8b)) and B = I − W into +the above equation and rearranging the recursion gives +xk+1 = 2Wxk − W2xk−1 − αW2(∇F(xk) − ∇F(xk−1)). +Following the same approach, we can also describe the xk +update for the GT algorithm (4) as above – see [14], [29]. Hence, +both methods are equivalent for static graph W. +Under Assumption 1, the fixed point of recursion (8), denoted +by (x⋆, z⋆), satisfies: +0 = αW2∇f(x⋆) + Bz⋆ +0 = Bx⋆. +(9) +where x⋆ = 1 ⊗ x⋆ and x⋆ is the optimal solution of (1). The +existence of z⋆ can be shown by using similar arguments as in +[30, Lemma 3.1] or [29, Lemma 1]. By introducing the notation +˜xk ≜ xk − x⋆, +˜z ≜ zk − z⋆, +(10) +using (8) and the fact (2W − I)x⋆ = x⋆, we can get the error +recursion: +� +˜xk+1 +˜zk+1 +� += +� +2W − I +−B +B +I +� � +˜xk +˜zk +� +− α +� +W2� +∇f(xk) − ∇f(x⋆) + vk� +0 +� +, +(11) +where vk ≜ ∇F(xk) − ∇f(xk). +Remark 1 (Alternative analysis approach). By describing +GT (4) in the alternative form (8), we are able to derive the +error recursion from the fixed point (11). This is similar to the +way Exact-diffusion/D2 is analyzed in [4], [12]. This alternative +approach allows us to derive tighter bounds compared with +existing GT works [9], [10], [13], [14]. +■ +Convergence analysis of (11) still remains difficult. We will +exploit the properties of the matrix W to transform recursion +(11) into a more suitable form for our analysis. To that end, +the following quantities are introduced: +¯xk ≜ 1 +n(1T +n ⊗ Id)xk = 1 +n +n +� +i=1 +xk +i , +(12a) +¯ek +x ≜ 1 +n(1T +n ⊗ Id)˜xk = ¯xk − x⋆, +(12b) +∇f(xk) ≜ 1 +n(1T +n ⊗ Id)∇f(xk) = 1 +n +n +� +i=1 +∇fi(xk +i ), +(12c) +¯vk ≜ 1 +n(1T +n ⊗ Id)vk. +(12d) +Under Assumption 1, the matrix W admits the following eigen- +decomposition: +W = UΣU−1 = � +1 ⊗ Id +ˆU +� +� +�� +� +U +� +Id +0 +0 +Λ +� +� +�� +� +Σ +� 1 +n1T ⊗ Id +ˆUT +� +� +�� +� +U−1 +(13) +where Λ is a diagonal matrix with eigenvalues strictly less than +one and ˆU is an dn × d(n − 1) matrix that satisfies +ˆUT ˆU = I, +(1T ⊗ Id) ˆU = 0 +(14a) +ˆU ˆUT = I − 1 +n11T ⊗ Id. +(14b) +Lemma 2 (Decomposed error recursion). Under Assump- +tion 1, there exists matrices ˆV and Γ to transform the error +recursion (11) into the following form: +¯ek+1 +x += ¯ek +x − α∇f(xk) + α¯vk, +(15a) +ˆxk+1 = Γˆxk − α ˆV−1 +l +Λ2 ˆUT� +∇f(xk) − ∇f(x⋆) + vk� +, +(15b) +where +ˆxk ≜ ˆV−1 +� ˆUT˜xk +ˆUT˜zk +� +, +(16) +and ˆV−1 +l +denotes the left block of ˆV−1 = [ ˆV−1 +l +ˆV−1 +r ]. Moreover, +the following bounds hold: +∥ ˆV∥2 ≤ 3, +∥ ˆV−1∥2 ≤ 9, +∥Γ∥ ≤ 1+λ +2 , +(17) +where λ = maxi∈{2,...,n} λi. +Proof. See Appendix A +The preceding result will serve as the starting point for deriv- +ing the bounds that will lead us to our conclusions. Specifically, +we can derive the following bounds from the above result. +Lemma 3 (Coupled error inequality). Suppose Assump- +tions 1–2 hold. Then, if α < +1 +4L, we have +E ∥¯ek+1 +x +∥2 ≤ (1 − µα) E ∥¯ek +x∥2 − α� +E f(¯xk) − f(x⋆)� ++ 3αc2 +1L +2n +E ∥ˆxk∥2 + α2σ2 +n +, +(18) +and +E ∥ˆxk+1∥2 ≤ γ E ∥ˆxk∥2 + α2c2 +2λ4 +(1 − γ) E ∥∇f(xk) − ∇f(x⋆)∥2 ++ α2c2 +2λ4nσ2, +(19) +where γ ≜ ∥Γ∥, c1 ≜ ∥ ˆV∥, and c2 = ∥ ˆV−1∥. +Proof. See Appendix B. +IV. Convergence Results +In this section, we present our main convergence results in +Theorems 1 and 2. We then discuss our results and highlight +the differences with existing bounds. +Theorem 1 (Convex case). Suppose that Assumptions 1-2 +are satisfied. Then, there exists a constant stepsize α such that +1 +K +K−1 +� +k=0 +� +E[f(¯xk) − f ⋆] + L +n E ∥xk − 1 ⊗ ¯xk∥2� +≤ σ∥¯e0 +x∥ +√ +nK ++ +� +Lλ4σ2 +1 − λ +�1/3 � +∥¯e0 +x∥2 +K +� 2 +3 + +4 ++ +� +Lλ2 +1 − λ∥¯e0 +x∥2 + +ς2 +⋆ +L(1 − λ) +� +C +K , +(20) +where ¯e0 +x ≜ ¯x0 − x⋆, ς2 +⋆ ≜ +1 +n +�n +i=1 ∥∇fi(x⋆)∥2, and C is an +absolute constant. +Proof. See Appendix C. +Theorem 2 (Strongly-convex case). Suppose that Assump- +tions 1-2 are satisfied. Then, there exists a constant stepsize α +such that +E ∥¯eK +x ∥2 + 1 +n∥xK − 1 ⊗ ¯xK∥2 ≤ ˜O +� +σ2 +nK + +σ2 +(1 − λ)K2 +� ++ ˜O +� +σ2 +(1 − λ)2nK3 + (a0 + ς2 +⋆) exp [−(1 − λ)K] +� +, +(21) +where a0 ≜ ∥¯x0 − x⋆∥2, ς2 +⋆ ≜ +1 +n +�n +i=1 ∥∇fi(x⋆)∥2, and the +notation ˜O(·) ignores logarithmic factors. +Proof. See Appendix D. +In comparison to [13], our results removes the log factor +O(log( +1 +1−λ)) and holds for any number of iteration K – see +Table I. Moreover, observe that for the strongly-convex case, +unlike [13], we do not have a network term 1/(1−λ) multiplying +the highest order exponential term exp(·). +Remark 2 (Improvement upon nonconvex GT rates). The +GT rates for convex and strongly-convex settings provided in +Theorems 1 and 2 improve upon the GT rates for non-convex +[13], [14] and PL condition [14] settings. For example, observe +from Table I that the GT rate under the PL condition [14] is +1 +nϵ + +� +1 +(1−λ)1/2 + +1 +(1−λ)√n +� +1 +√ϵ + +1 +1−λ log +� +(a0+ς2 +⋆) +ϵ +� +, which has +an additional term +1 +(1−λ)√n +1 +√ϵ compared to our strongly-convex +rate. +■ +Remark 3 (Comparison with Exact-diffusion/D2 [12]). For +the convex case, the difference with Exact-diffusion/D2 [12] is in +the highest order term. Exact-diffusion/D2 is +� +a0 +(1−λ) + ς2 +⋆ +� +1 +K +while GT is +� +a0 +(1−λ) + +ς2 +⋆ +(1−λ) +� +1 +K where GT has 1/(1 − λ) mul- +tiplied by ς2 +⋆, which is slightly worse than Exact-diffusion/D2. +A similar conclusion can be reached for the strongly-convex +scenario. +■ +V. Simulation results +This section will present several numerical simulations that +compare Gradient-tracking with centralized SGD (CSGD) and +decentralized SGD (DSGD). +Linear regression. We consider solving a strongly-convex +problem (1) with fi(x) = +1 +2E(aT +i x − bi)2 in which random +variable ai ∼ N(0, Id), bi = aT +i x⋆ +i + ni for some local so- +lution x⋆ +i ∈ Rd and ni ∼ N(0, σ2 +n). The stochastic gradient +is calculated as ∇Fi(x) = ai(aT +i x − bi). Each local solution +x⋆ +i = x⋆ +vi is generated using the formula x⋆ +i = x⋆ +vi, where +x⋆ ∼ N(0, Id) is a randomly generated global solution while +vi ∼ N(0, σ2 +vId) controls similarities between local solutions. +Generally speaking, a large σ2 +v will result in local solutions +{x⋆ +i }n +i=1 that are vastly different from one another. We used +d = 5, σ2 +n = 0.01, and σ2 +v = 1 in simulations. Experiments +are carried out on ring and exponential graphs of size n = 30, +respectively. Each algorithm’s stepsize (learning rate) is care- +fully tuned so that they all converge to the same relative mean- +square-error. Each simulation is run 30 times, with the solid line +representing average performance and the shadow representing +0 +100 +200 +300 +400 +iteration +10 +4 +10 +3 +10 +2 +10 +1 +100 +relative error +Exponential graph with 30 nodes +CSGD +DSGD +Gradient-Tracking +0 +250 +500 +750 +1000 1250 1500 1750 2000 +iteration +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +relative error +Ring graph with 30 nodes +CSGD +DSGD +Gradient-Tracking +Fig. 1: Comparison between different algorithms over exponential +and ring graphs when solving distributed linear regression with +heterogeneous data distributions. The spectral gap 1 − λ is 0.33 +and 0.0146 for exponential and ring graphs, respectively. +0 +200 +400 +600 +800 +iteration +10 +4 +10 +3 +10 +2 +10 +1 +100 +relative error +Exponential graph with 30 nodes +CSGD +DSGD +Gradient-Tracking +0 +200 +400 +600 +800 +1000 1200 1400 1600 +iteration +10 +4 +10 +3 +10 +2 +10 +1 +100 +relative error +Ring graph with 30 nodes +CSGD +DSGD +Gradient-Tracking +Fig. 2: Comparison between different algorithms over exponential +and ring graphs when solving distributed logistic regression. +standard deviation. The results are depicted in Fig. 1. The rela- +tive error is shown on the y-axis as 1 +n +�n +i=1 E∥xk +i −x⋆∥2/∥x⋆∥2. +When running over the exponential graph which has a well- +connected topology with 1 − λ = 0.33, it is observed that +both DSGD and Gradient-tracking perform similarly to CSGD. +However, when running over the ring graph which has a badly- +connected topology with 1 − λ = 0.0146, DSGD gets far +slower than CSGD due to its sensitivity to network topology. +In contrast, Gradient-tracking just gets a little bit slower than +CSGD and performs far better than DSGD. This phenomenon +coincides with our established complexity bound in Table I +showing that GT has a much weaker dependence on network +topology (i.e., 1 − λ). +Logistic regression. We next consider the logistic regres- +sion problem, which has fi(x) = E ln(1 + exp(−yihT +i x)) where +(hi, yi) represents the training dataset stored in node i with +hi ∈ Rd as the feature vector and yi ∈ −{1, +1} as the label. +This is a convex but not strongly-convex problem. Similar to +the linear regression experiments, we will first generate a local +solution x⋆ +i based on x⋆ +i = x⋆ + vi using vi ∼ N(0, σ2 +vId). We +can generate local data that follows distinct distributions using +x⋆ +i . To this end, we generate each feature vector hi ∼ N(0, Id) +at node i. To produce the corresponding label yi, we create a +random variable zi ∼ U(0, 1). If zi ≤ 1 + exp(−yihT +i x⋆ +i ), we +set yi = 1; otherwise yi = −1. Clearly, solution x⋆ +i controls +the distribution of the labels. By adjusting σ2 +v, we can easily +control data heterogeneity. The remaining parameters are the +same as in linear regression experiments. The performances +of each algorithm in logistic regression depicted in Fig. 2 are +consistent with that in linear regression, i.e., Gradient-tracking +performs well for both graphs while DSGD has a significantly +deteriorated performance over the ring graph due to its less +robustness to network topology. +Appendix A +Decomposed Error Recursion + +5 +Prof of Lemma 2 +Using the decomposition (13) and B = I − W: +W2 = UΣ2U−1 = � +1 ⊗ Id +ˆU +� � +Id +0 +0 +Λ2 +� � 1 +n1T ⊗ Id +ˆUT +� +(22a) +B = U(I − Σ)U−1 = � +1 ⊗ Id +ˆU +� � +0 +0 +0 +I − Λ +� � 1 +n1T ⊗ Id +ˆUT +� +, +(22b) +with I − Λ > 0. Substituting (22) into (11) and multiplying +both sides by blkdiag{U−1, U−1} on the left, we obtain +� +U−1˜xk+1 +U−1˜zk+1 +� += +� +2Σ2 − I +−(I − Σ) +I − Σ +I +� � +U−1˜xk +U−1˜zk +� +− α +� +Σ2U−1� +∇f(xk) − ∇f(x⋆) + vk� +0 +� +. +(23) +Since ˜zk always lies in the range space of B, we have (1T +n ⊗ +Id)˜zk = 0 for all k. Using, the structure of U from (13) and the +definitions (12), we have +U−1˜xk = +� +¯ek +x +ˆUT˜xk +� +, +U−1˜zk = +� +0 +ˆUT˜zk +� +U−1∇f(x) = +� +∇f(xk) +ˆUT∇f(x) +� +. +Thus, by using the structure of Σ2 and Σ2 +b given in (22), we +can rewrite (23) as +¯ek+1 +x += ¯ek +x − α� +∇f(xk) − ∇f(x⋆)� +(24a) +� ˆUT˜xk+1 +ˆUT˜zk+1 +� += +� +2Λ − I +−(I − Λ) +I − Λ +I +� � ˆUT˜xk +ˆUT˜zk +� +− α +� +Λ2 ˆUT� +∇f(xk) − ∇f(x⋆)vk� +0 +� +. +(24b) +Let +G ≜ +� +2Λ − I +−(I − Λ) +I − Λ +I +� +. +(25) +It is important to note that the matrix G is identical to the one +studied in [14] (for nonconvex case). Therefore, following the +same arguments used in [14, Appendix B], we can decompose it +as G = ˆVΓ ˆV−1 for matrices ˆV and Γ satisfying the conditions +in the lemma. Multiplying the second equation in (24) by ˆV−1, +we arrive at (15). +Appendix B +Coupled Error Inequalities +Proof of Lemma 3 +Proof of inequality (18) +The proof adjusts the argument from [31, Lemma 8]. Using +(15a) and Assumption 3, we have +E[∥¯ek+1 +x +∥2|F k] += ∥¯ek +x − α +n +�n +i=1(∇fi(xk +i ) − ∇fi(x⋆))∥2 + α2 E[∥¯vk∥2|F k] +≤ ∥¯ek +x − α +n +�n +i=1(∇fi(xk +i ) − ∇fi(x⋆))∥2 + α2σ2 +n += ∥¯ek +x∥2 + α2∥ 1 +n +n� +i=1 +(∇fi(xk +i ) − ∇fi(x⋆))∥2 +− 2α +n +n� +i=1 +� +∇fi(xk +i ), ¯ek +x +� ++ α2σ2 +n +, +(26) +where we used �n +i=1 ∇fi(x⋆) = 0. The second term on the right +can be bounded as follows: +α2∥ 1 +n +n� +i=1 +� +∇fi(xk +i ) − ∇fi(¯xk) + ∇fi(¯xk) − ∇fi(x⋆)� +∥2 +≤ 2α2∥ 1 +n +n� +i=1 +(∇fi(xk +i ) − ∇fi(¯xk))∥2 ++ 2α2∥ 1 +n +n� +i=1 +(∇fi(¯xk) − ∇fi(x⋆))∥2 +≤ 2α2 +n +n� +i=1 +∥∇fi(xk +i ) − ∇fi(¯xk)∥2 +(27) ++ 2α2∥∇f(¯xk) − ∇f(x⋆)∥2 +≤ 2α2L2 +n +∥xk − 1 ⊗ ¯xk∥2 + 2α2∥∇f(¯xk) − ∇f(x⋆)∥2 +≤ 2α2L2 +n +∥xk − 1 ⊗ ¯xk∥2 + 4Lα2(f(¯xk) − f(x⋆)), +(28) +where the first two inequalities follows from Jensen’s inequal- +ity. The third inequality follows from the Lipschitz gradient +assumption. In the last inequality, we used the L-smoothness +property of the aggregate function [32]: +∥∇f(¯xk) − ∇f(x⋆)∥2 ≤ 2L� +f(¯xk) − f(x⋆)� +. +Note that for L-smooth and µ-strongly-convex function f, it +holds that [32]: +f(x) − f(y) − L +2 ∥x − y∥2 ≤ ⟨∇f(y), (x − y)⟩ +(29a) +f(x) − f(y) + µ +2 ∥x − y∥2 ≤ ⟨∇f(x), (x − y)⟩. +(29b) +Using these inequalities, the cross term in (28) can be bounded +by +− 2α +n +n� +i=1 +⟨∇fi(xk +i ), ¯ek +x⟩ += 2α +n +n� +i=1 +� +− ⟨∇fi(xk +i ), ¯xk − xk +i ⟩ − ⟨∇fi(xk +i ), xk +i − x⋆⟩� +≤ 2α +n +n� +i=1 +� +− fi(¯xk) + fi(xk +i ) + L +2 ∥¯xk − xk +i ∥2 +− µ +2 ∥xk +i − x⋆∥2 − fi(xk +i ) + fi(x⋆) +� +≤ −2α� +f(¯xk) − f(x⋆)� ++ Lα +n +n� +i=1 +∥¯xk − xk +i ∥2 − µα∥¯xk − x⋆∥2 += −2α� +f(¯xk) − f(x⋆)� ++ Lα +n ∥xk − 1 ⊗ ¯xk∥2 − µα∥¯ek +x∥2, +(30) +where the last inequality holds due to − 1 +n +�n +i=1 ∥xk +i − x⋆∥2 ≤ +−∥ 1 +n +�n +i=1(xk +i −x⋆)∥2. Substituting (28) and (30) into (26) and +taking expectation, we obtain: +E ∥¯ek+1 +x +∥2 ≤ (1 − µα) E ∥¯ek +x∥2 − 2α(1 − 2Lα) E � +f(¯xk) − f(x⋆)� ++ αL +n (1 + 2αL) E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2 +n +≤ (1 − µα) E ∥¯ek +x∥2 − α� +E f(¯xk) − f(x⋆)� ++ 3Lα +2n E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2 +n +, +(31) +where the last step uses α +≤ +1 +4L. Using (14), we have +∥ ˆUT˜xk∥2 = ∥ ˆUT ˆU˜xk∥2 = ∥xk − 1 ⊗ ¯xk∥2. Hence, +∥xk − 1 ⊗ ¯xk∥2 (16) += ∥ ˆVˆxk∥2 − ∥ ˆUT˜zk∥2 ≤ ∥ ˆV∥2∥ˆxk∥2. +(32) +Substituting the above into (31) yields (18). + +6 +Proof of inequality (19) +From (15b), we have +E[∥ˆxk+1∥2|F k] += E +���Γˆxk − α ˆV−1 +l +Λ2 ˆUT� +∇f(xk) − ∇f(x⋆) + vk� +|F k��� +2 +(7a) += +���Γˆxk − α ˆV−1 +l +Λ2 ˆUT� +∇f(xk) − ∇f(x⋆)���� +2 ++ α2 E +��� ˆV−1 +l +Λ2 ˆUTvk��F k��� +2 +(7b) +≤ +���Γˆxk − α ˆV−1 +l +Λ2 ˆUT� +∇f(xk) − ∇f(x⋆)���� +2 ++ α2∥ ˆV−1 +l +∥2∥Λ2∥2∥ ˆUT∥2nσ2. +Now, for any vectors a and b, it holds from Jensen’s inequality +that ∥a + b∥2 ≤ 1 +θ ∥a∥2 + +1 +1−θ ∥b∥ for any θ ∈ (0, 1). Utilizing +this bound with θ = γ ≜ ∥Γ∥ on the first term of the previous +inequality, we get +E[∥ˆxk+1∥2|F k] +≤ γ∥ˆxk∥2 + +α2∥ ˆ +V−1 +l +∥2∥Λ2∥2∥ ˆ +UT∥2 +(1−γ) +∥∇f(xk) − ∇f(x⋆)∥2 ++ α2∥ ˆV−1 +l +∥2∥Λ2∥2∥ ˆUT∥2nσ2. +Taking expectation and using ∥ ˆUT∥ ≤ 1, ∥ ˆV−1 +l +∥2 ≤ ∥ ˆV−1∥2, +and ∥Λ2∥2 ≤ λ4 yield our result (19). +Appendix C +Proof of Theorem 1 +Using similar argument to (28) and (32), it holds that +∥∇f(xk) − ∇f(x⋆)∥2 +≤ 2∥∇f(1 ⊗ ¯xk) − ∇f(x⋆)∥2 + 2∥∇f(xk) − ∇f(1 ⊗ ¯xk)∥2 +≤ 4nL[f(¯xk) − f(x⋆)] + 2c2 +1L2∥ˆxk∥2. +Plugging the above bound into (19) gives +E ∥ˆxk+1∥2 ≤ +� +γ + +2α2c2 +1c2 +2L2λ4 +(1−γ) +� +E ∥ˆxk∥2 ++ +4α2c2 +2Lλ4n +(1−γ) +E ˜f(¯xk) + α2c2 +2λ4nσ2 +≤ ¯γ E ∥ˆxk∥2 + +4α2c2 +2Lλ4n +(1−γ) +E ˜f(¯xk) + α2c2 +2λ4nσ2, +where ˜f(¯xk) ≜ f(¯xk)−f(x⋆), ¯γ ≜ 1+γ +2 , and the last inequiality +holds when γ + +2α2c2 +1c2 +2L2λ4 +(1−γ) +≤ 1+γ +2 , which is satisfied for +α ≤ +1 − λ +4c1c2Lλ2 . +(33) +Iterating the last recursion (for any k = 1, 2, . . . ) gives +E ∥ˆxk∥2 ≤ ¯γk∥ˆx0∥2 + +4α2c2 +2Lλ4n +(1−γ) +k−1 +� +ℓ=0 +¯γk−1−ℓ E ˜f(¯xℓ) ++ +k−1 +� +ℓ=0 +¯γk−1−ℓ � +α2c2 +2λ4nσ2� +≤ ¯γk∥ˆx0∥2 + +4α2c2 +2Lλ4n +(1−γ) +k−1 +� +ℓ=0 +¯γk−1−ℓ E ˜f(¯xℓ) ++ +α2c2 +2λ4nσ2 +1−¯γ +. +(34) +In the last inequality we used �k−1 +ℓ=0 ¯γk−1−ℓ ≤ +1 +1−¯γ . Averaging +over k = 1, 2 . . . , K and using ¯γ = 1+γ +2 , it holds that +1 +K +K +� +k=1 +E ∥ˆxk∥2 +≤ +2∥ˆx0∥2 +(1−γ)K + +4α2c2 +2Lλ4n +(1−γ)K +K +� +k=1 +k−1 +� +ℓ=0 +� 1+γ +2 +�k−1−ℓ E ˜f(¯xℓ) + +2α2c2 +2λ4nσ2 +1−γ +≤ +2∥ˆx0∥2 +(1−γ)K + +8α2c2 +2Lλ4n +(1−γ)2K +K−1 +� +k=0 +E ˜f(¯xk) + +2α2c2 +2λ4nσ2 +1−γ +. +(35) +It follows that +1 +K +K−1 +� +k=0 +E ∥ˆxk∥2 ≤ +3∥ˆx0∥2 +(1 − γ)K + +8α2c2 +2Lλ4n +(1−γ)2K +K−1 +� +k=0 +E ˜f(¯xk) ++ 2α2c2 +2λ4nσ2 +1 − γ +. +(36) +where we added +∥ˆx0∥2 +(1−γ)K and used ∥ˆx0∥2 +K +≤ +∥ˆx0∥2 +(1−γ)K . Now when +µ = 0, we can rearrange (18) to get +E(f(¯xk) − f(x⋆)) ≤ 1 +α +� +E ∥¯ek +x∥2 − E ∥¯ek+1 +x +∥2� ++ 3c2 +1L +2n E ∥ˆxk∥2 + ασ2 +n . +(37) +Averaging over k = 0, . . . , K − 1 (K ≥ 1), it holds that +1 +K +K−1 +� +k=0 +E ˜f(¯xk) ≤ +∥¯e0 +x∥2 +αK ++ +3c2 +1L +2nK +K−1 +� +k=0 +E ∥ˆxk∥2 + ασ2 +n . +(38) +Multiplying inequality (36) by 2 × +3c2 +1L +2n , adding to (38), and +rearranging we obtain +� +1 − +24α2c2 +1c2 +2L2λ4 +(1−γ)2 +� 1 +K +K−1 +� +k=0 +E ˜f(¯xk) + +3c2 +1L +2nK +K−1 +� +k=0 +E ∥ˆxk∥2 +≤ ∥¯e0 +x∥2 +αK ++ 9c2 +1L∥ˆx0∥2 +(1 − γ)nK + ασ2 +n ++ 6α2c2 +1c2 +2Lλ4σ2 +1 − γ +. +(39) +Notice from (16) that +∥ˆx0∥2 ≤ ∥ ˆV−1∥2 � +∥ ˆUT˜x0∥2 + ∥ ˆUT˜z0∥2� +. +(40) +If we start from consensual initialization x0 = 1 ⊗ x0 and use +the fact z0 = 0, the above reduces to +∥ˆx0∥2 ≤ ∥ ˆV−1∥2∥ ˆUTz⋆∥2 ≤ α2c2 +2λ4 +(1 − λ)2 ∥ ˆUT∇f(x⋆)∥2, +(41) +where the last step holds by using (9) and (22), which implies +that ˆUTz⋆ = α(I − Λ)−1Λ2 ˆUT∇f(x⋆). Plugging the previous +inequality into (39) and setting 1 +2 ≤ 1 − +24α2c2 +1c2 +2L2λ4 +(1−γ)2 +, i.e., +α ≤ +1 − λ +4 +√ +6c1c2Lλ2 , +(42) +gives +1 +K +K−1 +� +k=0 +Ek ≤ ∥¯e0 +x∥2 +αK ++ a1α + a2α2 +� +�� +� +≜ΨK ++a⋆α2 +K +, +(43) +where we defined Ek ≜ 1 +2 E ˜f(¯xk) + +3c2 +1L +2n E ∥ˆxk∥2 and +a⋆ ≜ 18c2 +1c2 +2Lλ4∥ ˆUT∇f(x⋆)∥2 +(1 − λ)3n +(44a) +a1 ≜ σ2 +n +a2 ≜ 12c2 +1c2 +2Lλ4σ2 +1 − λ +. +(44b) +We now select the stepsize α to arrive at our result in a manner +similar to [31]. First note that the previous inequality holds for +α ≤ 1 +α ≜ min +� +1 +4L, +1 − λ +4 +√ +6c1c2Lλ2 +� +. +(45) + +7 +Setting α = min +�� +∥¯e0 +x∥2 +a1K +� 1 +2 , +� +∥¯e0 +x∥2 +a2K +� 1 +3 , 1 +α +� +≤ +1 +α we have +three cases: i) If α = 1 +α, which is smaller than both +� +∥¯e0 +x∥2 +a1K +� 1 +2 +and +� +∥¯e0 +x∥2 +a2K +� 1 +3 , then +ΨK = α∥¯e0 +x∥2 +K ++ a1 +α + a2 +α2 +≤ α∥¯e0 +x∥2 +K ++ +� +a1∥¯e0 +x∥2 +K +� 1 +2 ++ a +1 +3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 +; +ii) If α = +� +∥¯e0 +x∥2 +a1K +� 1 +2 < +� +∥¯e0 +x∥2 +a2K +� 1 +3 , then +ΨK ≤ 2 +� +a1∥¯e0 +x∥2 +K +� 1 +2 ++ a2 +� +∥¯e0 +x∥2 +a1K +� +≤ 2 +� +a1∥¯e0 +x∥2 +K +� 1 +2 ++ a +1 +3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 +; +iii) If α = +� +∥¯e0 +x∥2 +a2K +� 1 +3 < +� +∥¯e0 +x∥2 +a1K +� 1 +2 , then +ΨK ≤ 2a +1 +3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 ++ a1 +� +∥¯e0 +x∥2 +a2K +� 1 +3 +≤ 2a +1 +3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 ++ +� +a1∥¯e0 +x∥2 +K +� 1 +2 +. +Combining the above cases, we have +ΨK ≤ 2 +� +a1∥¯e0 +x∥2 +K +� 1 +2 ++ 2a1/3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 ++ α∥¯e0 +x∥2 +K +. +Therefore, substituting into (43) we conclude that +1 +K +K−1 +� +k=0 +Ek ≤ 2 +� +a1∥¯e0 +x∥2 +K +� 1 +2 + 2a +1 +3 +2 +� +∥¯e0 +x∥2 +K +� 2 +3 ++ +(α∥¯e0 +x∥2 + a⋆ +α2 ) +K +. +Plugging the constants (44) and the upper bound for α in (45), +and using ς2 +⋆ = 1 +n∥ ˆUT∇f(x⋆)∥2 = 1 +n +�n +i=1 ∥∇fi(x⋆)−∇f(x⋆)∥2 +yields our rate (20). +Appendix D +Proof of Theorem 2 +Substituting the bound +∥∇f(xk) − ∇f(x⋆)∥2 ≤ L2∥xk − x⋆∥2 +≤ 2L2∥xk − 1 ⊗ ¯xk∥2 + 2L2∥1 ⊗ ¯xk − x⋆∥2 +≤ 2L2c2 +1∥ˆxk∥2 + 2nL2∥¯ek +x∥2, +into (19), we get +E ∥ˆxk+1∥2 +≤ +� +γ + +2α2c2 +1c2 +2L2λ4 +(1−γ) +� +E ∥ˆxk∥2 + +2α2c2 +2L2λ4n +(1−γ) +∥¯ek +x∥2 + α2c2 +2λ4nσ2 +≤ +�1 + γ +2 +� +E ∥ˆxk∥2 + +2α2c2 +2L2λ4n +(1−γ) +∥¯ek +x∥2 + α2c2 +2λ4nσ2, +(46) +where we used condition (33) in the last inequality. Using +−α� +E f(¯xk) − f(x⋆)� +≤ 0 in (18) and combining with above, +it holds that +� E ∥¯ek+1 +x +∥2 +c2 +1 +n E ∥ˆxk+1∥2 +� +≤ +� +1 − µα +3 +2αL +2α2c2 +1c2 +2L2λ4 +(1−γ) +1+γ +2 +� +� +�� +� +≜A +� E ∥¯ek +x∥2 +c2 +1 +n E ∥ˆxk∥2 +� ++ +� +α2σ2 +n +α2c2 +1c2 +2λ4σ2 +� +� +�� +� +≜b +. +(47) +The spectral radius of the matrix A can be upper bounded by: +ρ(A) ≤ ∥A∥1 = max +� +1 − µα + +2c2 +1c2 +2α2L2λ4 +(1−γ) +, +1+γ +2 ++ 3 +2Lα +� +≤ 1 − µα +2 , +(48) +where the last inequality holds under the stepsize condition: +α ≤ min +� +µ(1 − γ) +4c2 +1c2 +2L2λ4 , 1 − γ +3L + µ +� +. +(49) +Since ρ(A) < 1, we can iterate inequality (47) to get +� E ∥¯ek +x∥2 +c2 +1 +n E ∥ˆxk∥2 +� +≤ Ak +� E ∥¯e0 +x∥2 +c2 +1 +n E ∥ˆx0∥2 +� ++ +k−1 +� +ℓ=0 +Aℓb +≤ Ak +� E ∥¯e0 +x∥2 +c2 +1 +n E ∥ˆx0∥2 +� ++ (I − A)−1b. +(50) +Taking the (induced) 1-norm, using the sub-multiplicative +properties of matrix induced norms, it holds that +E ∥¯ek +x∥2 + +c2 +1 +n E ∥ˆxk∥2 ≤ ∥Ak∥1˜a0 + +��(I − A)−1b +�� +1 +≤ ∥A∥k +1˜a0 + +��(I − A)−1b +�� +1 . +(51) +where ˜a0 = E ∥¯x0 − x⋆∥2 + +c2 +1 +n E ∥ˆx0∥2. We now bound the last +term by noting that +(I − A)−1b += +1 +det(I−A) +� +1−γ +2 +3 +2αL +2α2c2 +1c2 +2L2λ4 +(1−γ) +µα +� +b += +1 +αµ(1 − γ)( 1 +2 − +3α2c2 +1c2 +2L3λ4 +(1−γ)2µ +) +� +1−γ +2 +3 +2αL +2α2c2 +1c2 +2L2λ4 +(1−γ) +µα +� � +α2σ2 +n +α2c2 +1c2 +2λ4σ2 +� +≤ +4 +αµ(1 − γ) +� +� +(1−γ)α2σ2 +2n ++ 3 +2c2 +1c2 +2α3Lλ4σ2 +2α4c2 +1c2 +2L2λ4σ2 +n(1−γ) ++ α3c2 +1c2 +2µλ4σ2 +� +� , +where det(·) denotes the determinant operation. In the last step +we used 1 +2 − +3c2 +1c2 +2α2L3λ4 +(1−γ)2µ +≥ 1 +4 or α ≤ +√µ(1−γ) +2 +√ +3c1c2L3/2λ2 . Therefore, +from (51) +E ∥¯ek +x∥2 + +c2 +1 +n E ∥ˆxk∥2 +≤ (1 − αµ +2 )k˜a0 + +��(I − A)−1b +�� +1 +≤ (1 − αµ +2 )k˜a0 + 2σ2 +µn α ++ +6c2 +1c2 +2(L/µ)λ4σ2+4c2 +1c2 +2λ4σ2 +1−γ +α2 + +8c2 +1c2 +2L2λ4σ2 +µn(1−γ)2 +α3. +(52) +Using (1 − αµ +2 )K ≤ exp(− αµ +2 K) and (41), it holds that +E ∥¯eK +x ∥2 + +c2 +1 +n E ∥ˆxK∥2 +≤ exp(− αµ +2 K)(a0 + α2a⋆) + a1α + a2α2 + a3α3, +(53) +where +a0 ≜ E ∥¯x0 − x⋆∥2, +a⋆ ≜ +c2 +1c2 +2λ4 +(1−λ)2n∥ ˆUT∇f(x⋆)∥2 +(54a) +a1 ≜ 2σ2 +µn , +a2 ≜ 10c2 +1c2 +2Lλ4σ2 +µ(1 − γ) +(54b) +a3 ≜ 8c2 +1c2 +2L2λ4σ2 +µn(1 − γ)2 . +(54c) + +8 +Note that by combining all stepsize conditions, it is sufficient +to require +α ≤ 1 +α ≜ min +� +1 − λ +8L , µ(1 − λ) +8c2 +1c2 +2L2λ4 , +√µ(1 − λ) +4 +√ +3c1c2L3/2λ2 +� +. +(55) +We now select +α = min +� +ln +� +max +� +2, µ2(a0 + a⋆ +α2 ) K +a1 +�� +/µK, 1 +α +� +≤ 1 +α. (56) +Under this choice the exponential term in (53) can be upper +bounded as follows. i) If α = +ln(max{1,µ2(a0+a⋆/α2)K/a1}) +µK +≤ 1 +α +then +exp(− αµ +2 K)(a0 + α2a⋆) +≤ ˜O +� +(a0 + a⋆ +α2 ) exp +� +− ln +� +max +� +1, µ2(a0 + a⋆ +α2 )K/a1 +���� += O +� +a1 +µK +� +; +ii) Otherwise α = 1 +α ≤ +ln(max{1,µ2(a0+a⋆/α2)K/a1}) +µK +and +exp(− αµ +2 K)(a0 + α2a⋆) = exp +� +− µK +2α +� +(a0 + a⋆ +α2 ). +Therefore, under the stepsize condition (56) it holds that +E ∥¯eK +x ∥2 + +c2 +1 +n E ∥ˆxK∥2 +≤ exp(− αµ +2 K)(a0 + α2a⋆) + a1α + a2α2 + a3α3 +≤ ˜O +� +a1 +µK + +a2 +µ2K2 + +a3 +µ3K3 + (a0 + a⋆ +α2 ) exp +� +− K +α +�� +. +Plugging the constants (54) into the above inequality, using +(55) and (32) yields our rate (21). +References +[1] C. G. Lopes and A. H. Sayed, “Diffusion least-mean squares +over adaptive networks: Formulation and performance analy- +sis,” IEEE Transactions on Signal Processing, vol. 56, no. 7, +pp. 3122–3136, 2008. +[2] S. S. Ram, A. Nedic, and V. V. 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Stich, +“A unified theory of decentralized SGD with changing topology +and local updates,” in International Conference on Machine +Learning, pp. 5381–5393, 2020. +[32] Y. Nesterov, Introductory Lectures on Convex Optimization: A +Basic Course, vol. 87. Springer, 2013. + diff --git a/7tE1T4oBgHgl3EQfBwI8/content/tmp_files/load_file.txt b/7tE1T4oBgHgl3EQfBwI8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa0e2f8f8a52631dfc26a78837ee4ce2cf7f9164 --- /dev/null +++ b/7tE1T4oBgHgl3EQfBwI8/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf,len=639 +page_content='1 An Enhanced Gradient-Tracking Bound for Distributed Online Stochastic Convex Optimization Sulaiman A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Alghunaim and Kun Yuan Abstract—Gradient-tracking (GT) based decentralized methods have emerged as an effective and viable alterna- tive method to decentralized (stochastic) gradient descent (DSGD) when solving distributed online stochastic opti- mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Initial studies of GT methods implied that GT methods have worse network dependent rate than DSGD, contradicting experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' This dilemma has recently been resolved, and tighter rates for GT methods have been established, which improves upon DSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In this work, we establish more enhanced rates for GT methods under the online stochastic convex settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We present an alternative approach for analyzing GT methods for convex problems and over static graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' When compared to previous analyses, this approach allows us to establish enhanced network dependent rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Index Terms—Distributed stochastic optimization, decen- tralized learning, gradient-tracking, adapt-then-combine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Introduction We consider the multi-agent consensus optimization prob- lem, in which n agents work together to solve the following stochastic optimization problem: minimize x∈Rd f(x) = 1 n n � i=1 fi(x) fi(x) ≜ E[Fi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (1) Here, fi : Rd → R is the private cost function held by agent i, which is defined as the expected value of some loss function Fi(·, ξi) over local random variable ξi (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', data points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' An algorithm that solves (1) is said to be a decentralized method if its implementation requires the agents to communicate only with agents who are directly connected to them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', neighbors) based on the given network topology/graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' One of the most popular decentralized methods to solve prob- lem (1) is decentralized stochastic gradient descent (DSGD) [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' While DSGD is communication efficient and simple to implement, it converges slowly when the local functions/data are heterogeneous across nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Furthermore, because data heterogeneity can be amplified by large and sparse network topologies [4], DSGD performance is significantly degraded with these topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In this work, we analyze the performance of the gradient- tracking method [5], [6], which is another well-known decentral- ized method that solves problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' To describe the algorithm, we let wij ≥ 0 denote the weight used by agent i to scale information received from agent j with wij = 0 if j /∈ Ni where Ni is the neighborhood of agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The adapt-then-combine gradient-tracking (ATC-GT) method [5] is described as follows: xk+1 i = � j∈Ni wij(xk j − αgk j ) (2a) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Alghunaim (sulaiman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='alghunaim@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='kw) is with the Department of Electrical Engineering, Kuwait University, Kuwait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Yuan (kunyuan@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='cn) is with the Center for Machine Learning Research, Peking University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' gk+1 i = � j∈Ni wij � gk j + ∇Fj(xk+1 j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk+1 j ) − ∇Fj(xk j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk j )� (2b) with initialization g0 i = ∇Fi(x0 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξ0 i ) and arbitrary x0 i ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Here, ∇Fi(xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk i ) is the stochastic gradient and ξk i is the data sampled by agent i at iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Gradient-tracking can eliminate the impact of heterogeneity between local functions [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In massive numerical experi- ments reported in [9]–[12], GT can significantly outperform DSGD in the online stochastic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Initial studies on the convergence rate of GT methods are inadequate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' they provide loose convergence rates that are more sensitive to network topology than vanilla DSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' According to these findings, GT will converge slower than DSGD on large and sparse networks, which is counter-intuitive and contradicts numerical results published in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Recent works [13], [14] establish the first convergence rates for GT that are faster than DSGD and more robust to sparse topologies under stochastic and non- convex settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In this paper, we will provide additional en- hancements for GT under convex and strongly convex settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Related works Gradient-tracking (GT) methods, which utilize dynamic tracking mechanisms [15] to approximate the globally averaged gradient, have emerged as an alternative to decentralized gradi- ent descent (DGD) [1]–[3], [16], [17] with exact convergence for deterministic problems [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Since their inception, numerous works have investigated GT methods in a variety of contexts [9], [10], [18]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' However, all of these works provide convergence rates that can be worse than vanilla DSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In particular, these results indicate that GT is less robust to sparse topologies even if it can remove the influence of data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The work [14] established refined bounds for various methods including GT methods that improve upon DSGD under nonconvex set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Improved network dependent bounds for GT methods in both convex and non-convex settings are also provided in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In this work, we provide additional improvements over previous works in convex and strongly convex settings – see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' It should be noted that there are other methods that are different from GT methods but have been shown to have com- parable or superior performance – see [14], [29] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In contrast to these other methods, GT methods have been shown to converge in a variety of scenarios, such as directed graphs and time-varying graphs [18], [19], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We should also mention that there are modifications to GT ap- proaches that can improve the rate at the price of knowing addi- tional network information and/or more computation/memory [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' However, the focus of this study is on basic vanilla GT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Contributions We present an alternative approach for analyzing GT methods in convex and static graph settings, which may arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='02855v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='OC] 7 Jan 2023 2 TABLE I: Convergence rate to reach ϵ accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The strongly convex (SC) and PL condition rates ignores iteration logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The quantity λ = ρ(W − 1 n11T) ∈ (0, 1) is the mixing rate of the network where W is the network combination matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' a0 = ∥¯x0 − x⋆∥2, ς2 ⋆ = 1 n �n i=1 ∥∇fi(x⋆)∥2, ς2 0 = 1 n �n i=1 ∥∇fi(x0) − ∇f(x0)∥2, x0 is the initialization for all nodes, and x⋆ is an optimal solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Reference ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Iterations to ϵ accuracy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Remark ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Convex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='[13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ )1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ3/2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ )(a0+ς2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='0 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Rate holds only when iteration number K > ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Convex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Our work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ3/2 + (a0+ς2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='⋆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='SC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='√ϵ + C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='√ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='C depends on 1/(1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='PL∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='[10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='√ϵ + ˜C log 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='˜C depends on 1/(1 − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='SC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='[13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ )1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='√ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(a0+ς2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='0 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Rate holds only when iteration number K > ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='log( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='PL∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='[14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='nϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)1/2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(1−λ)√n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='√ϵ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1−λ log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='(a0+ς2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='⋆) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='Rate holds by tuning stepsize from [14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Theorem 2] SC Our work 1 nϵ + 1 (1−λ)1/2 1 √ϵ + 1 1 − λ log � (a0+ς2 ⋆) ϵ � – ∗ The PL condition is weaker than SC and can hold for nonconvex functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' any SC function satisfies the PL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' be useful for analyzing GT methods in other settings such as variance-reduced gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In stochastic and convex environments, our convergence rate improve and tighten existing GT bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We show, in particular, that under convex settings, GT methods have better dependence on network topologies than in nonconvex settings [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Also, our bounds removes the network dependent log factors in [13] – See Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ATC-GT and Main Assumption In this section, we describe the GT algorithm (2) in network notation and list all necessary assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We begin by defin- ing some network quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' GT in network notation We define xk i ∈ Rd as the estimated value of x ∈ Rd at agent i and iteration (time) k, and we introduce the augmented network quantities: xk ≜ col{xk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , xk n} ∈ Rdn f(xk) ≜ n � i=1 fi(xk i ) ∇f(xk) ≜ col{∇f1(xk 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , ∇fn(xk n)} ∇F(xk) ≜ col{∇F1(xk 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , ∇Fn(xk n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk n)} gk ≜ col{gk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , gk n} ∈ Rdn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Here, col{·} is an operation to stack all vectors on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In addition, we define W ≜ [wij] ∈ Rn×n, W ≜ W ⊗ Id, (3) where W is the network weight (or combination, mixing, gossip) matrix with elements wij, and symbol ⊗ denotes the Kronecker product operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Using the above quantities, the ATC-GT method (2) can be described as follows: xk+1 = W[xk − αgk] (4a) gk+1 = W[gk + ∇F(xk+1) − ∇F(xk)], (4b) with initialization g0 = ∇F(x0) and arbitrary x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Assumptions Here, we list the assumptions used in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Our first assumption is on the network graph stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Assumption 1 (Weight matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The network graph is as- sumed to be static and, the weight matrix W to be doubly stochastic and primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We further assume W to be symmetric and positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ■ It is important to note that assuming W to be positive semidefinite is not restrictive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' given any doubly stochastic and symmetric ˜ W, we can easily construct a positive semidefinite weight matrix by W = (I + ˜ W)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We also remark that, under Assumption 1, the mixing rate of the network is: λ ≜ ��W − 1 n11T�� = max i∈{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=',n} |λi| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (5) The next assumption is on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Assumption 2 (Objective function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Each function fi : Rd → R is L-smooth ∥∇fi(y) − ∇fi(z)∥ ≤ L∥y − z∥, ∀ y, z ∈ Rd (6) and (µ-strongly) convex for some L ≥ µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' As a result, the aggregate function f(x) = 1 n �n i=1 fi(x) is also L-smooth and (µ-strongly) convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (When µ = 0, then the objective functions are simply convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=') ■ We now state our final assumption related to the gradient noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Assumption 3 (Gradient noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' For all {i}n i=1 and k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', we assume the following inequalities hold E � ∇Fi(xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk i ) − ∇fi(xk i ) | F k� = 0, (7a) E � ∥∇Fi(xk i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ξk i ) − ∇fi(xk i )∥2 | F k� ≤ σ2, (7b) for some σ2 ≥ 0, where F k ≜ {x0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , xk} is the algorithm- generated filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We further assume that conditioned on F k, the random data {ξt i} are independent of one another for any {i}n i=1 and {t}t≤k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ■ III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Error Recursion To establish the convergence of (4), we will first derive an error recursion that will be key to our enhanced bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 3 Motivated by [14], the following result rewrites algorithm (4) in an equivalent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Lemma 1 (Equivalent GT form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Let x0 take any arbitrary value and z0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Then for static graphs, the update for xk in algorithm (4) is equivalent to following updates for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' xk+1 = (2W − I)xk − αW2∇F(xk) − Bzk (8a) zk+1 = zk + Bxk (8b) with initialization x1 = W(x0 − α∇F(x0)) and z1 = Bx0, and B = I − W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Clearly with the above initialization, both x1 are iden- tical for the updates (4) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Now, for k ≥ 1, it holds from (8a) that xk+1 − xk = (2W − I)(xk − xk−1) − B(zk − zk−1) − αW2(∇F(xk) − ∇F(xk−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Substituting zk − zk−1 = Bxk−1 ((8b)) and B = I − W into the above equation and rearranging the recursion gives xk+1 = 2Wxk − W2xk−1 − αW2(∇F(xk) − ∇F(xk−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Following the same approach, we can also describe the xk update for the GT algorithm (4) as above – see [14], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Hence, both methods are equivalent for static graph W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Under Assumption 1, the fixed point of recursion (8), denoted by (x⋆, z⋆), satisfies: 0 = αW2∇f(x⋆) + Bz⋆ 0 = Bx⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (9) where x⋆ = 1 ⊗ x⋆ and x⋆ is the optimal solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The existence of z⋆ can be shown by using similar arguments as in [30, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='1] or [29, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' By introducing the notation ˜xk ≜ xk − x⋆, ˜z ≜ zk − z⋆, (10) using (8) and the fact (2W − I)x⋆ = x⋆, we can get the error recursion: � ˜xk+1 ˜zk+1 � = � 2W − I −B B I � � ˜xk ˜zk � − α � W2� ∇f(xk) − ∇f(x⋆) + vk� 0 � , (11) where vk ≜ ∇F(xk) − ∇f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Remark 1 (Alternative analysis approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' By describing GT (4) in the alternative form (8), we are able to derive the error recursion from the fixed point (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' This is similar to the way Exact-diffusion/D2 is analyzed in [4], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' This alternative approach allows us to derive tighter bounds compared with existing GT works [9], [10], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ■ Convergence analysis of (11) still remains difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We will exploit the properties of the matrix W to transform recursion (11) into a more suitable form for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' To that end, the following quantities are introduced: ¯xk ≜ 1 n(1T n ⊗ Id)xk = 1 n n � i=1 xk i , (12a) ¯ek x ≜ 1 n(1T n ⊗ Id)˜xk = ¯xk − x⋆, (12b) ∇f(xk) ≜ 1 n(1T n ⊗ Id)∇f(xk) = 1 n n � i=1 ∇fi(xk i ), (12c) ¯vk ≜ 1 n(1T n ⊗ Id)vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (12d) Under Assumption 1, the matrix W admits the following eigen- decomposition: W = UΣU−1 = � 1 ⊗ Id ˆU � � �� � U � Id 0 0 Λ � � �� � Σ � 1 n1T ⊗ Id ˆUT � � �� � U−1 (13) where Λ is a diagonal matrix with eigenvalues strictly less than one and ˆU is an dn × d(n − 1) matrix that satisfies ˆUT ˆU = I, (1T ⊗ Id) ˆU = 0 (14a) ˆU ˆUT = I − 1 n11T ⊗ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (14b) Lemma 2 (Decomposed error recursion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Under Assump- tion 1, there exists matrices ˆV and Γ to transform the error recursion (11) into the following form: ¯ek+1 x = ¯ek x − α∇f(xk) + α¯vk, (15a) ˆxk+1 = Γˆxk − α ˆV−1 l Λ2 ˆUT� ∇f(xk) − ∇f(x⋆) + vk� , (15b) where ˆxk ≜ ˆV−1 � ˆUT˜xk ˆUT˜zk � , (16) and ˆV−1 l denotes the left block of ˆV−1 = [ ˆV−1 l ˆV−1 r ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Moreover, the following bounds hold: ∥ ˆV∥2 ≤ 3, ∥ ˆV−1∥2 ≤ 9, ∥Γ∥ ≤ 1+λ 2 , (17) where λ = maxi∈{2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=',n} λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' See Appendix A The preceding result will serve as the starting point for deriv- ing the bounds that will lead us to our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Specifically, we can derive the following bounds from the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Lemma 3 (Coupled error inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Suppose Assump- tions 1–2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Then, if α < 1 4L, we have E ∥¯ek+1 x ∥2 ≤ (1 − µα) E ∥¯ek x∥2 − α� E f(¯xk) − f(x⋆)� + 3αc2 1L 2n E ∥ˆxk∥2 + α2σ2 n , (18) and E ∥ˆxk+1∥2 ≤ γ E ∥ˆxk∥2 + α2c2 2λ4 (1 − γ) E ∥∇f(xk) − ∇f(x⋆)∥2 + α2c2 2λ4nσ2, (19) where γ ≜ ∥Γ∥, c1 ≜ ∥ ˆV∥, and c2 = ∥ ˆV−1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Convergence Results In this section, we present our main convergence results in Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We then discuss our results and highlight the differences with existing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Theorem 1 (Convex case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Suppose that Assumptions 1-2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Then, there exists a constant stepsize α such that 1 K K−1 � k=0 � E[f(¯xk) − f ⋆] + L n E ∥xk − 1 ⊗ ¯xk∥2� ≤ σ∥¯e0 x∥ √ nK + � Lλ4σ2 1 − λ �1/3 � ∥¯e0 x∥2 K � 2 3 4 + � Lλ2 1 − λ∥¯e0 x∥2 + ς2 ⋆ L(1 − λ) � C K , (20) where ¯e0 x ≜ ¯x0 − x⋆, ς2 ⋆ ≜ 1 n �n i=1 ∥∇fi(x⋆)∥2, and C is an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Theorem 2 (Strongly-convex case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Suppose that Assump- tions 1-2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Then, there exists a constant stepsize α such that E ∥¯eK x ∥2 + 1 n∥xK − 1 ⊗ ¯xK∥2 ≤ ˜O � σ2 nK + σ2 (1 − λ)K2 � + ˜O � σ2 (1 − λ)2nK3 + (a0 + ς2 ⋆) exp [−(1 − λ)K] � , (21) where a0 ≜ ∥¯x0 − x⋆∥2, ς2 ⋆ ≜ 1 n �n i=1 ∥∇fi(x⋆)∥2, and the notation ˜O(·) ignores logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In comparison to [13], our results removes the log factor O(log( 1 1−λ)) and holds for any number of iteration K – see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Moreover, observe that for the strongly-convex case, unlike [13], we do not have a network term 1/(1−λ) multiplying the highest order exponential term exp(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Remark 2 (Improvement upon nonconvex GT rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The GT rates for convex and strongly-convex settings provided in Theorems 1 and 2 improve upon the GT rates for non-convex [13], [14] and PL condition [14] settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' For example, observe from Table I that the GT rate under the PL condition [14] is 1 nϵ + � 1 (1−λ)1/2 + 1 (1−λ)√n � 1 √ϵ + 1 1−λ log � (a0+ς2 ⋆) ϵ � , which has an additional term 1 (1−λ)√n 1 √ϵ compared to our strongly-convex rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ■ Remark 3 (Comparison with Exact-diffusion/D2 [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' For the convex case, the difference with Exact-diffusion/D2 [12] is in the highest order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Exact-diffusion/D2 is � a0 (1−λ) + ς2 ⋆ � 1 K while GT is � a0 (1−λ) + ς2 ⋆ (1−λ) � 1 K where GT has 1/(1 − λ) mul- tiplied by ς2 ⋆, which is slightly worse than Exact-diffusion/D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' A similar conclusion can be reached for the strongly-convex scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ■ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Simulation results This section will present several numerical simulations that compare Gradient-tracking with centralized SGD (CSGD) and decentralized SGD (DSGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We consider solving a strongly-convex problem (1) with fi(x) = 1 2E(aT i x − bi)2 in which random variable ai ∼ N(0, Id), bi = aT i x⋆ i + ni for some local so- lution x⋆ i ∈ Rd and ni ∼ N(0, σ2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The stochastic gradient is calculated as ∇Fi(x) = ai(aT i x − bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Each local solution x⋆ i = x⋆ +vi is generated using the formula x⋆ i = x⋆ +vi, where x⋆ ∼ N(0, Id) is a randomly generated global solution while vi ∼ N(0, σ2 vId) controls similarities between local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Generally speaking, a large σ2 v will result in local solutions {x⋆ i }n i=1 that are vastly different from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We used d = 5, σ2 n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='01, and σ2 v = 1 in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Experiments are carried out on ring and exponential graphs of size n = 30, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Each algorithm’s stepsize (learning rate) is care- fully tuned so that they all converge to the same relative mean- square-error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Each simulation is run 30 times, with the solid line representing average performance and the shadow representing 0 100 200 300 400 iteration 10 4 10 3 10 2 10 1 100 relative error Exponential graph with 30 nodes CSGD DSGD Gradient-Tracking 0 250 500 750 1000 1250 1500 1750 2000 iteration 10 5 10 4 10 3 10 2 10 1 100 relative error Ring graph with 30 nodes CSGD DSGD Gradient-Tracking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 1: Comparison between different algorithms over exponential and ring graphs when solving distributed linear regression with heterogeneous data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The spectral gap 1 − λ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='33 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='0146 for exponential and ring graphs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 0 200 400 600 800 iteration 10 4 10 3 10 2 10 1 100 relative error Exponential graph with 30 nodes CSGD DSGD Gradient-Tracking 0 200 400 600 800 1000 1200 1400 1600 iteration 10 4 10 3 10 2 10 1 100 relative error Ring graph with 30 nodes CSGD DSGD Gradient-Tracking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 2: Comparison between different algorithms over exponential and ring graphs when solving distributed logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The rela- tive error is shown on the y-axis as 1 n �n i=1 E∥xk i −x⋆∥2/∥x⋆∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' When running over the exponential graph which has a well- connected topology with 1 − λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='33, it is observed that both DSGD and Gradient-tracking perform similarly to CSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' However, when running over the ring graph which has a badly- connected topology with 1 − λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='0146, DSGD gets far slower than CSGD due to its sensitivity to network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In contrast, Gradient-tracking just gets a little bit slower than CSGD and performs far better than DSGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' This phenomenon coincides with our established complexity bound in Table I showing that GT has a much weaker dependence on network topology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', 1 − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We next consider the logistic regres- sion problem, which has fi(x) = E ln(1 + exp(−yihT i x)) where (hi, yi) represents the training dataset stored in node i with hi ∈ Rd as the feature vector and yi ∈ −{1, +1} as the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' This is a convex but not strongly-convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Similar to the linear regression experiments, we will first generate a local solution x⋆ i based on x⋆ i = x⋆ + vi using vi ∼ N(0, σ2 vId).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We can generate local data that follows distinct distributions using x⋆ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' To this end, we generate each feature vector hi ∼ N(0, Id) at node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' To produce the corresponding label yi, we create a random variable zi ∼ U(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' If zi ≤ 1 + exp(−yihT i x⋆ i ), we set yi = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' otherwise yi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Clearly, solution x⋆ i controls the distribution of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' By adjusting σ2 v, we can easily control data heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The remaining parameters are the same as in linear regression experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The performances of each algorithm in logistic regression depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 2 are consistent with that in linear regression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', Gradient-tracking performs well for both graphs while DSGD has a significantly deteriorated performance over the ring graph due to its less robustness to network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Appendix A Decomposed Error Recursion 5 Prof of Lemma 2 Using the decomposition (13) and B = I − W: W2 = UΣ2U−1 = � 1 ⊗ Id ˆU � � Id 0 0 Λ2 � � 1 n1T ⊗ Id ˆUT � (22a) B = U(I − Σ)U−1 = � 1 ⊗ Id ˆU � � 0 0 0 I − Λ � � 1 n1T ⊗ Id ˆUT � , (22b) with I − Λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Substituting (22) into (11) and multiplying both sides by blkdiag{U−1, U−1} on the left, we obtain � U−1˜xk+1 U−1˜zk+1 � = � 2Σ2 − I −(I − Σ) I − Σ I � � U−1˜xk U−1˜zk � − α � Σ2U−1� ∇f(xk) − ∇f(x⋆) + vk� 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (23) Since ˜zk always lies in the range space of B, we have (1T n ⊗ Id)˜zk = 0 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Using, the structure of U from (13) and the definitions (12), we have U−1˜xk = � ¯ek x ˆUT˜xk � , U−1˜zk = � 0 ˆUT˜zk � U−1∇f(x) = � ∇f(xk) ˆUT∇f(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Thus, by using the structure of Σ2 and Σ2 b given in (22), we can rewrite (23) as ¯ek+1 x = ¯ek x − α� ∇f(xk) − ∇f(x⋆)� (24a) � ˆUT˜xk+1 ˆUT˜zk+1 � = � 2Λ − I −(I − Λ) I − Λ I � � ˆUT˜xk ˆUT˜zk � − α � Λ2 ˆUT� ∇f(xk) − ∇f(x⋆)vk� 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (24b) Let G ≜ � 2Λ − I −(I − Λ) I − Λ I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (25) It is important to note that the matrix G is identical to the one studied in [14] (for nonconvex case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Therefore, following the same arguments used in [14, Appendix B], we can decompose it as G = ˆVΓ ˆV−1 for matrices ˆV and Γ satisfying the conditions in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Multiplying the second equation in (24) by ˆV−1, we arrive at (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Appendix B Coupled Error Inequalities Proof of Lemma 3 Proof of inequality (18) The proof adjusts the argument from [31, Lemma 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Using (15a) and Assumption 3, we have E[∥¯ek+1 x ∥2|F k] = ∥¯ek x − α n �n i=1(∇fi(xk i ) − ∇fi(x⋆))∥2 + α2 E[∥¯vk∥2|F k] ≤ ∥¯ek x − α n �n i=1(∇fi(xk i ) − ∇fi(x⋆))∥2 + α2σ2 n = ∥¯ek x∥2 + α2∥ 1 n n� i=1 (∇fi(xk i ) − ∇fi(x⋆))∥2 − 2α n n� i=1 � ∇fi(xk i ), ¯ek x � + α2σ2 n , (26) where we used �n i=1 ∇fi(x⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The second term on the right can be bounded as follows: α2∥ 1 n n� i=1 � ∇fi(xk i ) − ∇fi(¯xk) + ∇fi(¯xk) − ∇fi(x⋆)� ∥2 ≤ 2α2∥ 1 n n� i=1 (∇fi(xk i ) − ∇fi(¯xk))∥2 + 2α2∥ 1 n n� i=1 (∇fi(¯xk) − ∇fi(x⋆))∥2 ≤ 2α2 n n� i=1 ∥∇fi(xk i ) − ∇fi(¯xk)∥2 (27) + 2α2∥∇f(¯xk) − ∇f(x⋆)∥2 ≤ 2α2L2 n ∥xk − 1 ⊗ ¯xk∥2 + 2α2∥∇f(¯xk) − ∇f(x⋆)∥2 ≤ 2α2L2 n ∥xk − 1 ⊗ ¯xk∥2 + 4Lα2(f(¯xk) − f(x⋆)), (28) where the first two inequalities follows from Jensen’s inequal- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' The third inequality follows from the Lipschitz gradient assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In the last inequality, we used the L-smoothness property of the aggregate function [32]: ∥∇f(¯xk) − ∇f(x⋆)∥2 ≤ 2L� f(¯xk) − f(x⋆)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Note that for L-smooth and µ-strongly-convex function f, it holds that [32]: f(x) − f(y) − L 2 ∥x − y∥2 ≤ ⟨∇f(y), (x − y)⟩ (29a) f(x) − f(y) + µ 2 ∥x − y∥2 ≤ ⟨∇f(x), (x − y)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (29b) Using these inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' the cross term in (28) can be bounded by − 2α n n� i=1 ⟨∇fi(xk i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ¯ek x⟩ = 2α n n� i=1 � − ⟨∇fi(xk i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ¯xk − xk i ⟩ − ⟨∇fi(xk i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' xk i − x⋆⟩� ≤ 2α n n� i=1 � − fi(¯xk) + fi(xk i ) + L 2 ∥¯xk − xk i ∥2 − µ 2 ∥xk i − x⋆∥2 − fi(xk i ) + fi(x⋆) � ≤ −2α� f(¯xk) − f(x⋆)� + Lα n n� i=1 ∥¯xk − xk i ∥2 − µα∥¯xk − x⋆∥2 = −2α� f(¯xk) − f(x⋆)� + Lα n ∥xk − 1 ⊗ ¯xk∥2 − µα∥¯ek x∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (30) where the last inequality holds due to − 1 n �n i=1 ∥xk i − x⋆∥2 ≤ −∥ 1 n �n i=1(xk i −x⋆)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Substituting (28) and (30) into (26) and taking expectation, we obtain: E ∥¯ek+1 x ∥2 ≤ (1 − µα) E ∥¯ek x∥2 − 2α(1 − 2Lα) E � f(¯xk) − f(x⋆)� + αL n (1 + 2αL) E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2 n ≤ (1 − µα) E ∥¯ek x∥2 − α� E f(¯xk) − f(x⋆)� + 3Lα 2n E ∥xk − 1 ⊗ ¯xk∥2 + α2σ2 n , (31) where the last step uses α ≤ 1 4L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Using (14), we have ∥ ˆUT˜xk∥2 = ∥ ˆUT ˆU˜xk∥2 = ∥xk − 1 ⊗ ¯xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Hence, ∥xk − 1 ⊗ ¯xk∥2 (16) = ∥ ˆVˆxk∥2 − ∥ ˆUT˜zk∥2 ≤ ∥ ˆV∥2∥ˆxk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (32) Substituting the above into (31) yields (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' 6 Proof of inequality (19) From (15b), we have E[∥ˆxk+1∥2|F k] = E ���Γˆxk − α ˆV−1 l Λ2 ˆUT� ∇f(xk) − ∇f(x⋆) + vk� |F k��� 2 (7a) = ���Γˆxk − α ˆV−1 l Λ2 ˆUT� ∇f(xk) − ∇f(x⋆)���� 2 + α2 E ��� ˆV−1 l Λ2 ˆUTvk��F k��� 2 (7b) ≤ ���Γˆxk − α ˆV−1 l Λ2 ˆUT� ∇f(xk) − ∇f(x⋆)���� 2 + α2∥ ˆV−1 l ∥2∥Λ2∥2∥ ˆUT∥2nσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Now, for any vectors a and b, it holds from Jensen’s inequality that ∥a + b∥2 ≤ 1 θ ∥a∥2 + 1 1−θ ∥b∥ for any θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Utilizing this bound with θ = γ ≜ ∥Γ∥ on the first term of the previous inequality, we get E[∥ˆxk+1∥2|F k] ≤ γ∥ˆxk∥2 + α2∥ ˆ V−1 l ∥2∥Λ2∥2∥ ˆ UT∥2 (1−γ) ∥∇f(xk) − ∇f(x⋆)∥2 + α2∥ ˆV−1 l ∥2∥Λ2∥2∥ ˆUT∥2nσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Taking expectation and using ∥ ˆUT∥ ≤ 1, ∥ ˆV−1 l ∥2 ≤ ∥ ˆV−1∥2, and ∥Λ2∥2 ≤ λ4 yield our result (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Appendix C Proof of Theorem 1 Using similar argument to (28) and (32), it holds that ∥∇f(xk) − ∇f(x⋆)∥2 ≤ 2∥∇f(1 ⊗ ¯xk) − ∇f(x⋆)∥2 + 2∥∇f(xk) − ∇f(1 ⊗ ¯xk)∥2 ≤ 4nL[f(¯xk) − f(x⋆)] + 2c2 1L2∥ˆxk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Plugging the above bound into (19) gives E ∥ˆxk+1∥2 ≤ � γ + 2α2c2 1c2 2L2λ4 (1−γ) � E ∥ˆxk∥2 + 4α2c2 2Lλ4n (1−γ) E ˜f(¯xk) + α2c2 2λ4nσ2 ≤ ¯γ E ∥ˆxk∥2 + 4α2c2 2Lλ4n (1−γ) E ˜f(¯xk) + α2c2 2λ4nσ2, where ˜f(¯xk) ≜ f(¯xk)−f(x⋆), ¯γ ≜ 1+γ 2 , and the last inequiality holds when γ + 2α2c2 1c2 2L2λ4 (1−γ) ≤ 1+γ 2 , which is satisfied for α ≤ 1 − λ 4c1c2Lλ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (33) Iterating the last recursion (for any k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ) gives E ∥ˆxk∥2 ≤ ¯γk∥ˆx0∥2 + 4α2c2 2Lλ4n (1−γ) k−1 � ℓ=0 ¯γk−1−ℓ E ˜f(¯xℓ) + k−1 � ℓ=0 ¯γk−1−ℓ � α2c2 2λ4nσ2� ≤ ¯γk∥ˆx0∥2 + 4α2c2 2Lλ4n (1−γ) k−1 � ℓ=0 ¯γk−1−ℓ E ˜f(¯xℓ) + α2c2 2λ4nσ2 1−¯γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (34) In the last inequality we used �k−1 ℓ=0 ¯γk−1−ℓ ≤ 1 1−¯γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Averaging over k = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , K and using ¯γ = 1+γ 2 , it holds that 1 K K � k=1 E ∥ˆxk∥2 ≤ 2∥ˆx0∥2 (1−γ)K + 4α2c2 2Lλ4n (1−γ)K K � k=1 k−1 � ℓ=0 � 1+γ 2 �k−1−ℓ E ˜f(¯xℓ) + 2α2c2 2λ4nσ2 1−γ ≤ 2∥ˆx0∥2 (1−γ)K + 8α2c2 2Lλ4n (1−γ)2K K−1 � k=0 E ˜f(¯xk) + 2α2c2 2λ4nσ2 1−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (35) It follows that 1 K K−1 � k=0 E ∥ˆxk∥2 ≤ 3∥ˆx0∥2 (1 − γ)K + 8α2c2 2Lλ4n (1−γ)2K K−1 � k=0 E ˜f(¯xk) + 2α2c2 2λ4nσ2 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (36) where we added ∥ˆx0∥2 (1−γ)K and used ∥ˆx0∥2 K ≤ ∥ˆx0∥2 (1−γ)K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Now when µ = 0, we can rearrange (18) to get E(f(¯xk) − f(x⋆)) ≤ 1 α � E ∥¯ek x∥2 − E ∥¯ek+1 x ∥2� + 3c2 1L 2n E ∥ˆxk∥2 + ασ2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (37) Averaging over k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' , K − 1 (K ≥ 1), it holds that 1 K K−1 � k=0 E ˜f(¯xk) ≤ ∥¯e0 x∥2 αK + 3c2 1L 2nK K−1 � k=0 E ∥ˆxk∥2 + ασ2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (38) Multiplying inequality (36) by 2 × 3c2 1L 2n , adding to (38), and rearranging we obtain � 1 − 24α2c2 1c2 2L2λ4 (1−γ)2 � 1 K K−1 � k=0 E ˜f(¯xk) + 3c2 1L 2nK K−1 � k=0 E ∥ˆxk∥2 ≤ ∥¯e0 x∥2 αK + 9c2 1L∥ˆx0∥2 (1 − γ)nK + ασ2 n + 6α2c2 1c2 2Lλ4σ2 1 − γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (39) Notice from (16) that ∥ˆx0∥2 ≤ ∥ ˆV−1∥2 � ∥ ˆUT˜x0∥2 + ∥ ˆUT˜z0∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (40) If we start from consensual initialization x0 = 1 ⊗ x0 and use the fact z0 = 0, the above reduces to ∥ˆx0∥2 ≤ ∥ ˆV−1∥2∥ ˆUTz⋆∥2 ≤ α2c2 2λ4 (1 − λ)2 ∥ ˆUT∇f(x⋆)∥2, (41) where the last step holds by using (9) and (22), which implies that ˆUTz⋆ = α(I − Λ)−1Λ2 ˆUT∇f(x⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Plugging the previous inequality into (39) and setting 1 2 ≤ 1 − 24α2c2 1c2 2L2λ4 (1−γ)2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=', α ≤ 1 − λ 4 √ 6c1c2Lλ2 , (42) gives 1 K K−1 � k=0 Ek ≤ ∥¯e0 x∥2 αK + a1α + a2α2 � �� � ≜ΨK +a⋆α2 K , (43) where we defined Ek ≜ 1 2 E ˜f(¯xk) + 3c2 1L 2n E ∥ˆxk∥2 and a⋆ ≜ 18c2 1c2 2Lλ4∥ ˆUT∇f(x⋆)∥2 (1 − λ)3n (44a) a1 ≜ σ2 n a2 ≜ 12c2 1c2 2Lλ4σ2 1 − λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (44b) We now select the stepsize α to arrive at our result in a manner similar to [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' First note that the previous inequality holds for α ≤ 1 α ≜ min � 1 4L, 1 − λ 4 √ 6c1c2Lλ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (45) 7 Setting α = min �� ∥¯e0 x∥2 a1K � 1 2 , � ∥¯e0 x∥2 a2K � 1 3 , 1 α � ≤ 1 α we have three cases: i) If α = 1 α, which is smaller than both � ∥¯e0 x∥2 a1K � 1 2 and � ∥¯e0 x∥2 a2K � 1 3 , then ΨK = α∥¯e0 x∥2 K + a1 α + a2 α2 ≤ α∥¯e0 x∥2 K + � a1∥¯e0 x∥2 K � 1 2 + a 1 3 2 � ∥¯e0 x∥2 K � 2 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ii) If α = � ∥¯e0 x∥2 a1K � 1 2 < � ∥¯e0 x∥2 a2K � 1 3 , then ΨK ≤ 2 � a1∥¯e0 x∥2 K � 1 2 + a2 � ∥¯e0 x∥2 a1K � ≤ 2 � a1∥¯e0 x∥2 K � 1 2 + a 1 3 2 � ∥¯e0 x∥2 K � 2 3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' iii) If α = � ∥¯e0 x∥2 a2K � 1 3 < � ∥¯e0 x∥2 a1K � 1 2 , then ΨK ≤ 2a 1 3 2 � ∥¯e0 x∥2 K � 2 3 + a1 � ∥¯e0 x∥2 a2K � 1 3 ≤ 2a 1 3 2 � ∥¯e0 x∥2 K � 2 3 + � a1∥¯e0 x∥2 K � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Combining the above cases, we have ΨK ≤ 2 � a1∥¯e0 x∥2 K � 1 2 + 2a1/3 2 � ∥¯e0 x∥2 K � 2 3 + α∥¯e0 x∥2 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Therefore, substituting into (43) we conclude that 1 K K−1 � k=0 Ek ≤ 2 � a1∥¯e0 x∥2 K � 1 2 + 2a 1 3 2 � ∥¯e0 x∥2 K � 2 3 + (α∥¯e0 x∥2 + a⋆ α2 ) K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Plugging the constants (44) and the upper bound for α in (45), and using ς2 ⋆ = 1 n∥ ˆUT∇f(x⋆)∥2 = 1 n �n i=1 ∥∇fi(x⋆)−∇f(x⋆)∥2 yields our rate (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Appendix D Proof of Theorem 2 Substituting the bound ∥∇f(xk) − ∇f(x⋆)∥2 ≤ L2∥xk − x⋆∥2 ≤ 2L2∥xk − 1 ⊗ ¯xk∥2 + 2L2∥1 ⊗ ¯xk − x⋆∥2 ≤ 2L2c2 1∥ˆxk∥2 + 2nL2∥¯ek x∥2, into (19), we get E ∥ˆxk+1∥2 ≤ � γ + 2α2c2 1c2 2L2λ4 (1−γ) � E ∥ˆxk∥2 + 2α2c2 2L2λ4n (1−γ) ∥¯ek x∥2 + α2c2 2λ4nσ2 ≤ �1 + γ 2 � E ∥ˆxk∥2 + 2α2c2 2L2λ4n (1−γ) ∥¯ek x∥2 + α2c2 2λ4nσ2, (46) where we used condition (33) in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Using −α� E f(¯xk) − f(x⋆)� ≤ 0 in (18) and combining with above, it holds that � E ∥¯ek+1 x ∥2 c2 1 n E ∥ˆxk+1∥2 � ≤ � 1 − µα 3 2αL 2α2c2 1c2 2L2λ4 (1−γ) 1+γ 2 � � �� � ≜A � E ∥¯ek x∥2 c2 1 n E ∥ˆxk∥2 � + � α2σ2 n α2c2 1c2 2λ4σ2 � � �� � ≜b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (47) The spectral radius of the matrix A can be upper bounded by: ρ(A) ≤ ∥A∥1 = max � 1 − µα + 2c2 1c2 2α2L2λ4 (1−γ) , 1+γ 2 + 3 2Lα � ≤ 1 − µα 2 , (48) where the last inequality holds under the stepsize condition: α ≤ min � µ(1 − γ) 4c2 1c2 2L2λ4 , 1 − γ 3L + µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (49) Since ρ(A) < 1, we can iterate inequality (47) to get � E ∥¯ek x∥2 c2 1 n E ∥ˆxk∥2 � ≤ Ak � E ∥¯e0 x∥2 c2 1 n E ∥ˆx0∥2 � + k−1 � ℓ=0 Aℓb ≤ Ak � E ∥¯e0 x∥2 c2 1 n E ∥ˆx0∥2 � + (I − A)−1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (50) Taking the (induced) 1-norm, using the sub-multiplicative properties of matrix induced norms, it holds that E ∥¯ek x∥2 + c2 1 n E ∥ˆxk∥2 ≤ ∥Ak∥1˜a0 + ��(I − A)−1b �� 1 ≤ ∥A∥k 1˜a0 + ��(I − A)−1b �� 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (51) where ˜a0 = E ∥¯x0 − x⋆∥2 + c2 1 n E ∥ˆx0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' We now bound the last term by noting that (I − A)−1b = 1 det(I−A) � 1−γ 2 3 2αL 2α2c2 1c2 2L2λ4 (1−γ) µα � b = 1 αµ(1 − γ)( 1 2 − 3α2c2 1c2 2L3λ4 (1−γ)2µ ) � 1−γ 2 3 2αL 2α2c2 1c2 2L2λ4 (1−γ) µα � � α2σ2 n α2c2 1c2 2λ4σ2 � ≤ 4 αµ(1 − γ) � � (1−γ)α2σ2 2n + 3 2c2 1c2 2α3Lλ4σ2 2α4c2 1c2 2L2λ4σ2 n(1−γ) + α3c2 1c2 2µλ4σ2 � � , where det(·) denotes the determinant operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' In the last step we used 1 2 − 3c2 1c2 2α2L3λ4 (1−γ)2µ ≥ 1 4 or α ≤ √µ(1−γ) 2 √ 3c1c2L3/2λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Therefore, from (51) E ∥¯ek x∥2 + c2 1 n E ∥ˆxk∥2 ≤ (1 − αµ 2 )k˜a0 + ��(I − A)−1b �� 1 ≤ (1 − αµ 2 )k˜a0 + 2σ2 µn α + 6c2 1c2 2(L/µ)λ4σ2+4c2 1c2 2λ4σ2 1−γ α2 + 8c2 1c2 2L2λ4σ2 µn(1−γ)2 α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (52) Using (1 − αµ 2 )K ≤ exp(− αµ 2 K) and (41), it holds that E ∥¯eK x ∥2 + c2 1 n E ∥ˆxK∥2 ≤ exp(− αµ 2 K)(a0 + α2a⋆) + a1α + a2α2 + a3α3, (53) where a0 ≜ E ∥¯x0 − x⋆∥2, a⋆ ≜ c2 1c2 2λ4 (1−λ)2n∥ ˆUT∇f(x⋆)∥2 (54a) a1 ≜ 2σ2 µn , a2 ≜ 10c2 1c2 2Lλ4σ2 µ(1 − γ) (54b) a3 ≜ 8c2 1c2 2L2λ4σ2 µn(1 − γ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (54c) 8 Note that by combining all stepsize conditions, it is sufficient to require α ≤ 1 α ≜ min � 1 − λ 8L , µ(1 − λ) 8c2 1c2 2L2λ4 , √µ(1 − λ) 4 √ 3c1c2L3/2λ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (55) We now select α = min � ln � max � 2, µ2(a0 + a⋆ α2 ) K a1 �� /µK, 1 α � ≤ 1 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' (56) Under this choice the exponential term in (53) can be upper bounded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' i) If α = ln(max{1,µ2(a0+a⋆/α2)K/a1}) µK ≤ 1 α then exp(− αµ 2 K)(a0 + α2a⋆) ≤ ˜O � (a0 + a⋆ α2 ) exp � − ln � max � 1, µ2(a0 + a⋆ α2 )K/a1 ���� = O � a1 µK � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' ii) Otherwise α = 1 α ≤ ln(max{1,µ2(a0+a⋆/α2)K/a1}) µK and exp(− αµ 2 K)(a0 + α2a⋆) = exp � − µK 2α � (a0 + a⋆ α2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf'} +page_content=' Therefore, under the stepsize condition (56) it holds that E ∥¯eK x ∥2 + c2 1 n E 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b/8dAzT4oBgHgl3EQfgfzo/content/tmp_files/2301.01471v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fc845e3be7a53cc99c58a565d808d3f323f6fef --- /dev/null +++ b/8dAzT4oBgHgl3EQfgfzo/content/tmp_files/2301.01471v1.pdf.txt @@ -0,0 +1,726 @@ +Freeform Islamic Geometric Patterns +Rebecca Lin1,2 and Craig S. Kaplan3 +1 Massachusetts Institute of Technology, Cambridge, MA, USA +2 University of British Columbia, Vancouver, BC, Canada +3 University of Waterloo, Waterloo, ON, Canada +Abstract. Islamic geometric patterns are a rich and venerable orna- +mental tradition. Many classic designs feature periodic arrangements of +rosettes: star shapes surrounded by rings of hexagonal petals. We present +a new technique for generating ‘freeform’ compositions of rosettes: finite +designs that freely mix rosettes of unusual sizes while retaining the aes- +thetics of traditional patterns. We use a circle packing as a scaffolding +for developing a patch of polygons and fill each polygon with a motif +based on established constructions from Islamic art. +Keywords: Islamic geometric patterns · modular design · circle packing +1 +Introduction +Islamic geometric patterns are a rich and venerable ornamental tradition [6]. The +most iconic of these patterns involve periodic arrangements of star shapes, with +gaps between them filled by additional polygons with less symmetry (Fig. 2). +Often, the concave corners of stars are filled with rings of petal-shaped hexagons, +yielding composite shapes called rosettes [25] (Fig. 1). +We refer to the number of arms in a star or rosette as its order. These motifs +appear in patterns in several standard orders: multiples of three and four are the +most common due to their compatibility with rotations in periodic symmetry +groups. It is challenging to create patterns that incorporate unusual orders or +unusual combinations of orders. For example, considerable geometric sleight-of- +hand is required for orders such as 11 and 13 (Fig. 2b), which are incompatible +with crystallographic symmetries [3, Pg. 484]. +Whether designers employ stars and rosettes of standard or unusual orders, +they typically construct periodic compositions repeating in two directions in the +plane. Repetition is one of the hallmarks of ornamentation: surface decoration on +walls and floors, clothing, and objects should be appealing but not distracting. +When presented with a periodic pattern, we visually ‘factor’ for a non-repeating +kernel and a rule for filling the plane with copies of that kernel. Thus the eye +may casually appreciate the pattern without being overwhelmed by it. Accord- +ing to Gombrich [18], ‘aesthetic delight lies somewhere between boredom and +confusion’, a sentiment echoed by many others [21,2,11]. In decorative contexts, +a measure of boredom helps a pattern recede from conscious attention. +arXiv:2301.01471v1 [cs.GR] 4 Jan 2023 + +2 +R. Lin and C.S. Kaplan +Fig. 1. Freeform designs composed of rosettes of many different orders. +On the other hand, art benefits from a larger dose of confusion. An artwork +like a painting is a finite composition that rewards careful study, and so every +part of that composition can bear some measure of novelty. In contrast, an +infinite Islamic pattern that pleases the eye when elaborated over a wall might +lose its appeal if cropped, framed, and hung on that same wall. As an artwork, +it would have no natural boundary—no broad composition to guide the eye. +This article presents a technique for constructing ‘freeform’ Islamic geomet- +ric patterns: finite, non-repetitive arrangements of rosettes intended as self- +contained compositions rather than as ornamental textures. A few sample com- +positions appear in Fig. 1. Our freeform designs give us significant flexibility to +mix and match unusual rosette orders. We move along Gombrich’s continuum, +away from the boredom of ornamentation and towards the confusion of art. The +resulting visual experiment allows us to reimagine the canonical motifs of Islamic +geometric patterns in a highly non-traditional setting. +We construct motifs based on an initial polygonal patch using a hybrid of +standard techniques (Sec. 3). We define the overall arrangement of rosettes from +a circle packing derived from a triangulation (Sec. 4). We show how any circle +packing can be converted into a patch of connected polygons (Sec. 4.3). We then +inscribe a motif in each polygon and join the motifs together to form the final +pattern (Sec. 4.4). This technique is robust over a wide range of rosette orders. +The designer can control the final pattern by starting with a triangulation of +their choosing. We support a few additional special effects via ‘gadgets’ that +perform local surgery on the computed circle packing (Sec. 5). We also adapt +our technique to periodic patterns via toroidal circle packings (Sec. 6). + +rosetteFreeform Islamic Geometric Patterns +3 +2 +Related Work +Artists and mathematicians use many strategies to disrupt the potential monoto- +ny of ornamental Islamic patterns. For example, an artist often introduces mild +variations in colours, decorative fills, or calligraphic inscriptions in periodic pat- +terns of otherwise identical stars or rosettes. They also sometimes alter the geom- +etry at the centres of selected rosettes while maintaining outward compatibility +with the rest of the pattern. +As the practice of Islamic geometric patterns grew in sophistication, artists +sought to incorporate stars or rosettes of unusual orders into their work. The +Topkapı scroll, a 15th-century visual guide to the drawing of Islamic ornament, +included a number of patterns with unusual combinations of stars. Cromwell [12] +analyzed these patterns and articulated rules for their construction. Later, he +presented a robust method for assembling patterns from irregular stars with dif- +ferent numbers of points [13]. That work demonstrated the wheel construction +(Sec. 3), which we will detail and use in our method. More recently, Gailiunas [16] +studied the amount of geometric error that accumulates when juxtaposing oth- +erwise incompatible stars. +Bonner [3] presented a comprehensive treatment of the modular construction +of Islamic patterns. His polygonal technique, also known as polygons-in-contact +(PIC) after Hankin [20], builds a motif in every tile of a polygonal tiling (Fig. 2). +Bonner’s book includes a vast collection of patterns with different combinations +of stars, including some ‘non-systematic’ patterns that feature stars or rosettes +with unusual orders, such as 7, 9, 11, 13, and 14. +Fig. 2. Illustration of polygons-in-contact, with examples of rosettes highlighted. (a) A +classical Islamic geometric pattern derived from a tiling by regular decagons, pentagons, +and barrel-shaped hexagons. (b) A non-systematic pattern with an underlying tiling +by regular 11-gons, 13-gons, and irregular pentagons. +Another means of achieving irregularity is to move away from the Euclidean +plane. In Islamic architecture, domes are often decorated with specialized geo- + +(a) +(b)4 +R. Lin and C.S. Kaplan +metric patterns adjusted to varying curvature [6]. Kaplan and Salesin [24] demon- +strated adapting PIC to produce patterns on the sphere and in the hyperbolic +plane. While repetitive in the mathematical sense, hyperbolic patterns are nec- +essarily distorted when projected into the Euclidean plane. Kaplan [22] later +presented a more general method for mapping planar patterns with sufficient +symmetry, including many Islamic patterns, onto arbitrary surfaces in 3D. +A Moroccan zellij design typically features a large central star surrounded +by radially symmetric constellations of smaller modules [7]. These modules are +formed from a standard set of individual tile shapes derived from an 8-pointed +star. The result is a monumental work containing substantial visual novelty +and appeal. The puzzle of creating such designs is more combinatorial than +geometric: the artist seeks new discrete configurations of a fixed set of shapes. +Recently, Kaplan [23] presented an algorithm for the procedural generation of +small zellij compositions, which shares some aesthetic goals with our work. +Modern mathematics allows us to produce patterns that are orderly without +being periodic. Many techniques have been proposed that use substitution tilings +or quasiperiodicity to guide the placement of Islamic motifs [4,27,8,9,29]. Some +researchers have even credited ancient designers with an explicit understanding +of quasiperiodicity [1,26], though such claims are controversial [14]. Non-periodic +patterns with long-range organization occupy an aesthetic sweet spot: they ad- +vertise global structure, but the precise nature of that structure is not trivially +unravelled by the eye. +In the broader world of computer graphics, researchers have explored some +interactive and automated techniques for laying out small motifs to create or- +namental patterns [17]. Practical numerical algorithms for constructing circle +packings are relatively new [10], so circle packings have not received much at- +tention as an organizing tool for pattern design. A notable exception is the work +of Hamekasi and Samavati [19], who use circle packings to guide the placement of +motifs in Persian floral designs. Most recently, Brewer et al. derived circle pack- +ings from k-uniform tilings and used them as a framework in which to inscribe +Islamic motifs [5]. Their technique overlaps somewhat with ours, though they +are restricted to arrangements that can arise naturally from the vertex types +and polygon orders of the tilings they use as a starting point. +3 +Modular Motif Construction +Many standard techniques for constructing Islamic patterns are modular: they +decompose the canvas into disjoint regions such as disks or polygons and define +a procedure for filling every region with a motif. This section summarizes two +motif construction techniques that will form the basis of our method. +In the polygons-in-contact technique (PIC), the canvas is subdivided into +polygons that meet edge-to-edge. We choose a contact angle θ ∈ (0, π/2). For +every edge of a polygon P in the subdivision, we construct the two rays that grow +from the edge’s midpoint towards the interior of P, rotated by ±θ relative to +the edge. A motif is formed by truncating these rays where they meet rays from + +Freeform Islamic Geometric Patterns +5 +other edges. In simple cases, we need only compute intersections with rays from +neighbouring edges (Fig. 3a), or from two edges away (Fig. 3b). A more robust +construction requires heuristics to decide how to truncate rays, such as minimiz- +ing the total length of the motif’s line segments [3, Sec. 4.4.2]. Fig. 2 shows two +patterns created by constructing motifs for every polygon in a subdivision. +In the wheel construction, the modules are circles, each tangent to neigh- +bouring circles in a larger pattern. The construction inscribes a star in every +circle. Given a circle C of radius r, we first identify a set of points S on its +boundary, including the points where C meets its neighbours. We also choose a +smaller circle C′ with radius αr for a given α ∈ (0, 1), lying in the interior of +C. Let p and q be two points in S. We construct the perpendicular bisector of +chord pq and find the intersection of that bisector with C′. Then we draw line +segments from p and q to the intersection. Fig. 3 shows stars that emerge when +this process is repeated for all pairs of p and q in S that are consecutive (c) or +non-consecutive (d). Here we can control the sharpness of the star by varying +the scaling ratio α between the radii of the outer and inner circles. +Fig. 3. Regular 9-pointed stars constructed using PIC (left) and the wheel construction +(right). Using PIC, we truncate rays at their first (a) or second (b) intersections with a +contact angle of 2π/5. Using the wheel construction, we draw zig-zag paths connecting +consecutive points (c) or every other point (d) on the outer circle. +Both of these constructions can produce symmetric n-pointed stars. For PIC, +a symmetric star is produced when P is regular; for the wheel construction, we +require the points in S to be distributed evenly around C, and for C and C′ to +be concentric. We can convert between PIC’s θ and the wheel construction’s α +in this case. Stars (a) and (c) in Fig. 3 are related by +α = 1 − +sin (π/n) sin θ +sin (π(n + 2)/2n − θ) +(1) +and stars (b) and (d) are related by +α = 1 − 2 sin (π/n) sin (π(n − 2)/2n) sin (θ − π/n) +sin (π/2 + 2π/n − θ) +. +(2) +Empirically, the wheel construction works well when forming a star whose +points lie on a common circle: it degrades gracefully as the point distribution + +(a) +(b) +(d)6 +R. Lin and C.S. Kaplan +becomes uneven. PIC is a better choice for small polygons whose irregularity +is harder to characterize. We shall use both in our method. Note that neither +of these techniques explicitly constructs rosettes. Although explicit rosette con- +structions exist [25], we will allow rosettes to emerge as a by-product of the +polygonal decompositions we use as a basis for motif construction, as in the +examples of Fig. 2. +4 +Freeform Designs +In this section, we present the steps that make up our main technique for con- +structing finite, freeform compositions of rosettes. The steps are visualized in +Fig. 4. We begin with an arbitrary simplicial complex (a), which induces a circle +packing (b). Based on the circle packing, we construct a polygonal patch (c), +comprising large cyclic polygons separated by smaller irregular pentagons. Fi- +nally, we use a combination of PIC and the wheel construction to define motifs +for each polygon (d), and optionally render the design (e). In the following sub- +sections, we describe each of these steps in detail. +Fig. 4. Our method takes a complex (a) and computes a circle packing (b). Then it +forms a freeform patch of polygons (c), from which it develops motifs that form a +seamless constellation (d) that may be styled (e). + +(a) +(b) +(e) +(d) +cFreeform Islamic Geometric Patterns +7 +4.1 +Complex +The main input to our technique is a complex, more formally a planar, simply +connected, pure simplicial 2-complex K. In simpler terms, we may regard K as a +collection of non-overlapping triangles in the plane, meeting edge-to-edge. The +union of the triangles defines a region the plane—a simple polygon. We will +refer to the vertices, edges, and faces of the complex. We distinguish between its +boundary vertices, which lie on the simple polygon, and interior vertices, which +lie interior to the polygon. +We may construct input complexes in numerous ways. It is easy to author +them manually by placing and connecting vertices. They can also be generated +procedurally, such as by computing the Delaunay triangulation of a point set. +4.2 +Circle Packing +Let K be a complex with n vertices. A circle packing for K is a collection of +non-overlapping circles {C1, . . . , Cn} whose tangencies echo the combinatorial +structure of K. Each circle Ci corresponds with vertex vi of the complex, and +two circles Ci and Cj are externally tangent if and only if vi and vj are con- +nected by an edge in K. The Discrete Uniformization Theorem guarantees that a +circle packing exists for any given complex K [28]. Although the circle packing’s +connectivity will be identical to that of its complex, they will generally not be +equivalent geometrically: the locations and sizes of the circles are not directly +related to the locations of the vertices in the complex, or to the shapes of its +triangles. +Collins and Stephenson [10] describe a simple numerical algorithm that com- +putes circle packings through iterative adjustments of an initial assignment of +radii to the Ci. The radii of boundary circles must be further constrained with +additional boundary conditions. The simple Python script by Eppstein [15] ac- +cepts explicit values for boundary radii. Given a boundary vertex of degree n, our +implementation chooses a radius r for a circle that would be perfectly surrounded +by 2n − 2 unit circles, giving r = (1 − sin φ)/ sin φ, where φ = π/(2n − 2). +4.3 +Polygonal Patch +A patch is a finite set of polygons with disjoint interiors whose union is a topo- +logical disk. Given a circle packing, we construct a patch that has a large cyclic +polygon (i.e., a polygon whose vertices lie on a common circle) associated with +each circle, separated from other cyclic polygons by haloes of pentagonal ‘filler +polygons’. By design, these polygons can serve as scaffolding for building motifs +typical in Islamic geometric patterns. +Let C be an interior circle in a circle packing, and let k be the degree of the +vertex associated with C in the complex. As illustrated in Fig. 5a, we construct +a cyclic 2k-gon P in the interior of C. To begin, we set the vertices of P to be the +k points of tangency between C and its neighbours, together with the midpoints +of the minor arcs of C connecting adjacent tangency points. Now let τ ∈ (0, 1) + +8 +R. Lin and C.S. Kaplan +be a user-selected scaling factor. Scale P relative to the centre of C by a factor +of τ, and add the scaled polygon to the patch. By default, we use τ = 0.8, a +choice that we discuss in Sec. 7. +The gaps between circles in the packing are triangular regions bounded by +arcs of three mutually tangent circles. Let Ci, Cj, and Ck be one such trio +of circles. We divide the space between their cyclic polygons into three new +pentagons, as shown in Fig. 5b, by drawing edges connecting vertices of cyclic +polygons. Three outer line segments pass through the pairwise tangencies of +the circles. Three inner segments connect arc midpoints to a new point o, the +incentre of the triangle formed from the centres of Ci, Cj, and Ck. +Fig. 5. Constructing a polygonal patch from a circle packing: we create a cyclic polygon +for every circle (a), and fill the gaps between three mutually tangent circles with trios +of irregular pentagons (b). +4.4 +Motif Construction +The final step in our process is to construct a motif for every polygon in the +patch produced in the previous step. Here we apply both the wheel construction +and PIC, depending on the type of polygon being decorated. Our large cyclic +polygons yield motifs that garner attention. We safeguard the quality of these +motifs by exploiting the robustness of the wheel construction in their develop- +ment (Fig. 6a). We then use PIC for the more unpredictable filler pentagons +(Fig. 6b). Optionally, we remove motif segments around the boundary of the +resulting composition, paring it down to a core of whole rosettes (Fig. 6c). Our +construction depends on a single global contact angle θ, as described in Sec. 3. +By default, we use θ = 2π/5, the angle for which PIC would inscribe a perfect +pentacle in a regular pentagon. +Let C be an interior circle in the packing with centre o and k points of +tangency. Let P be the cyclic 2k-gon associated with C in the patch. We use the +wheel construction to build a star centred at o whose outer points lie at the edge + +(a) +(b)Freeform Islamic Geometric Patterns +9 +Fig. 6. To construct a design from a patch, we use the wheel construction to create a +star in every cyclic polygon (a) and apply PIC to build motifs for filler polygons (b). +Optionally, we remove the outer layers of geometry to extract an arrangement of whole +rosettes (c). +midpoints of P. Generally, these midpoints do not lie on a common circle, but the +wheel construction is tolerant of small deviations in their distances from o. Let rC +be the radius of C, and define r to be rC cos(π/2k). The value r approximates the +radius of an inscribed circle meeting P’s edge midpoints, an approximation that +converges on the correct value when P is regular. Now compute α by plugging +the user-supplied contact angle θ into Eqn. 2, and let C′ be a circle with center +o and radius αr. The radius of C′ is chosen to ensure that the contact angles at +the points of the star approximate θ. We apply the wheel construction using the +edge midpoints of P and the inner circle C′, connecting every other star point +as in Fig. 3d. +It remains to build motifs for the filler pentagons. Let Q be one such pen- +tagon. As in PIC, construct a pair of rays emanating from the midpoint of every +edge of Q, and truncate them where they intersect rays growing from neigh- +bouring edges. If an edge e of Q is adjacent to a cyclic polygon, then we choose +contact angles that yield rays parallel to the star edges meeting across e (Fig. 6b, +red). These angles may not be symmetric across the perpendicular bisector of e, +but the discrepancy is small in practice. If e is adjacent to another pentagon, on +the other hand, then we use θ as the contact angle for its rays (Fig. 6b, purple). +In summary, our method uses a patch to construct a constellation of localized +motifs that combine to form familiar visual elements: rosettes. By our application +of the Discrete Uniformization Theorem, each rosette corresponds to a vertex in +the triangulation K, and two rosettes are adjacent if and only if their vertices +share an edge in K. The order of a rosette is twice the degree of its associated +vertex. +5 +Gadgets +The basic technique of the previous section can produce a wide variety of freeform +designs with combinations of rosettes of different orders. However, some config- +urations found in traditional Islamic geometric patterns remain out of reach, + +(a) +(b) +C10 +R. Lin and C.S. Kaplan +most obviously because we define only one way to fill the triangular gaps be- +tween cyclic polygons. In this section, we introduce two gadgets that help us +recover some of that variety, increasing the visual intrigue of our designs. Gad- +gets are small subgraphs with labelled vertices that can be incorporated into a +complex. These vertices then determine local clusters of polygons during patch +construction, overriding the polygons of Sec. 4.3. +A square gadget is a 5-vertex subgraph with a central vertex a of degree 4, +as shown in Fig. 8a. Given a complex containing a copy of the square gadget, +we obtain a circle packing containing a cluster of circles like the one shown in +Fig. 8b, where circle A is associated with vertex a. When building the patch, +we remove A from the circle packing and tile the hole left behind with four +pentagons, as shown in Fig. 8c. The new point o is the mean of vertices i ,j, k, +and ℓ. Our motif construction will produce a squarish region surrounded by four +rosettes containing a central octagon. +Fig. 7. The square gadget (a) produces a circle packing (b) from which we derive four +filler pentagons (c). +Fig. 8. The bowtie gadget (a) produces a circle packing (b) from which we derive a +cluster (c) of four filler pentagons and a barrel-shaped hexagon. +A bowtie gadget is a 6-vertex subgraph with two central vertices a and b +of degree 4, as shown in Fig. 7a. As with the square gadget, we remove the +corresponding circles A and B from the circle packing and fill the void with a +new configuration of tiles. First, when constructing a cyclic polygon for circle D +associated with vertex d, we divide the minor arc between the tangencies with +C and E into three equal pieces instead of the usual two, yielding vertices j and + +C +a +A +D +B +h +d +E +(a) +(b) +Cd +D +10 +a +E +A +B +e +F +(a) +(b) +cFreeform Islamic Geometric Patterns +11 +j′ in Fig. 7c. Similarly, we divide F’s arc into three, which gives us vertices ℓ +and ℓ′. We then construct a bowtie-shaped arrangement of four pentagons and +one barrel-shaped hexagon, as illustrated in Fig. 7c, where o is the mean of i, j, +and ℓ and o′ is the mean of j′, k, and ℓ′. +When the barrel-shaped hexagon in the centre of the +bowtie gadget becomes too thin, it can produce a motif +with a small region of overlap at its centre, as shown in red +in the inset. When these overlaps occur, we replace the +red segments with a perfect ‘X’, shown in blue. The blue +segments alter the contact angles with the edges of the hexagon; we propagate +any changes to the hexagon’s four neighbouring pentagons. +Recall that without gadgets, our construction was limited to +rosettes of even orders. But when a bowtie gadget appears in +a complex, vertices d and f each contribute three edges to their +corresponding cyclic polygons. Therefore, if a complex vertex acts +as d or f in one such gadget, as in the central vertex of the +subgraph in the inset, that vertex will yield a rosette of odd order. More generally, +we may hang any odd number of suitably oriented bowtie gadgets from a vertex +to obtain an odd-order rosette. +Fig. 9 shows a freeform design constructed from a random arrangement of +bowtie and square gadgets. In future work, we hope to explore gadgets beyond +these two. +Fig. 9. A composition based on a square grid, where every square is randomly subdi- +vided with a diagonal, a square gadget, or a bowtie gadget. + +12 +R. Lin and C.S. Kaplan +6 +Periodic Patterns +While the focus of our technique is the creation of finite, freeform compositions, +we have also examined its ability to produce more orderly designs. For example, a +finite subset of a periodic arrangement of bowtie gadgets (Fig. 10, left) yields an +approximation of the decagonal design in Fig. 2a (Fig. 10, right). Other periodic +arrangements of triangles and gadgets can reproduce different classic designs. +However, because of flexibility in the circle packing algorithm, these freeform +designs could contain rosettes of continuously varying scales. +We can extend our technique to generate truly periodic patterns by gener- +alizing the Discrete Uniformization Theorem beyond the Euclidean plane. In +particular, if K is embedded on a torus, then the theorem guarantees the ex- +istence of a circle packing in the torus’s intrinsic metric [28, Ch. 9]. The circle +packing algorithm is, in some sense, even simpler in this case because there is no +longer any need for explicit boundary conditions: every circle is completely sur- +rounded. The torus can then be cut open and unrolled, yielding a finite collection +of circles that can be stamped out to produce an infinite periodic packing. +Fig. 11 gives an example of a periodic pattern generated from a simplicial +complex embedded on a torus. The light grey disks in Fig. 11b should be inter- +preted as translated copies of the dark grey disks with the same indices. The +numerical circle packing algorithm yields a layout that tiles the plane by trans- +lation (Fig. 11c), from which we can create a periodic pattern with rosettes of +orders 10, 12, 14, and 16. +Future work could explore the analogous extensions of this technique to +other spaces, such as the sphere and the Poincar´e disk model of the hyperbolic +plane [24]. +Fig. 10. A periodic arrangement of bowtie gadgets (left) can be used to generate a +freeform version of the pattern in Fig. 2a (right). + +Freeform Islamic Geometric Patterns +13 +Fig. 11. A triangulation drawn on a square with periodic boundary conditions (a) is +used to generate a circle packing (b) that covers the plane through translation (c). We +construct motifs to obtain a periodic Islamic geometric pattern (d). +7 +Discussion +In this section, we discuss some of the details of our technique, including al- +ternative approaches that we considered during its development. Some of these +alternatives may offer opportunities for future work. +Selecting interior circles. We typically do not generate a motif for every +circle in the packing. Boundary circles, and circles adjacent to them, can dif- +fer substantially in size from their neighbours. These variations can propagate +through the rest of the construction and produce unacceptably distorted motifs, +such as uneven rosette petals (Fig. 12). Future work could consider ways to op- +timize the geometry of the circle packing to serve patch and motif construction. +For now, we omit outer layers of circles in our final designs. Note that this ap- +proach may separate the design into multiple connected components, in which +case we simply keep the largest component. +Beyond these technicalities, we can be selective for aesthetic reasons. Hav- +ing the freedom to craft the shape of a design provides opportunities to create +interesting compositions (Fig. 15). + +(b) +a +(d)14 +R. Lin and C.S. Kaplan +Fig. 12. Circles near the boundary of a packing can lead to distorted stars and rosettes +(right). We discard outer circles, which can sometimes partition a design into multiple +connected components (left). +Choosing a scale factor. Recall that the +parameter τ controls the scale of each cyclic +polygon relative to its circle, which in turn affects the shapes of filler pentagons. +As shown in the inset, the quality of a motif generated within a filler pentagon +decreases as that pentagon deviates from regularity. Thus we seek to choose τ +to minimize the total deviation across a design. +To gauge the deviation of a pentagon Q from regularity, we adopt a contin- +uous symmetry measure by Zabrodsky et al. [30], which quantifies the minimal +distance that the vertices of Q must travel to form a regular pentagon. Let the +error of a freeform patch be the average deviation of its pentagons from regular- +ity. We can compute this error for a range of closely-spaced τ values and choose +the one with minimal error (Fig. 13a). Over a range of circle packings, we see +significant deviation outside the range (0.7, 0.9) and find that τ = 0.8 produces +satisfactory results, as shown throughout this work. +Fig. 13. A patch with pentagons coloured by their deviations (red) from regularity +(green). In (a), cyclic polygons are scaled by τ = 0.7, 0.8, and 0.9, showing that 0.8 +produces good quality overall. In (b), they are offset by a fixed amount, with less +consistent results. + +a) +b +RFreeform Islamic Geometric Patterns +15 +In the future, we hope to investigate other measurements of polygon quality +in order to produce patches that are closer to ideal. For example, PIC can often +produce a satisfactory motif in a polygon that has lower-order symmetries while +not being fully regular. +As an alternative to treating τ as a scaling factor, we also considered offsetting +cyclic polygons by a constant inward distance τ. However, we found that this +approach was not as successful in producing high-quality pentagons (Fig. 13b). +With either interpretation of τ, the quality is the poorest for pentagons adjacent +to two cyclic polygons of widely different radii. Hamekasi and Samavati note this +issue as well [19], and mitigate it by avoiding complexes containing neighbouring +vertices of widely varying degrees. In future work, we would like to develop a +global optimization that chooses a different scaling factor for every cyclic polygon +so as to maximize the overall quality of all filler pentagons. +Cyclic vs. regular polygons. It is tempt- +ing to construct regular polygons in place of +cyclic polygons, as these would yield perfectly +symmetric stars as motifs. Using the aforemen- +tioned regularity measurement [30], we fit a reg- +ular polygon ˆP to each cyclic polygon P gen- +erated in Sec. 4.3, and centre ˆP on the circum- +centre of P. The result for τ = 0.8 is shown in +the inset. This approach prioritizes the quality of large, prominent stars. How- +ever, it yields distorted pentagons whose motifs self-intersect. By choosing cyclic +polygons rather than regular polygons, our algorithm sacrifices some quality in +large stars for the sake of creating feasible connections between them. +Fig. 14. Generative designs constructed from Delaunay triangulations of random points +sets, without (left) and with (right) bowtie gadgets. + +orders +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +816 +R. Lin and C.S. Kaplan +8 +Results +We demonstrate the versatility of our technique by presenting a range of freeform +designs. For stylized results such as the filled composition in Fig. 14 and the +interlaced design in Fig. 17, we adapt the rendering algorithms described by +Kaplan in Bonner’s text [3, Sec. 4.5]. +Our method places no constraints on the input complex, giving users consid- +erable control over the output design. Fully generative designs can be created +using Delaunay triangulations of random point sets, leading to arrangements of +rosettes with various orders (Fig. 14, left). We can further increase the number +of possible charges and broaden the expressiveness of our technique by insert- +ing random gadgets (Fig. 14, right). Of course, an artist may select a subset of +rosettes in a generative design to craft a desired high-level composition (Fig. 15). +Fig. 15. A generative design in which the user has manually chosen to keep a subset +of rosettes from an initial arrangement, producing a more dynamic composition with +an irregular boundary and internal voids. +On the other hand, we can begin with a highly structured complex and obtain +a repetitive final design (Figs. 10 and 17), or use a toroidal complex to produce a +truly periodic pattern (Fig. 11d). In principle, these approaches could be used to +produce exactly or approximately periodic drawings of many historical Islamic +geometric patterns. However, we have not attempted to catalogue exactly which +ones are possible because existing construction techniques are much better suited +to the task of drawing them. +In between the extremes of full control and generative randomness, we can in- +sert carefully constructed subgraphs into a complex to create a single high-order +rosette (Fig. 16a), or create appealing local arrangements of rosettes (Fig. 16b,c). + +Freeform Islamic Geometric Patterns +17 +Another way to balance order and chaos is to place random gadgets within an +otherwise ordered grid (Fig. 9). +Fig. 16. A tuned design with a high-order rosette (a), for which τ = 0.96 and α = 0.75, +and a composition (c) incorporating multiple instances of a web-like sub-complex (b). +The high-order rosette in Fig. 16a is a special case, in that it requires hand- +tuning. Recall that adjacent circles with widely varying radii can produce dis- +torted motifs (Fig. 12). To produce a satisfactory large rosette, we manually set +the τ and α values for its cyclic polygon for a better fit with the surrounding +geometry. +Close examination of many of our results reveals small geometric discrepan- +cies of the kind illustrated starkly in Fig. 12. When rosettes have arms that vary +too dramatically in width or length, they disrupt the elegance of a pattern and +the feeling of ‘inevitability’ in its construction. There are several places in our +work where we choose global constants like τ that produce acceptable results in +general without adapting to the detailed geometry of local parts of individual +designs. The large rosette in Fig. 16a gives one clear example of where local ad- +justments can improve a design. In future work, we would like to explore more +fine-grained constructions that can enhance the quality of every rosette based +on the configuration of the circle packing in its immediate neighbourhood. +9 +Conclusion +We presented a robust method for constructing freeform Islamic geometric pat- +terns comprising rosettes of unusual orders. Our technique relies on the theory +of circle packings, giving us a principled geometric scaffolding from which to +develop a polygonal patch and then motifs. The user controls the initial com- +plex and any gadgets in it, allowing for significant creative freedom in the design + +(b) +(a)18 +R. Lin and C.S. Kaplan +process. Our results are more organic and less repetitive than existing patterns +and suggest many ideas for further exploration. We believe they communicate +the aesthetic of Islamic geometric patterns while also interpreting them in a non- +traditional context. They still manage to convey the ‘aesthetic delight’ that Gom- +brich discussed [18], but with slightly less boredom and more confusion, paving +the way for more artistic applications of these designs. The work enhances our +understanding of traditional patterns and reveals new opportunities—freeform +or otherwise—for both ornamentation and art-making. +Fig. 17. A highly structured composition (right) generated from a finite subset of a +conceptually periodic complex (left). Although the circle packing is not constructed in +a toroidal domain as in Sec. 6, the resulting composition appears close to periodic. +Acknowledgements +This research was supported by the Natural Sciences and Engineering Research +Council of Canada and the Cheriton School of Computer Science at the Univer- +sity of Waterloo. +References +1. R.A. Al Ajlouni, The global long-range order of quasi-periodic patterns in islamic +architecture, Acta Crystallographica Section A 68 (2012), pp. 235–243. 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Gailiunas, Near-miss Star Patterns, in Proceedings of Bridges 2020: Mathemat- +ics, Art, Music, Architecture, Education, Culture, C. Yackel, R. Bosch, E. Torrence, +and K. Fenyvesi, eds., Phoenix, Arizona. Tessellations Publishing, 2020, pp. 27–34. +Available at http://archive.bridgesmathart.org/2020/bridges2020-27.html. +17. L. Gieseke, P. Asente, R. Mˇech, and M. Benes Bedrich an d Fuchs, A survey of +control mechanisms for creative pattern generation, Computer Graphics Forum 40 +(2021), pp. 585–609. Available at https://onlinelibrary.wiley.com/doi/abs/ +10.1111/cgf.142658. +18. E. Gombrich, The Sense of Order: A study in the psychology of decorative art, +second revised ed., Phaidon Press, Oxford, 1994. +19. N. Hamekasi and F. Samavati, Designing Persian Floral Patterns using Circle +Packing, in Proceedings of the International Conference on Computer Graphics +Theory and Applications and International Conference on Information Visual- +ization Theory and Applications (GRAPP/IVAPP), P. Richard, M. Kraus, R.S. +Laramee, and J. Braz, eds., Feb 24-26. SciTePress, 2012, pp. 135–142. Available at +http://dx.doi.org/10.5220/0003850101350142. + +20 +R. Lin and C.S. Kaplan +20. E.H. Hankin, The Drawing of Geometric Patterns in Saracenic Art, Memoirs of +the Archaeological Society of India Vol. 15, Government of India, Calcutta, 1925. +21. F. +Hutcheson, +An +inquiry +into +the +original +of +our +ideas +of +beauty +and +virtue, Printed by J. Darby in Bartholomew Close, London, 1725, Avail- +able at https://oll.libertfund.org/title/leidhold-an-inquiry-into-the- +original-of-our-ideas-of-beauty-and-virtue-1726-2004#lf1458_head_013. +22. C.S. Kaplan, Semiregular patterns on surfaces, in NPAR ’09: Proceedings of the +7th international symposium on Non-photorealistic animation and rendering, New +York. ACM Press, 2009, pp. 35–39. +23. C.S. Kaplan, Generative Zellij, in Proceedings of Bridges 2022: Mathematics, +Art, Music, Architecture, Culture, D. Reimann, D. Norton, and E. Torrence, +eds., Phoenix, Arizona. Tessellations Publishing, 2022, pp. 285–288. Available at +http://archive.bridgesmathart.org/2022/bridges2022-285.html. +24. C.S. Kaplan and D.H. Salesin, Islamic star patterns in absolute geometry, ACM +Transactions on Graphics 23 (2004), pp. 97–119. +25. A. Lee, Islamic star patterns, Muqarnas 4 (1987), pp. 182–197. +26. P.J. Lu and P.J. Steinhardt, Decagonal and quasi-crystalline tilings in medieval +islamic architecture, Science 315 (2007), pp. 1106–1110. +27. M. Pelletier and J. Bonner, A 7-Fold System for Creating Islamic Geometric Pat- +terns Part 2: Contemporary Expression, in Proceedings of Bridges 2012: Mathemat- +ics, Music, Art, Architecture, Culture, R. Bosch, D. McKenna, and R. Sarhangi, +eds., Phoenix, Arizona. Tessellations Publishing, 2012, pp. 149–156. Available at +http://archive.bridgesmathart.org/2012/bridges2012-149.html. +28. K. Stephenson, Introduction to circle packing: The theory of discrete analytic func- +tions, Cambridge University Press, New York, 2005. +29. B. Wichmann and J. Rigby, A penrose-type islamic interlacing pattern, Vi- +sual Mathematics 9 (2007). Avaiable at http://symmetry-us.com/Journals/ +wichmann2007/penrose.html, accessed 12 August 2021. +30. H. Zabrodsky, S. Peleg, and D. Avnir, Continuous symmetry measures, Journal of +the American Chemical Society 114 (1992), pp. 7843–7851. + diff --git a/8dAzT4oBgHgl3EQfgfzo/content/tmp_files/load_file.txt b/8dAzT4oBgHgl3EQfgfzo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58824760b66bcec02288707f01e61c034beedb0d --- /dev/null +++ b/8dAzT4oBgHgl3EQfgfzo/content/tmp_files/load_file.txt @@ -0,0 +1,696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf,len=695 +page_content='Freeform Islamic Geometric Patterns Rebecca Lin1,2 and Craig S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan3 1 Massachusetts Institute of Technology, Cambridge, MA, USA 2 University of British Columbia, Vancouver, BC, Canada 3 University of Waterloo, Waterloo, ON, Canada Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Islamic geometric patterns are a rich and venerable orna- mental tradition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Many classic designs feature periodic arrangements of rosettes: star shapes surrounded by rings of hexagonal petals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We present a new technique for generating ‘freeform’ compositions of rosettes: finite designs that freely mix rosettes of unusual sizes while retaining the aes- thetics of traditional patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We use a circle packing as a scaffolding for developing a patch of polygons and fill each polygon with a motif based on established constructions from Islamic art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Keywords: Islamic geometric patterns · modular design · circle packing 1 Introduction Islamic geometric patterns are a rich and venerable ornamental tradition [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The most iconic of these patterns involve periodic arrangements of star shapes, with gaps between them filled by additional polygons with less symmetry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Often, the concave corners of stars are filled with rings of petal-shaped hexagons, yielding composite shapes called rosettes [25] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We refer to the number of arms in a star or rosette as its order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' These motifs appear in patterns in several standard orders: multiples of three and four are the most common due to their compatibility with rotations in periodic symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' It is challenging to create patterns that incorporate unusual orders or unusual combinations of orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For example, considerable geometric sleight-of- hand is required for orders such as 11 and 13 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2b), which are incompatible with crystallographic symmetries [3, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 484].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Whether designers employ stars and rosettes of standard or unusual orders, they typically construct periodic compositions repeating in two directions in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Repetition is one of the hallmarks of ornamentation: surface decoration on walls and floors, clothing, and objects should be appealing but not distracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' When presented with a periodic pattern, we visually ‘factor’ for a non-repeating kernel and a rule for filling the plane with copies of that kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Thus the eye may casually appreciate the pattern without being overwhelmed by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Accord- ing to Gombrich [18], ‘aesthetic delight lies somewhere between boredom and confusion’, a sentiment echoed by many others [21,2,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In decorative contexts, a measure of boredom helps a pattern recede from conscious attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='01471v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='GR] 4 Jan 2023 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Freeform designs composed of rosettes of many different orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' On the other hand, art benefits from a larger dose of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' An artwork like a painting is a finite composition that rewards careful study, and so every part of that composition can bear some measure of novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In contrast, an infinite Islamic pattern that pleases the eye when elaborated over a wall might lose its appeal if cropped, framed, and hung on that same wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As an artwork, it would have no natural boundary—no broad composition to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' This article presents a technique for constructing ‘freeform’ Islamic geomet- ric patterns: finite, non-repetitive arrangements of rosettes intended as self- contained compositions rather than as ornamental textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A few sample com- positions appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our freeform designs give us significant flexibility to mix and match unusual rosette orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We move along Gombrich’s continuum, away from the boredom of ornamentation and towards the confusion of art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The resulting visual experiment allows us to reimagine the canonical motifs of Islamic geometric patterns in a highly non-traditional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We construct motifs based on an initial polygonal patch using a hybrid of standard techniques (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We define the overall arrangement of rosettes from a circle packing derived from a triangulation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We show how any circle packing can be converted into a patch of connected polygons (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We then inscribe a motif in each polygon and join the motifs together to form the final pattern (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' This technique is robust over a wide range of rosette orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The designer can control the final pattern by starting with a triangulation of their choosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We support a few additional special effects via ‘gadgets’ that perform local surgery on the computed circle packing (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We also adapt our technique to periodic patterns via toroidal circle packings (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' rosetteFreeform Islamic Geometric Patterns 3 2 Related Work Artists and mathematicians use many strategies to disrupt the potential monoto- ny of ornamental Islamic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For example, an artist often introduces mild variations in colours, decorative fills, or calligraphic inscriptions in periodic pat- terns of otherwise identical stars or rosettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' They also sometimes alter the geom- etry at the centres of selected rosettes while maintaining outward compatibility with the rest of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As the practice of Islamic geometric patterns grew in sophistication, artists sought to incorporate stars or rosettes of unusual orders into their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The Topkapı scroll, a 15th-century visual guide to the drawing of Islamic ornament, included a number of patterns with unusual combinations of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Cromwell [12] analyzed these patterns and articulated rules for their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Later, he presented a robust method for assembling patterns from irregular stars with dif- ferent numbers of points [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' That work demonstrated the wheel construction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3), which we will detail and use in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' More recently, Gailiunas [16] studied the amount of geometric error that accumulates when juxtaposing oth- erwise incompatible stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Bonner [3] presented a comprehensive treatment of the modular construction of Islamic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' His polygonal technique, also known as polygons-in-contact (PIC) after Hankin [20], builds a motif in every tile of a polygonal tiling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Bonner’s book includes a vast collection of patterns with different combinations of stars, including some ‘non-systematic’ patterns that feature stars or rosettes with unusual orders, such as 7, 9, 11, 13, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Illustration of polygons-in-contact, with examples of rosettes highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' (a) A classical Islamic geometric pattern derived from a tiling by regular decagons, pentagons, and barrel-shaped hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' (b) A non-systematic pattern with an underlying tiling by regular 11-gons, 13-gons, and irregular pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Another means of achieving irregularity is to move away from the Euclidean plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In Islamic architecture, domes are often decorated with specialized geo- (a) (b)4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan metric patterns adjusted to varying curvature [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan and Salesin [24] demon- strated adapting PIC to produce patterns on the sphere and in the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' While repetitive in the mathematical sense, hyperbolic patterns are nec- essarily distorted when projected into the Euclidean plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan [22] later presented a more general method for mapping planar patterns with sufficient symmetry, including many Islamic patterns, onto arbitrary surfaces in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A Moroccan zellij design typically features a large central star surrounded by radially symmetric constellations of smaller modules [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' These modules are formed from a standard set of individual tile shapes derived from an 8-pointed star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The result is a monumental work containing substantial visual novelty and appeal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The puzzle of creating such designs is more combinatorial than geometric: the artist seeks new discrete configurations of a fixed set of shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Recently, Kaplan [23] presented an algorithm for the procedural generation of small zellij compositions, which shares some aesthetic goals with our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Modern mathematics allows us to produce patterns that are orderly without being periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Many techniques have been proposed that use substitution tilings or quasiperiodicity to guide the placement of Islamic motifs [4,27,8,9,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Some researchers have even credited ancient designers with an explicit understanding of quasiperiodicity [1,26], though such claims are controversial [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Non-periodic patterns with long-range organization occupy an aesthetic sweet spot: they ad- vertise global structure, but the precise nature of that structure is not trivially unravelled by the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In the broader world of computer graphics, researchers have explored some interactive and automated techniques for laying out small motifs to create or- namental patterns [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Practical numerical algorithms for constructing circle packings are relatively new [10], so circle packings have not received much at- tention as an organizing tool for pattern design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A notable exception is the work of Hamekasi and Samavati [19], who use circle packings to guide the placement of motifs in Persian floral designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Most recently, Brewer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' derived circle pack- ings from k-uniform tilings and used them as a framework in which to inscribe Islamic motifs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Their technique overlaps somewhat with ours, though they are restricted to arrangements that can arise naturally from the vertex types and polygon orders of the tilings they use as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3 Modular Motif Construction Many standard techniques for constructing Islamic patterns are modular: they decompose the canvas into disjoint regions such as disks or polygons and define a procedure for filling every region with a motif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' This section summarizes two motif construction techniques that will form the basis of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In the polygons-in-contact technique (PIC), the canvas is subdivided into polygons that meet edge-to-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We choose a contact angle θ ∈ (0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For every edge of a polygon P in the subdivision, we construct the two rays that grow from the edge’s midpoint towards the interior of P, rotated by ±θ relative to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A motif is formed by truncating these rays where they meet rays from Freeform Islamic Geometric Patterns 5 other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In simple cases, we need only compute intersections with rays from neighbouring edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3a), or from two edges away (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A more robust construction requires heuristics to decide how to truncate rays, such as minimiz- ing the total length of the motif’s line segments [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2 shows two patterns created by constructing motifs for every polygon in a subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In the wheel construction, the modules are circles, each tangent to neigh- bouring circles in a larger pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The construction inscribes a star in every circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Given a circle C of radius r, we first identify a set of points S on its boundary, including the points where C meets its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We also choose a smaller circle C′ with radius αr for a given α ∈ (0, 1), lying in the interior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let p and q be two points in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We construct the perpendicular bisector of chord pq and find the intersection of that bisector with C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Then we draw line segments from p and q to the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3 shows stars that emerge when this process is repeated for all pairs of p and q in S that are consecutive (c) or non-consecutive (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Here we can control the sharpness of the star by varying the scaling ratio α between the radii of the outer and inner circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Regular 9-pointed stars constructed using PIC (left) and the wheel construction (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Using PIC, we truncate rays at their first (a) or second (b) intersections with a contact angle of 2π/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Using the wheel construction, we draw zig-zag paths connecting consecutive points (c) or every other point (d) on the outer circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Both of these constructions can produce symmetric n-pointed stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For PIC, a symmetric star is produced when P is regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' for the wheel construction, we require the points in S to be distributed evenly around C, and for C and C′ to be concentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We can convert between PIC’s θ and the wheel construction’s α in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Stars (a) and (c) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3 are related by α = 1 − sin (π/n) sin θ sin (π(n + 2)/2n − θ) (1) and stars (b) and (d) are related by α = 1 − 2 sin (π/n) sin (π(n − 2)/2n) sin (θ − π/n) sin (π/2 + 2π/n − θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' (2) Empirically, the wheel construction works well when forming a star whose points lie on a common circle: it degrades gracefully as the point distribution (a) (b) (d)6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan becomes uneven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' PIC is a better choice for small polygons whose irregularity is harder to characterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We shall use both in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Note that neither of these techniques explicitly constructs rosettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Although explicit rosette con- structions exist [25], we will allow rosettes to emerge as a by-product of the polygonal decompositions we use as a basis for motif construction, as in the examples of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4 Freeform Designs In this section, we present the steps that make up our main technique for con- structing finite, freeform compositions of rosettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The steps are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We begin with an arbitrary simplicial complex (a), which induces a circle packing (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Based on the circle packing, we construct a polygonal patch (c), comprising large cyclic polygons separated by smaller irregular pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fi- nally, we use a combination of PIC and the wheel construction to define motifs for each polygon (d), and optionally render the design (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In the following sub- sections, we describe each of these steps in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our method takes a complex (a) and computes a circle packing (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Then it forms a freeform patch of polygons (c), from which it develops motifs that form a seamless constellation (d) that may be styled (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' (a) (b) (e) (d) cFreeform Islamic Geometric Patterns 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='1 Complex The main input to our technique is a complex, more formally a planar, simply connected, pure simplicial 2-complex K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In simpler terms, we may regard K as a collection of non-overlapping triangles in the plane, meeting edge-to-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The union of the triangles defines a region the plane—a simple polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We will refer to the vertices, edges, and faces of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We distinguish between its boundary vertices, which lie on the simple polygon, and interior vertices, which lie interior to the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We may construct input complexes in numerous ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' It is easy to author them manually by placing and connecting vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' They can also be generated procedurally, such as by computing the Delaunay triangulation of a point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='2 Circle Packing Let K be a complex with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A circle packing for K is a collection of non-overlapping circles {C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' , Cn} whose tangencies echo the combinatorial structure of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Each circle Ci corresponds with vertex vi of the complex, and two circles Ci and Cj are externally tangent if and only if vi and vj are con- nected by an edge in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The Discrete Uniformization Theorem guarantees that a circle packing exists for any given complex K [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Although the circle packing’s connectivity will be identical to that of its complex, they will generally not be equivalent geometrically: the locations and sizes of the circles are not directly related to the locations of the vertices in the complex, or to the shapes of its triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Collins and Stephenson [10] describe a simple numerical algorithm that com- putes circle packings through iterative adjustments of an initial assignment of radii to the Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The radii of boundary circles must be further constrained with additional boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The simple Python script by Eppstein [15] ac- cepts explicit values for boundary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Given a boundary vertex of degree n, our implementation chooses a radius r for a circle that would be perfectly surrounded by 2n − 2 unit circles, giving r = (1 − sin φ)/ sin φ, where φ = π/(2n − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='3 Polygonal Patch A patch is a finite set of polygons with disjoint interiors whose union is a topo- logical disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Given a circle packing, we construct a patch that has a large cyclic polygon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=', a polygon whose vertices lie on a common circle) associated with each circle, separated from other cyclic polygons by haloes of pentagonal ‘filler polygons’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' By design, these polygons can serve as scaffolding for building motifs typical in Islamic geometric patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let C be an interior circle in a circle packing, and let k be the degree of the vertex associated with C in the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 5a, we construct a cyclic 2k-gon P in the interior of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' To begin, we set the vertices of P to be the k points of tangency between C and its neighbours, together with the midpoints of the minor arcs of C connecting adjacent tangency points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Now let τ ∈ (0, 1) 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan be a user-selected scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Scale P relative to the centre of C by a factor of τ, and add the scaled polygon to the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' By default, we use τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='8, a choice that we discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The gaps between circles in the packing are triangular regions bounded by arcs of three mutually tangent circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let Ci, Cj, and Ck be one such trio of circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We divide the space between their cyclic polygons into three new pentagons, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 5b, by drawing edges connecting vertices of cyclic polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Three outer line segments pass through the pairwise tangencies of the circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Three inner segments connect arc midpoints to a new point o, the incentre of the triangle formed from the centres of Ci, Cj, and Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Constructing a polygonal patch from a circle packing: we create a cyclic polygon for every circle (a), and fill the gaps between three mutually tangent circles with trios of irregular pentagons (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='4 Motif Construction The final step in our process is to construct a motif for every polygon in the patch produced in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Here we apply both the wheel construction and PIC, depending on the type of polygon being decorated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our large cyclic polygons yield motifs that garner attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We safeguard the quality of these motifs by exploiting the robustness of the wheel construction in their develop- ment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We then use PIC for the more unpredictable filler pentagons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Optionally, we remove motif segments around the boundary of the resulting composition, paring it down to a core of whole rosettes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our construction depends on a single global contact angle θ, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' By default, we use θ = 2π/5, the angle for which PIC would inscribe a perfect pentacle in a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let C be an interior circle in the packing with centre o and k points of tangency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let P be the cyclic 2k-gon associated with C in the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We use the wheel construction to build a star centred at o whose outer points lie at the edge (a) (b)Freeform Islamic Geometric Patterns 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' To construct a design from a patch, we use the wheel construction to create a star in every cyclic polygon (a) and apply PIC to build motifs for filler polygons (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Optionally, we remove the outer layers of geometry to extract an arrangement of whole rosettes (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' midpoints of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Generally, these midpoints do not lie on a common circle, but the wheel construction is tolerant of small deviations in their distances from o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let rC be the radius of C, and define r to be rC cos(π/2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The value r approximates the radius of an inscribed circle meeting P’s edge midpoints, an approximation that converges on the correct value when P is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Now compute α by plugging the user-supplied contact angle θ into Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2, and let C′ be a circle with center o and radius αr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The radius of C′ is chosen to ensure that the contact angles at the points of the star approximate θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We apply the wheel construction using the edge midpoints of P and the inner circle C′, connecting every other star point as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' It remains to build motifs for the filler pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let Q be one such pen- tagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As in PIC, construct a pair of rays emanating from the midpoint of every edge of Q, and truncate them where they intersect rays growing from neigh- bouring edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' If an edge e of Q is adjacent to a cyclic polygon, then we choose contact angles that yield rays parallel to the star edges meeting across e (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6b, red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' These angles may not be symmetric across the perpendicular bisector of e, but the discrepancy is small in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' If e is adjacent to another pentagon, on the other hand, then we use θ as the contact angle for its rays (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6b, purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In summary, our method uses a patch to construct a constellation of localized motifs that combine to form familiar visual elements: rosettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' By our application of the Discrete Uniformization Theorem, each rosette corresponds to a vertex in the triangulation K, and two rosettes are adjacent if and only if their vertices share an edge in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The order of a rosette is twice the degree of its associated vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 5 Gadgets The basic technique of the previous section can produce a wide variety of freeform designs with combinations of rosettes of different orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' However, some config- urations found in traditional Islamic geometric patterns remain out of reach, (a) (b) C10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan most obviously because we define only one way to fill the triangular gaps be- tween cyclic polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In this section, we introduce two gadgets that help us recover some of that variety, increasing the visual intrigue of our designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Gad- gets are small subgraphs with labelled vertices that can be incorporated into a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' These vertices then determine local clusters of polygons during patch construction, overriding the polygons of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A square gadget is a 5-vertex subgraph with a central vertex a of degree 4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Given a complex containing a copy of the square gadget, we obtain a circle packing containing a cluster of circles like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 8b, where circle A is associated with vertex a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' When building the patch, we remove A from the circle packing and tile the hole left behind with four pentagons, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The new point o is the mean of vertices i ,j, k, and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our motif construction will produce a squarish region surrounded by four rosettes containing a central octagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The square gadget (a) produces a circle packing (b) from which we derive four filler pentagons (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The bowtie gadget (a) produces a circle packing (b) from which we derive a cluster (c) of four filler pentagons and a barrel-shaped hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A bowtie gadget is a 6-vertex subgraph with two central vertices a and b of degree 4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As with the square gadget, we remove the corresponding circles A and B from the circle packing and fill the void with a new configuration of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' First, when constructing a cyclic polygon for circle D associated with vertex d, we divide the minor arc between the tangencies with C and E into three equal pieces instead of the usual two, yielding vertices j and C a A D B h d E (a) (b) Cd D 10 a E A B e F (a) (b) cFreeform Islamic Geometric Patterns 11 j′ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Similarly, we divide F’s arc into three, which gives us vertices ℓ and ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We then construct a bowtie-shaped arrangement of four pentagons and one barrel-shaped hexagon, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7c, where o is the mean of i, j, and ℓ and o′ is the mean of j′, k, and ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' When the barrel-shaped hexagon in the centre of the bowtie gadget becomes too thin, it can produce a motif with a small region of overlap at its centre, as shown in red in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' When these overlaps occur, we replace the red segments with a perfect ‘X’, shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The blue segments alter the contact angles with the edges of the hexagon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' we propagate any changes to the hexagon’s four neighbouring pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Recall that without gadgets, our construction was limited to rosettes of even orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' But when a bowtie gadget appears in a complex, vertices d and f each contribute three edges to their corresponding cyclic polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Therefore, if a complex vertex acts as d or f in one such gadget, as in the central vertex of the subgraph in the inset, that vertex will yield a rosette of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' More generally, we may hang any odd number of suitably oriented bowtie gadgets from a vertex to obtain an odd-order rosette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 9 shows a freeform design constructed from a random arrangement of bowtie and square gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In future work, we hope to explore gadgets beyond these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A composition based on a square grid, where every square is randomly subdi- vided with a diagonal, a square gadget, or a bowtie gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan 6 Periodic Patterns While the focus of our technique is the creation of finite, freeform compositions, we have also examined its ability to produce more orderly designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For example, a finite subset of a periodic arrangement of bowtie gadgets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 10, left) yields an approximation of the decagonal design in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2a (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 10, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Other periodic arrangements of triangles and gadgets can reproduce different classic designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' However, because of flexibility in the circle packing algorithm, these freeform designs could contain rosettes of continuously varying scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We can extend our technique to generate truly periodic patterns by gener- alizing the Discrete Uniformization Theorem beyond the Euclidean plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In particular, if K is embedded on a torus, then the theorem guarantees the ex- istence of a circle packing in the torus’s intrinsic metric [28, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The circle packing algorithm is, in some sense, even simpler in this case because there is no longer any need for explicit boundary conditions: every circle is completely sur- rounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The torus can then be cut open and unrolled, yielding a finite collection of circles that can be stamped out to produce an infinite periodic packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 11 gives an example of a periodic pattern generated from a simplicial complex embedded on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The light grey disks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 11b should be inter- preted as translated copies of the dark grey disks with the same indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The numerical circle packing algorithm yields a layout that tiles the plane by trans- lation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 11c), from which we can create a periodic pattern with rosettes of orders 10, 12, 14, and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Future work could explore the analogous extensions of this technique to other spaces, such as the sphere and the Poincar´e disk model of the hyperbolic plane [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A periodic arrangement of bowtie gadgets (left) can be used to generate a freeform version of the pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 2a (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Freeform Islamic Geometric Patterns 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A triangulation drawn on a square with periodic boundary conditions (a) is used to generate a circle packing (b) that covers the plane through translation (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We construct motifs to obtain a periodic Islamic geometric pattern (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7 Discussion In this section, we discuss some of the details of our technique, including al- ternative approaches that we considered during its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Some of these alternatives may offer opportunities for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Selecting interior circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We typically do not generate a motif for every circle in the packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Boundary circles, and circles adjacent to them, can dif- fer substantially in size from their neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' These variations can propagate through the rest of the construction and produce unacceptably distorted motifs, such as uneven rosette petals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Future work could consider ways to op- timize the geometry of the circle packing to serve patch and motif construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For now, we omit outer layers of circles in our final designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Note that this ap- proach may separate the design into multiple connected components, in which case we simply keep the largest component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Beyond these technicalities, we can be selective for aesthetic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Hav- ing the freedom to craft the shape of a design provides opportunities to create interesting compositions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' (b) a (d)14 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Circles near the boundary of a packing can lead to distorted stars and rosettes (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We discard outer circles, which can sometimes partition a design into multiple connected components (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Choosing a scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Recall that the parameter τ controls the scale of each cyclic polygon relative to its circle, which in turn affects the shapes of filler pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As shown in the inset, the quality of a motif generated within a filler pentagon decreases as that pentagon deviates from regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Thus we seek to choose τ to minimize the total deviation across a design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' To gauge the deviation of a pentagon Q from regularity, we adopt a contin- uous symmetry measure by Zabrodsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' [30], which quantifies the minimal distance that the vertices of Q must travel to form a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Let the error of a freeform patch be the average deviation of its pentagons from regular- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We can compute this error for a range of closely-spaced τ values and choose the one with minimal error (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 13a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Over a range of circle packings, we see significant deviation outside the range (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='9) and find that τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='8 produces satisfactory results, as shown throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A patch with pentagons coloured by their deviations (red) from regularity (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In (a), cyclic polygons are scaled by τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='8, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='9, showing that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='8 produces good quality overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In (b), they are offset by a fixed amount, with less consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' a) b RFreeform Islamic Geometric Patterns 15 In the future, we hope to investigate other measurements of polygon quality in order to produce patches that are closer to ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For example, PIC can often produce a satisfactory motif in a polygon that has lower-order symmetries while not being fully regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' As an alternative to treating τ as a scaling factor, we also considered offsetting cyclic polygons by a constant inward distance τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' However, we found that this approach was not as successful in producing high-quality pentagons (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' With either interpretation of τ, the quality is the poorest for pentagons adjacent to two cyclic polygons of widely different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Hamekasi and Samavati note this issue as well [19], and mitigate it by avoiding complexes containing neighbouring vertices of widely varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In future work, we would like to develop a global optimization that chooses a different scaling factor for every cyclic polygon so as to maximize the overall quality of all filler pentagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Cyclic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' regular polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' It is tempt- ing to construct regular polygons in place of cyclic polygons, as these would yield perfectly symmetric stars as motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Using the aforemen- tioned regularity measurement [30], we fit a reg- ular polygon ˆP to each cyclic polygon P gen- erated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='3, and centre ˆP on the circum- centre of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The result for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='8 is shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' This approach prioritizes the quality of large, prominent stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' How- ever, it yields distorted pentagons whose motifs self-intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' By choosing cyclic polygons rather than regular polygons, our algorithm sacrifices some quality in large stars for the sake of creating feasible connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Generative designs constructed from Delaunay triangulations of random points sets, without (left) and with (right) bowtie gadgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' orders 18 17 16 15 14 13 12 11 10 9 816 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan 8 Results We demonstrate the versatility of our technique by presenting a range of freeform designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' For stylized results such as the filled composition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 14 and the interlaced design in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 17, we adapt the rendering algorithms described by Kaplan in Bonner’s text [3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our method places no constraints on the input complex, giving users consid- erable control over the output design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fully generative designs can be created using Delaunay triangulations of random point sets, leading to arrangements of rosettes with various orders (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 14, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We can further increase the number of possible charges and broaden the expressiveness of our technique by insert- ing random gadgets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 14, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Of course, an artist may select a subset of rosettes in a generative design to craft a desired high-level composition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A generative design in which the user has manually chosen to keep a subset of rosettes from an initial arrangement, producing a more dynamic composition with an irregular boundary and internal voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' On the other hand, we can begin with a highly structured complex and obtain a repetitive final design (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 10 and 17), or use a toroidal complex to produce a truly periodic pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 11d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In principle, these approaches could be used to produce exactly or approximately periodic drawings of many historical Islamic geometric patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' However, we have not attempted to catalogue exactly which ones are possible because existing construction techniques are much better suited to the task of drawing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In between the extremes of full control and generative randomness, we can in- sert carefully constructed subgraphs into a complex to create a single high-order rosette (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 16a), or create appealing local arrangements of rosettes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 16b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Freeform Islamic Geometric Patterns 17 Another way to balance order and chaos is to place random gadgets within an otherwise ordered grid (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A tuned design with a high-order rosette (a), for which τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='96 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='75, and a composition (c) incorporating multiple instances of a web-like sub-complex (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The high-order rosette in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 16a is a special case, in that it requires hand- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Recall that adjacent circles with widely varying radii can produce dis- torted motifs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' To produce a satisfactory large rosette, we manually set the τ and α values for its cyclic polygon for a better fit with the surrounding geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Close examination of many of our results reveals small geometric discrepan- cies of the kind illustrated starkly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' When rosettes have arms that vary too dramatically in width or length, they disrupt the elegance of a pattern and the feeling of ‘inevitability’ in its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' There are several places in our work where we choose global constants like τ that produce acceptable results in general without adapting to the detailed geometry of local parts of individual designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The large rosette in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 16a gives one clear example of where local ad- justments can improve a design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' In future work, we would like to explore more fine-grained constructions that can enhance the quality of every rosette based on the configuration of the circle packing in its immediate neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 9 Conclusion We presented a robust method for constructing freeform Islamic geometric pat- terns comprising rosettes of unusual orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our technique relies on the theory of circle packings, giving us a principled geometric scaffolding from which to develop a polygonal patch and then motifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The user controls the initial com- plex and any gadgets in it, allowing for significant creative freedom in the design (b) (a)18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Lin and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Kaplan process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Our results are more organic and less repetitive than existing patterns and suggest many ideas for further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' We believe they communicate the aesthetic of Islamic geometric patterns while also interpreting them in a non- traditional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' They still manage to convey the ‘aesthetic delight’ that Gom- brich discussed [18], but with slightly less boredom and more confusion, paving the way for more artistic applications of these designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' The work enhances our understanding of traditional patterns and reveals new opportunities—freeform or otherwise—for both ornamentation and art-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' A highly structured composition (right) generated from a finite subset of a conceptually periodic complex (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Although the circle packing is not constructed in a toroidal domain as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 6, the resulting composition appears close to periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Acknowledgements This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Cheriton School of Computer Science at the Univer- sity of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' Al Ajlouni, The global long-range order of quasi-periodic patterns in islamic architecture, Acta Crystallographica Section A 68 (2012), pp.' 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+page_content=' Avnir, Continuous symmetry measures, Journal of the American Chemical Society 114 (1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} +page_content=' 7843–7851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfgfzo/content/2301.01471v1.pdf'} diff --git a/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/2301.13614v1.pdf.txt b/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/2301.13614v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9990adf62102653c51dfbbfc74d084ea2eb005b9 --- /dev/null +++ b/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/2301.13614v1.pdf.txt @@ -0,0 +1,1117 @@ +Synchronized states in dissipatively coupled harmonic oscillator networks +Juan N. Moreno,1, ∗ Christopher W. W¨achtler,1, 2, † and Alexander Eisfeld1, 3, ‡ +1Max Planck Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany +2Department of Physics, University of California, Berkeley, California 94720, USA +3Universit¨at Potsdam, Institut f¨ur Physik und Astronomie, +Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Deutschland +The question under which conditions oscillators with slightly different frequencies synchronize +appears in various settings. +We show that synchronization can be achieved even for harmonic +oscillators that are bilinearly coupled via a purely dissipative interaction. By appropriately tuned +gain/loss stable dynamics may be achieved where for the cases studied in this work all oscillators +are synchronized. These findings are interpreted using the complex eigenvalues and eigenvectors of +the non-Hermitian matrix describing the dynamics of the system. +I. +INTRODUCTION +Synchronization is a fascinating phenomenon, which +can be interpreted as a display of cooperative behavior +appearing in many complex systems [1, 2]. Since the first +observation by Huygens in the late 1600s [3], it has been +studied in diverse communities, where it plays an im- +portant role in our understanding for example in electric +networks in engineering, circadian rhythms in biology, +pattern formation in statistical mechanics, and chemical +reactions in chemistry [4–6]. By now, it is seen as a uni- +versal phenomenon that is important both in fundamen- +tal studies and in technical applications, ranging from +laser networks [7], to phase-locked loops [8], Josephson +junction arrays [9, 10], spin-torque resonators [11], and +power grids [12]. +Even today, the originally observed +phenomenon of clock synchronization remains a crucial +application for modern communication networks [13, 14]. +Typically synchronization is viewed in terms of the ad- +justment of rhythms of autonomous oscillators, which at- +tain stable periodic orbits without active regulation from +the outside [15] and thus require nonlinearities in the +governing equations of motion. Far less common is the +investigation of synchronization in models that are lin- +ear in both the oscillators and the couplings. Without +dissipation, coupled harmonic oscillators form collective +eigenmodes, where the individual oscillators perform mo- +tion with a fixed phase relation. However, a system not +initialized in an eigenmode usually stays in a superposi- +tion of several eigenmodes with different eigenfrequencies +resulting in a beating pattern. Moreover, if the number +of coupled oscillators is large, the system dynamics does +not need to exhibit perfect revivals in general and syn- +chronized motion is absent. +Hence in a closed system +of oscillators, only for an eigenmode as initial condition +one obtains a time-independent phase relation between +the oscillators. However if the system is not closed, but +subject to gain and loss, the open system dynamics allow +∗ moreno@pks.mpg.de +† cwwaechtler@berkeley.edu +‡ eisfeld@pks.mpg.des +for a situation where all eigenmodes but one are damped. +Then, synchronization is possible as long as the respec- +tive eigenstate is present in the initial state. However, in +order to achieve a situation where all but one mode are +damped, one needs to carefully balance gain and loss. +In contrast to a self-sustained system where the non- +linearity counteracts the dissipation (or gain) in order to +stabilize periodic orbits, a single linear harmonic oscilla- +tor only exhibits the following dynamics in the absence +of periodic driving: Either the dissipation exceeds the +gain, such that the amplitude of the dissipative systems +shrinks and eventually reaches a single point in phase +space, or in other way around, where the gain exceeds +the dissipation, the oscillation amplitude infinitely grows. +In the special case where both are equivalent the system +is effectively described by closed system dynamics with +infinitely many closed orbits in phase space depending +on the initial energy of system. However, when coupling +between linear oscillators are introduced, many more so- +lutions are possible. +Here, we investigate a network of linear harmonic os- +cillators subject to gain and loss. Generally, one would +consider each oscillator to couple to its own environment +and direct coupling between two or more entities in the +network. +However, a purely dissipative coupling leads +to intriguing phenomena also for self-sustained oscilla- +tors like for example oscillator death [1]. In our model +of linear oscillators, it allows for the emergence of dissi- +pation free subspaces in parameter space. Within these +subspaces we find periodic motion of all oscillators in the +network, that is starting from an (nearly) arbitrary initial +state the system reaches a regime during time propaga- +tion in which all oscillators exhibit synchronized motion +for a long time. At this point, let us specify the notion +of synchronization we use throughout this work: +— With ’long time’ we mean times long compared to the +eigenfrequencies of the individual oscillators and we fo- +cus on the case where all oscillators have small deviations +from a common ’mean frequency’. In the ideal case they +oscillate forever. +— With ’synchronoized’ we mean that the oscillators +have a fixed phase relation. +Ideally we want that all +oscillators have the same amplitude. If this is the case, +arXiv:2301.13614v1 [nlin.CD] 30 Jan 2023 + +2 +then we denote it as full synchronization. If the system +is not in a fully synchronized state, we will characterize +its degree of synchronization by a suitable measure. +— With ’arbitrary’ initial state we mean that for most +initial states synchronization is achieved, yet there exist +some special initial conditions that do not lead to syn- +chronization. +We +note +that +within +the +above +definitions +for +uncoupled oscillators one only finds synchronization, +when there is no gain and loss and all oscillators have +the same frequency. +The remainder of the paper is organized as follows: In +Sec. II A we summarize some general considerations of +synchronization for linearly coupled harmonic oscillators +important for our work, followed by the specific model +under investigation in Sec. II B. In the subsequent Sec. III +we discuss our results, which includes the special case of +two coupled oscillators in Sec. III A and the more general +case of many oscillators in Sec. III B. Finally, we conclude +in Sec. IV. +II. +MODEL AND BASIC FORMALISM +A. +General considerations of synchronization in +linear oscillator models +To introduce the basic concepts and notation, we con- +sider N harmonic oscillators in a network, each labeled +by a subscript n = 1,...,N. The motional state of each +oscillator is characterized by a time dependent complex +amplitude an(t) = ∣an(t)∣eiφn(t). If all oscillators in the +network oscillate with a common real frequency ωsyn +while their relative amplitudes remain constant, we will +refer to it as synchronization. Using a vector notation +⃗a(t) = [a1(t),...,aN(t)]⊺, such synchronized motion may +be expressed as +⃗a(t) = f(t)⃗asyne−iωsynt, +(1) +where f(t) is a real function that takes into account the +possibility that the amplitudes decay (or grow) over time, +which we will discuss in Sec. II B in more detail. In the +case of f(t) = 1 the motion represents a periodic steady +state, which we refer to as ideal synchronized motion. +The above notion is not sufficient to fully characterize +synchronized motion as for example a single oscillatory +site in the network (while all other oscillators are at rest) +also fulfills Eq. (1). It is thus necessary to also quan- +tify the degree of synchronization of a vector ⃗a, which we +denote by S(⃗a). To this end, we use the inverse partici- +pation ratio [16] +S(⃗a) = +1 +∑N +n=1 ∣an∣4 , +(2) +which takes values between 1 and N. Here, a value of +S = 1 corresponds to the aforementioned case of a single +oscillator in motion, whereas a value of S = N indicates +FIG. 1. +Illustration of potentially attainable synchronized +motion in a network of N = 3 oscillators. The inverse partici- +pation ratio S(⃗a) increases from top to bottom in accordance +with the transition from partially to fully synchronized mo- +tion. +fully synchronized motion, i.e. all nodes have the same +amplitude (without phase). Values of S = ˜N < N cor- +respond to partial synchronization of approximately ˜N +oscillators. In Fig. 1, we illustrate different degrees of +synchronization and their respective dynamics in a net- +work of three oscillators. +The time evolution of a linearly coupled network of +harmonic oscillators in the presence of gain and loss is +generally expressed as +d +dt⃗a = −iW ⃗a, +(3) +where we assume the non-Hermitian matrix W to be +time-independent. Then, the state of the system at time +t is simply given by +⃗a(t) = e−iW t⃗a(0), +(4) +where ⃗a(0) denotes the initial state at time t = 0. Thus, +the dynamics of the network is fully characterized by the +matrix W, in particular by its eigenvalues and eigenvec- +tors. Since W is (in general) non-Hermitian, there exist +right and left eigenvectors defined via +W ⃗cj =wj⃗cj +and +⃗z† +jW = ⃗z† +jwj. +(5) + +0003 +Here, † indicates the complex conjugated and transpose, +and the eigenvectors are normalized according to +⃗c† +j⃗cj = 1 +and +⃗z† +j′⃗cj = δj′j. +(6) +Note, that in general ⃗c† +j ≠ ⃗z† +j. The matrix W can now be +expressed as W = ∑j wj⃗cj ⃗z† +j, such that the time evolution +of Eq. (4) is conveniently given by +⃗a(t) = ∑ +j +⃗cje−iwjt ⃗z† +j⃗a(0), +(7) +where ⃗z† +j⃗a(0) is the initial weight of the eigenstate j. +While the real part of the complex eigenvalue wj deter- +mines the oscillation frequency of eigenmode j, the imag- +inary part Im[wj] determines whether the oscillatory mo- +tion is damped (Im[wj] < 0), growing (Im[wj] > 0) or +oscillates forever (Im[wj] = 0). +In order to obtain a time evolution of the form of +Eq. (1) with f(t) = 1 after some initial transient time, i.e. +dynamically reach the eigenstate with Im[wsync] = 0, the +initial state needs to have non-vanishing overlap with the +synchronized eigenstate [⃗z† +synca(0) ≠ 0]. Furthermore, all +other eigenstates present in the initial state need to have +Im[wj] < 0, such that they are damped. In the following, +we will therefore search for conditions and parameters +under which one eigenstate fulfills Im[wsync] = 0 while +all other eigenstates fulfill Im[wj] < 0. Subsequently, we +will characterize the degree of synchronization of the re- +sulting state in terms of S; cf. Eq. (2). +B. +Linear oscillators with purely dissipative +coupling +After +the general considerations of the previous +Sec. II A, let us now specify the network of interest +throughout the remainder of this work: The individual +oscillators have frequencies Ωn ∈ R and are arranged on +a ring. Each oscillator is subject to gain/loss mediated +via the rate γ ∈ R and interacts with its two nearest +neighbors via a purely dissipative coupling v ∈ R. For +simplicity we assume that the coupling and dissipation +is equal for all oscillators; we are interested in the pos- +sibility of synchronization when the frequency of each +oscillator is different, which corresponds to the notion of +synchronization as an adjustment of rhythms due to the +presence of interactions. The equation of motion of the +n-th oscillator is then given by +d +dtan =(−iΩn − γ)an − v(an+1 + an−1), +(8) +with a0 ≡ aN and aN+1 ≡ a1 to fulfill periodic boundary +conditions. Note that positive values of γ represent loss +whereas negative values correspond to gain. To simplify +notation we express all energies in units of v and take v +to be positive (the case of negative v will be discussed +later), i.e. ωn = Ωn/v, g = γ/v and τ = tv. Furthermore, +we parameterize the frequencies as ωn = ¯ω + ∆n. Then, +Eq. (8) becomes +d +dτ an =[−i(¯ω + ∆n) − g]an − (an+1 + an−1). +(9) +Our goal in the following is to determine the values of +g for a given set of frequency differences ∆n, such that +the oscillators perform synchronized motion in the sense +discussed in Sec. II A. +As the term (−i¯ω − g) is independent of the oscilla- +tor index n, it only trivially contributes to the overall +dynamics; specifically oscillations with frequency ¯ω and +damping/growing with rate g. In matrix representation, +Eq. (9) can be written in the form of Eq. (3) with t → τ +and W = (¯ω − ig)I + M, where +M = +⎛ +⎜⎜⎜⎜⎜ +⎝ +∆1 +−i +0 +... +−i +−i +∆2 +−i +... +0 +0 +−i +⋮ +−i +0 +... +−i ∆N +⎞ +⎟⎟⎟⎟⎟ +⎠ +(10) +Note, that the (left and right) eigensvectors of W and +M are identical and their eigenvalues are simply shifted, +i.e., if M⃗cj = λj⃗cj then W ⃗cj = wj⃗cj with +wj = ¯ω + Re[λj] + i(−g + Im[λj]), +v > 0. +(11) +Moreover, as M only depends on ∆n, the eigenvectors +and thus the degree of synchronization S(⃗c) is indepen- +dent of g. +Let us summarize the general conditions of the pre- +vious Sec. II A for synchronized motion tailored to the +specifics of our system discussed here: +(i) There exists a single eigenstate ⃗csync of W with +purely real eigenvalue. This corresponds to a state +⃗csync that fulfills −g+Im[λsync] = 0, where M⃗csync = +λsync⃗csync. +(ii) All other eigenstates of W have negative imaginary +part for the set of parameters determined in (i). +That corresponds to −g+Im[λj] < 0 for all j ≠ sync. +(iii) The synchronization measure S(⃗csync) should be as +large as possible. Ideally S(⃗csync) = N. +So far, we have taken v to be positive. For negative +values of v we define the scaled energies in terms of −v +such that ωn = ¯ω + ∆n = −Ωn/v, g = −γn/v, and τ = −tv. +Then, Eq. (9) becomes +d +dτ an = [−i(¯ω + ∆n) − g]an + (an+1 + an−1), +(12) +where the first term remains identical while the sign +changes in front of the oscillator couplings. As a result, +the eigenvalues of W [cf. Eqs. (10) and (11)] are given +by +wj = ¯ω + Re[λj] + i(−g − Im[λj]), +v < 0. +(13) + +4 +Here, the real part of the eigenvalues (as well as the cor- +responding eigenstates and thus the measure S) remains +unchanged, while the imaginary part simply changes its +sign. Thus, eigenstates that are decaying for v > 0, are +growing for v < 0 and vice versa. +III. +RESULTS +In the following we first discuss the case of N = 2 in +Sec. III A, which provides a clear picture of the basic +mechanism underlying synchronization of linear oscilla- +tors interacting via dissipative couplings. Subsequently +in Sec. III B, we consider a ring of N > 2 oscillators and +show that also in this case synchronized motion may be +achieved and follows similar arguments as before. +A. +Two coupled oscillators (N = 2) +Without loss of generality, we may choose the scaled +frequency differences of the two oscillators to be ∆1 = ++∆ and ∆2 = −∆, such that matrix M governing the +dynamics [cf. Eq. (10)] is given by +M = (∆ +−i +−i −∆) +(14) +Here, we have chosen v > 0. However, from the discussion +in Sec. II B we know that a negative value of v simply +results in a change of sign of the imaginary part of the +eigenvalues. The two eigenvalues and corresponding right +eigenvectors of M are given by +λ± = ± +√ +∆2 − 1 +(15) +⃗c± = +1 +√ +1 + ∣∆ ± +√ +∆2 − 1∣2 +(i(∆ ± +√ +∆2 − 1) +1 +) +(16) +If ∣∆∣ < 1 ( ∣∆∣ > 1) the eigenvalues λ± are both purely +imaginary (real) and non-degenerate. +In contrast, for +∆ = ±1 not only are the eigenstates degenerate but also +the corresponding eigenvectors coalesce, i.e., these values +of ∆ correspond to exceptional points. The impact of +exceptional points on synchronization goes beyond the +scope of the present work and we will focus in the follow- +ing on the cases ∣∆∣ > 1 and ∣∆∣ < 1. +a. +Overview: +As discussed in Sec. II B, the eigenen- +ergies w± = ¯ω+Re[λ±]+i(−g+Im[λ±]) describe the overall +possibility of long lasting synchronized motion in terms +of oscillation frequency and damping, while S quantifies +the degree of synchronization. Let us start by consid- +ering the imaginary part of the eigenenergies w± given +by Im[w±] = −g + Im[λ±], which determines the (expo- +nential) damping or growing. In Figs. 2 (a) and (b) we +show Im(w−) and Im(w+), respectively, as a function of +the frequency difference ∆ and the dissipation strength +g. Note, that ∆ as well as g can take on positive and +FIG. 2. Top row: Density plots of the imaginary part Im(w±) +as a function of the frequency difference ∆ and the dissipation +strength g: (a) w− and (b) w+. Dissipation-free synchroniza- +tion is found along the white line. Middle row: Correspond- +ing real part (c) Re(w−) and (d) Re(w+) as a function of ∆, +which corresponds to the oscillation frequency of the respec- +tive eigenvector. Last row: Degree of synchronization S as +function of ∆ of the eigenvalue (e) ⃗c− and (f) ⃗c+. The largest +value is found for ∣∆∣ < 1 corresponding to fully synchronized +motion. +negative values. The red areas in Fig. 2(a) and (b) indi- +cate positive values corresponding to amplitude growth +whereas the blue areas indicate negative values and thus +amplitude damping. The two regions are separated by +a white region, where amplitudes neither increase nor +decrease. We discuss this most relevant region for dissi- +pation free synchronization in more detail below. +As expected from the discussion above, quite different +behavior of Im[w±] is observed depending on whether +∣∆∣ > 1 or ∣∆∣ < 1. +Similarly, a pronounced difference +is found in the behavior of the real part Re[w±] = ¯ω + +Re[λ±], which describes the oscillation frequency of the +eigenmodes and is shown in Fig. 2(c) and (d). For ∣∆∣ < 1 +the frequency remains unchanged and both eigenstates +oscillate with the mean frequency ¯ω. However, for ∣∆∣ > 1 +the frequency of the − state [cf. Fig. 2(c)] is decreasing, +while that of the + state [cf. Fig. 2(d)] is increasing. Both +follow the functional form of a square-root with opposite +sign, cf. Eq. (15). Lastly, in Fig. 2(e) and (f) we show +the degree of synchronization S as function of ∆, which + +-state ++state +2 +2.0 +(a) +(b) +1 +1.0 +Im(W=) +0 +0.0 +-1 +-1.0 +-2 +-2.0 +2 +(c) +(d) +0 +-2 ++ +t +2.5 +(e) +(f) +2.0 +1.5 +1.0 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +V +V5 +is given by [cf. Eq. (16)] +S(⃗c±,∆) = { 2 +, ∣∆∣ < 1 +2 +∆2 +2∆2−1 +, ∣∆∣ > 1 . +(17) +As expected, the maximum value lies within the range +of ∣∆∣ < 1 and rapidly decreases as ∣∆∣ increases, indi- +cating the absence of synchronization. After this broad +overview we will in the following discuss in more detail +the potential of synchronized motion in the system of +N = 2 oscillators, focusing on the three criteria (i)–(iii) +formulated in Sec. II B. +b. +Detailed discussion of the regime ∣∆∣ > 1: +In this +case, the eigenvalues λ± become purely real [cf. Eq. (15)], +such that the eigenenergies take the simple form w± = +(¯ω± +√ +∆2 − 1)−ig. Most importantly, the imaginary part +is solely given by −g for both states and is independent +of ∆, which can also be seen in Figs. 2(a) and (b). Thus, +both eigenstates show the same dynamical response to +dissipation, i.e., either both are dissipation free (g = 0) or +the amplitudes decay/increase with the same rate given +by −g. Although there exists a dissipation free subspace +for g = 0, and thus requirement (i) is fulfilled, require- +ment (ii) cannot be fulfilled simultaneously. The reasons +is that both states have different oscillation frequencies +¯ω ± +√ +∆2 − 1 and none of them is decaying, resulting in a +beating pattern. We show an example of such a time evo- +lution of the real amplitudes Re(an) governed by Eq. (9) +in Fig. 3(a) for ∆ = 1.1 and g = 0. +c. +Detailed discussion of the regime ∣∆∣ < 1: +Af- +ter we have ruled out the possibility of synchroniza- +tion [according to our conditions (i)–(iii)] in the previ- +ous regime, we now discuss the case of ∣∆∣ < 1, where +dissipation free synchronized motion is indeed possible. +For ∣∆∣ < 1 the eigenvalues λ± are purely imaginary [cf. +Eq. (15)] and dissipation free states are determined by +0 = −g ± +√ +∣1 − ∆2∣, such that condition (i) may be ful- +filled. In contrast to the previous case, we need to differ- +entiate between the two states: Dissipation vanishes for +the + state if g = g+ ≡ +√ +∣1 − ∆2∣, and for the − state if +g = g− ≡ − +√ +∣1 − ∆2∣. Each of these solutions describes a +half circle with radius one, cf. Figs. 2(a) and (b). +We now examine whether condition (ii) is also ful- +filled in this regime. +When the − state is dissipation +free, the amplitude of the + state is growing exponen- +tially as Im[w+(g−)] = −g− + +√ +1 − ∆2 = 2 +√ +1 − ∆2 > 0. +This is also verified by Fig. 2: Along the white region +in panel (a) within the regime ∣∆∣ < 1, the area in panel +(b) is red. In contrast, along the white region in panel +(b), the area in panel (a) is blue, i.e. while the + state +is dissipation free, the − state is damped. Specifically, +Im[w−(g+)] = −g+ − +√ +1 − ∆2 = −2 +√ +1 − ∆2 < 0. +Thus, +synchronized motion for ∣∆∣ < 1 is found whenever the +condition g = +√ +1 − ∆2 is fulfilled. Moreover, this state +has a degree of synchronization of S = 2 and is therefore +fully synchronized for all ∣∆∣ < 1. +In Fig. 3(b) we show the dynamics for the parameters +∆ = 0.6 and g = 0.8 when starting in the initial state +FIG. 3. Examples of different dissipation free dynamics found +for the case of N = 2 oscillators. We plot the real amplitude +Re(an(τ)) of the first oscillator in red (n = 1) and the second +one in blue (n = 2). +(a) For ∆ = 1.1 and g = 0, the pres- +ence of two oscillation frequencies within the dissipation free +subspace leads to beating. (b) For ∆ = 0.6 and g = 0.8, only +a single eigenstate with its respective oscillation frequency is +dissipation free, while the other is damped leading to a pe- +riodic steady state of both oscillators, i.e., synchronization. +Parameters: +¯ω = 10, ⃗a(0) = (1, 0)⊺. +These results are ob- +tained by direct integration of the differential equation. +It +agrees perfectly with the results obtained via diagonalization. +⃗a(0) = (1,0)⊺. +As discussed previously, we expect to +find synchronized motion for these parameters. Indeed, +after a short transient time of τ ≳ 2 a stationary oscil- +latory motion emerges where both oscillators have the +same amplitude. Note the phase shift between the two +oscillators, which may be understood as follows: Consid- +ering the + state ⃗c+ [cf. Eq. (16)], the long time dynamics +is given by ⃗async(t) = ⃗c+ exp[−iω+t]; cf. Eq. (7). Then, +Re[⃗async(t)] =N (cos(ω+t + φ) +cos(ω+t) ), +(18) +where the phase difference φ fulfills tan(φ) = − +√ +1 − ∆2/∆ +and N = (1 + ∣∆ + +√ +∆2 − 1∣2)−1/2 is the normalization +constant from Eq. (16). +B. +Many coupled oscillators on a ring +In this section, we generalize our results from the pre- +vious Sec. III A for the case of two coupled oscillators to +large numbers of oscillators arranged on a ring. Also for +the case of N oscillators, the dynamics is governed by +Eqs. (9)–(11). In the following we will first discuss the +case of equal frequencies of all oscillators. Afterwards, we +discuss the more relevant case of frequency differences. + +5.0 +(a) +Re(ai) +Re(a2) +2.5 +Re(ai) +0.0 +-2.5 +-5.0 +2 +(b) +Re(ai) +Re(a2) +1 +Re(ai) +188888888888888 +0 +-1 +-2 +0 +2 +4 +6 +8 +10 +26 +1. +Identical frequencies of all oscillators +To gain a basic understanding of the eigenstates and +eigenvector structure we now consider the case when all +frequencies are identical, i.e. ∆n = ∆. Then, the eigen- +values and (right) eigenvectors of W are given by +wj = (¯ω + ∆) − i(g ± 2cos(2πj +N )), +v ≷ 0, +(19) +⃗cj = +1 +√ +N +N +∑ +n=1 +ei 2π +N jn⃗en, +(20) +where ⃗en is the nth unit-vector. As all eigenstates are +independent of ∆ or g. One sees that most eigenstates +are degenerate. +For even N only the eigenstates with +j = N and j = N/2 are not degenerate; for odd N only +the state with j = N is not degenerate. Moreover, the real +part of the eigenenergies wj, i.e. the oscillation frequen- +cies, is simply shifted by ∆ for all eigenstates. However, +the imaginary part of wj, which dictates the dissipation +and more importantly the possibility of dissipation free +dynamics, requires a more careful analysis. +a. +Positive v: +The imaginary part of the jth eigen- +value Im[wj] = 0 if g = gj ≡ −2cos(2πj/N). Then, all +other eigenvalues wj′ with j′ ≠ j have imaginary part +given by +Im[wj′(gj)] = 2cos(2πj +N ) − 2cos(2πj′ +N ). +(21) +Furthermore, we need to distinguish the two cases of +odd and even N: +For an odd number of oscillators +and j ≠ (N ± 1)/2 there is always at least one j′ with +Im[wj′(gj)] > 0, and thus condition (ii) is not fulfilled. +On the other hand, if j = (N ± 1)/2 all other eigenstates +are damped except for j′ = j ∓ 1. Yet, this state is also +dissipation free and condition (ii) cannot be fulfilled. For +even N, however, there exists a non-degenerate eigen- +state j = N/2 that fulfills (i) and (ii). Then, g = 2 and +⃗csyn ≡ ⃗cN/2 = +1 +√ +N (−1,1...,−1,1)⊺, which corresponds +to anti-phase synchronization between nearest neighbors +with the same frequency ¯ω + ∆. +b. +Negative v: +In contrast to the previous case. the +imaginary part of the jth eigenstate now is equal to zero +if g = gj ≡ +2cos(2πj/N) and thus Eq. (21) becomes +Im[wj′(gj)] = −2cos(2πj +N ) + 2cos(2πj′ +N ) +(22) +for all other eigenvalues wj′ with j′ ≠ j. Here, only if +j = N are all other states damped and conditions (i) +and (ii) fulfilled. The corresponding eigenstate is ⃗csyn ≡ +⃗cN = +1 +√ +N (1,...,1)⊺, i.e., in-phase synchronization of all +oscillators with frequency ¯ω + ∆. +2. +Oscillators with different frequencies +In this section, we discuss the case of arbitrary fre- +quency differences ∆n for each oscillator on the ring. In +this case, the matrix M [cf. +Eq. (10)] can no longer +be diagonalized analytically. Therefore, we discuss the +basic behavior along a few examples of ∆n and solve +the eigenvalue problem numerically. Yet, these examples +demonstrate that dissipation free synchronized motion +also exists in such a general setup. +A convenient way to investigate how the properties +of synchronization are affected by changes of ∆n, is to +parametrize the frequency difference according to +∆n = sn∆, +(23) +and analyze the behavior of the eigenvalues and eigen- +vectors of W as a function of ∆ for a given (and fixed) +set of sn. Furthermore, we choose v to be negative, such +that for ∆ = 0 there exists a fully synchronized eigenstate +if g = 2 (see the discussion in Sec. III B 1b). Note that a +negative value of v implies gj = Im[λj]. +In the following we consider as example the case of +N = 5 oscillators and show in Fig. 4 the results of the nu- +merical diagonalization of the matrix M for three differ- +ent realizations of ⃗s = (s1,...,s5) (different columns). We +choose the largest difference between neighboring values +of sn to be equal to one, i.e. max[sn−sn+1] = 1. Then, for +∆ < 1 all frequency differences between neighboring os- +cillators are always smaller than the dissipative coupling +between them (which has magnitude one). +The case of N = 2 in our network of oscillators allows us +to represent the full parameter space as shown in Fig. 2 +and identify the dissipation free subspaces and synchro- +nization within. However, for larger system sizes (as con- +sidered now) a representation similar to Fig. 2 becomes +very space consuming. Yet, a dissipation free subspace is +always necessary for synchronization, which corresponds +to the white lines in Figs. 2(a) and (b). +Thus, in or- +der to determine whether conditions (i)–(iii) are fulfilled, +it is sufficient to only search along the parameters for +which each eigenstate becomes dissipation free. In par- +ticular, the relevant information of Fig. 2(a) and (b) may +be conveniently combined to contain only g± = Im[λ±] as +function of ∆. Accordingly, the top row of Fig. 4 shows +the imaginary part of all eigenvalues Im[λj] as function +of the parameter ∆ and the middle row shows the re- +spective real parts Re[λj]. +Lastly, in the bottom row +we plot the degree of synchronization S of each eigen- +vector also as function of ∆. The eigenvalues of M are +sorted in descending order of their imaginary parts, i.e. +Im[λ1] > Im[λ2] > ⋅⋅⋅ > Im[λN]. +In the following we discuss different regimes of ∆ and +its impact on the possibility of synchronized motion in +accordance with conditions (i)–(iii). +We focus on the +eigenstate ⃗c1 with largest imaginary part Im[λ1] (high- +lighted as thick blue lines in Fig. 4). The reason is that +for g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free +while all other eigenstates are simultaneously damped. In +contrast, if we would choose g such that another eigen- +state ⃗cj≠1 would become dissipation free, there is at least +one eigenstate that is exponentially growing. It is thus + +7 +2 +1 +0 +1 +2 +Im( i) +1 +2 +3 +4 +5 +10 +5 +0 +5 +10 +Re( i) +0 +1 +2 +3 +4 +5 +1 +2 +3 +4 +5 +(ci) +s=[0.14, 0.2, 1.2, -0.46, -1.1] +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +s=[-0.24, -0.07, 0.93, -0.08, -0.54] +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +s=[-0.04, 0.22, 1.22, -0.77, -0.63] +FIG. 4. Examples of dissipation free and (fully) synchronized dynamics in a ring of N = 5 oscillators with random frequency +disorder. The three different columns correspond to three different set of (scaled) frequency realizations ⃗s. The value of v is +taken to be negative. In the top row we show the imaginary part Im[λj] of the eigenvalues λj of the matrix M as a function +∆. The middle row shows the corresponding real part Re[λj] and the bottom row the degree of synchronization S(⃗cj) of the +corresponding eigenstates ⃗cj. For all three considered realizations, there exists an eigenstate (blue) with the maximum value of +S (bottom row) for small values of ∆ ≲ 1. This eigenstate also has the largest imaginary part of its associated eigenvalue (top +row), which allows the tuning g in such a way that it becomes dissipation free while all other eigenstates are damped. +sufficient to only analyze the possibility of synchroniza- +tion of ⃗c1 in the following. +a. +No frequency difference (∆ = 0): +This means +that there are no variations in the oscillator frequen- +cies and the situation is exactly the same as discussed +in Sec. III B 1b. Consequently, the eigenvalues of W are +given by Eq. (19). From the discussion in Sec. III B 1b, +we know that if g = 2 = Im[λsyn] there exists a dissi- +pation free synchronized state ⃗csyn ≡ +1 +√ +5(1,...,1)⊺ with +associated real eigenvalue wsyn = ¯ω, i.e. all oscillators are +in phase and oscillate with frequency ¯ω. This is exactly +what we observe in Fig. 4: the eigenvalue with largest +imaginary part has imaginary part Im[λ1] = 2 (blue thick +lines in the top row). Note that Im[λ2] = Im[λ3] and +Im[λ4] = Im[λ5]. Furthermore, Re[λj] = 0 (middle row) +which implies an oscillation frequency of ¯ω. +b. +Small frequency differences (0 < ∆ < 1): +In this +regime, the disorder in the frequency differences between +nearest neighboring oscillators always remains smaller +than the coupling between them (which is 1). We thus +expect that the degree of synchronization also remains +large [S(⃗c1) ≈ N], i.e. the full delocalization of the eigen- +state ⃗c1 persists. In the bottom row of Fig. 4 we observe +exactly this behavior of the thick blue line correspond- +ing to ⃗c1: For small values of ∆, S(⃗c1) is maximal and +slowly decreases as ∆ approaches the value of 1. Thus, +the synchronized state remains close to be fully synchro- +nized within this regime [condition (iii)]. Note, that the +values for which S(⃗c1) starts to decrease depends on the +specific realization of disorder ⃗s. +The imaginary part of the corresponding eigenvalue +(top row) continues to be the largest value of all eigen- +values (blue thick line), Im[λ1] > Im[λj≠1]. +Thus, for +g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free +while all other eigenstates are damped, i.e. conditions (i) +and (ii) are fulfilled. As ∆ increases, Im[λ1] decreases +resulting from the larger amount of frequency disorder. +Simultaneously, the real part Re[λ1] remains close to 0 +such that the oscillation frequency of the synchronized +state ⃗c1 also continues to be close ¯ω. Note, the value of +Re[λ1] only affects the oscillation frequency. +c. +Large frequency differences (∆ ≥ 1): +As ∆ is +increased further, the frequency difference exceeds the +nearest neighbor interaction such that – similar to (An- +derson) localization in finite systems [17] – the degree of +synchronization S(⃗c1) of the previously delocalized eigen- +state ⃗c1 rapidly decreases as ∆ increases; see blue thick + +8 +lines in the bottom row of Fig. 4. Hence, only partial +synchronization is possible in this regime and condition +(iii) is not fulfilled. +At the same time, the largest imaginary value Im[λ1] +continues to decrease as function of ∆. +Yet, close to +∆ = 1 it remains well separated from the second largest +imaginary value Im[λ2] such that a suitable choice of g +still allows for dissipation free dynamics with a sinlge os- +cillation frequency. However, Im[λ1] may coalesce with +Im[λ2] for larger values of ∆ depending on the specific +realization of ⃗s. An example of such a degeneracy is ob- +served for ∆ ≈ 1.6 in the top right panel of Fig. 4. As a re- +sult, both eigenstates would be dissipation free resulting +in the beating pattern discussed previously in Sec. III A. +However, as mentioned above, only partial synchroniza- +tion is possible in this regime anyways. +d. +Very large frequency differences (∆ ≫ 1): +In the +regime of very large frequency differences, we expect that +the degree of synchronization takes its minimum value +S(⃗cj) = 1 for all eigenstates j since the scaling follows +∆ ≫ v. This implies that the values ∆n = ∆sn are much +larger than the dissipative coupling strength v. Then, M +is approximately diagonal with eigenvectors ⃗cj nearly lo- +calized. Note that in this limit there is no synchronized +state. +We have checked numerically that for ∆ larger +than the smallest difference between the sn the synchro- +nization measure of all eigenstates approaches one, as +expected (not shown here). +Lastly, to demonstrate that the dynamics of the sys- +tem of oscillators is consistent with our discussion of +the different regimes above (obtained from analyzing the +eigenvectors and eigenfrequencies), we show in Fig. 5 ex- +amples of Re[an(τ)] as a function of the scaled time τ +for ⃗s = (1.14,0.20,1.20,−0.46,−1.1) (corresponding to the +first column of Fig. 4) for three different values of ∆. In +all cases, we choose the initial state ⃗a0 = (1,1,2,−1,−1). +Panel (a) corresponds to the case of vanishing fre- +quency difference, i.e. ∆ = 0. We choose the dissipation +g = 2 such that only the eigenstate with largest imagi- +nary part is dissipation free. As expected after a short +transient time of τ ≈ 2.5 all oscillators are in-phase syn- +chronized. +In panel (b), we increase the frequency difference to +be ∆ = 0.5. Hence, the synchronized state is dissipation +free for g = 1.91. +Analogues to the previous case (a), +all oscillators are synchronized after a transient time of +τ ≈ 2.5, yet with a small phase shift. Importantly, all +oscillators have the same amplitude consistent with the +finding of Fig. 4 that the degree of synchronization is +maximal [S(⃗c1) = 5 for this value of ∆]. +Contrarily, in panel (c) where ∆ = 1.1 (and g = 1.51 +to match the condition of dissipation free dynamics) the +amplitudes vary among the oscillators. This is in accor- +dance with S(⃗c1) < 5. However, still only a single oscil- +lation frequency is present (after some transient time). +This is an example of partial synchronization. +FIG. 5. Dynamical behavior of Re(ai(τ)) given by Eq.(14) +for different values of the scaling factor ∆. +In all three +cases the mean frequency of the oscillators is ¯ω = 10 and +the disorder is the same of the first panel of Fig. (4), namely +⃗s = (1.14, 0.20, 1.20, −0.46, −1.1) The coupling strength v +is taken to be negative and all frequencies are given in units +of ∣v∣. The initial condition is ⃗a0 = (1, 1, 2, −1, −1). Panels (a) +and (b) show fully synchronized motion, while panel (c) is an +example of partial synchronization. +IV. +CONCLUSIONS +In this work we have investigated the possibility of +long-lived synchronized motion in networks of harmonic +oscillators, which are subject to gain/loss and interact +via nearest neighbor dissipative couplings. In this con- +text, we refer to synchronization as the existence of a +single eigenstate of the dynamical matrix, which is dis- +sipation free. +Furthermore, if it attains the maximum +value of the (inverse) participation ratio we refer to it +as ‘fully synchronized’. We find that in the case of only +two coupled oscillators, synchronization may always be +achieved by tuning the gain appropriately as long as the +frequency difference between the two oscillators is smaller +than their interaction strength. +A similar behavior may be observed in larger net- +works, i.e. many oscillators arranged on a ring with near- +est neighbor interactions, yet the possibility of synchro- +nization then depends on the specifics of the system at +hand: If all oscillators are identical, synchronized col- + +1 +2 +3 +4 +5 +2 +(a) +A=0 g=2 +Re(ai) +2 +(b) +A=0.5 +Re( +3 +A=1.1 g=1.51 +2 +Re(ai) +0 +-2 +二 +0 +2 +4 +6 +8 +10 +12 +149 +lective motion may be achieved for an even number of +sites with repulsive dissipative couplings (v positive) or +an odd number of sites with attractive dissipative in- +teractions (v negative). For small frequency differences +compared to the coupling between the oscillators, this +behavior remains, which we show specifically for the case +of N = 5, yet it should also hold for larger networks. +However, as the number of coupled oscillators increases, +it becomes increasingly difficult to achieve full synchro- +nization and may only be observed for very small fre- +quency differences. For larger frequency differences, the +(inverse) participation ratio decreases significantly such +that only partial synchronization may be achieved. This +is in accordance with Anderson localization, where on- +site disorder results in localized eigenstates. +However, +as the dynamical matrix in this work is non-Hermitian, +Anderson localization is not directly applicable. +Here, +future work is needed to study the interplay of synchro- +nization and localization, in particular in the thermody- +namic limit and arbitrary small frequency perturbations. +Synchronization as discussed in this work is intimately +related to the existence of dissipation free dynamics and +thus isolated points/submanifolds in parameter space. +Hence, they require a very precise tuning of gain and +loss in order to obtain periodic steady states. +This is +however hard to achieve in any realistic experiment and +the synchronized state will experience some gain or loss. +We can relax the condition Im[wj] = 0 by solely requir- +ing ∣Im[wj]∣ ≪ ∣Re[wj]∣, which means that the change of +amplitude of oscillation is small over many oscillations. +In addition, we then require Im[wj] ≪ Im[wsync], which +means that all other eigenstates decay much faster than +the ’synchronized’ one. In principle, one may relax the +condition even further and demand that there exists only +one state with Im[wj] > 0, while all other states fulfill +Im[wi] ≤ 0. +Then the synchronized state would grow +while all other states are exponentially damped. +ACKNOWLEDGMENTS +C.W.W. acknowledges support from the Max-Planck +Gesellschaft via the MPI-PKS Next Step fellowship and is +financially supported by the Deutsche Forschungsgemein- +schaft (DFG, German Research Foundation) – Project +No. 496502542 (WA 5170/1-1). A.E. acknowledges sup- +port from the DFG via a Heisenberg fellowship (Grant +No EI 872/10-1). +[1] A. Pikovsky, J. Kurths, M. Rosenblum, and J. Kurths, +Synchronization: a universal concept in nonlinear sci- +ences (Cambridge university press, Cambridge Univer- +sity Press, 2003). +[2] S. H. Strogatz, Nonlinear dynamics and chaos: with ap- +plications to physics, biology, chemistry, and engineering +(CRC Press, 2018). +[3] M. Bennett, M. F. Schatz, H. Rockwood, and K. Wiesen- +feld, Proc. R. Soc. Lond. A 458, 563 (2002). +[4] S. H. Strogatz and I. Stewart, Sci. Am. 269, 102 (1993). +[5] M. Rosenblum and A. Pikovsky, Contemp. Phys. 44, 401 +(2003). +[6] A. Arenas, A. D´ıaz-Guilera, J. Kurths, Y. Moreno, and +C. Zhou, Physics reports 469, 93 (2008). +[7] K. Thornburg, M. M¨oller, R. Roy, T. Carr, R.-D. Li, and +T. Erneux, Phys. Rev. E 55, 3865 (1997). +[8] J. J. Lynch and R. A. York, IEEE Microw. Guide Wave +Lett. 5, 213 (1995). +[9] A. +Cawthorne, +P. +Barbara, +S. +Shitov, +C. +Lobb, +K. Wiesenfeld, and A. Zangwill, Phys. Rev. B 60, 7575 +(1999). +[10] R. Fazio and H. Van Der Zant, Phys. Rep. 355, 235 +(2001). +[11] A. Slavin, Nature Nanotech. 4, 479 (2009). +[12] T. Nishikawa and A. E. Motter, New J. Phys. 17, 015012 +(2015). +[13] J. C. Bellamy, IEEE Commun. Mag. 33, 70 (1995). +[14] L. Narula and T. E. Humphreys, IEEE J. Sel. Top. Signal +Process. 12, 749 (2018). +[15] A. Jenkins, Phys. Rep. 525, 167 (2013). +[16] B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 +(1993). +[17] S. M¨obius, S. Vlaming, V. Malyshev, J. Knoester, and +A. Eisfeld, arXiv:1404.4475 [cond-mat.dis-nn] (2014). + diff --git a/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/load_file.txt b/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd39502c5e4606b3aa2e01fd9eafccd671647276 --- /dev/null +++ b/8tFRT4oBgHgl3EQfpzcC/content/tmp_files/load_file.txt @@ -0,0 +1,615 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf,len=614 +page_content='Synchronized states in dissipatively coupled harmonic oscillator networks Juan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Moreno,1, ∗ Christopher W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' W¨achtler,1, 2, † and Alexander Eisfeld1, 3, ‡ 1Max Planck Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 38, 01187 Dresden, Germany 2Department of Physics, University of California, Berkeley, California 94720, USA 3Universit¨at Potsdam, Institut f¨ur Physik und Astronomie, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 24-25, 14476 Potsdam, Deutschland The question under which conditions oscillators with slightly different frequencies synchronize appears in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We show that synchronization can be achieved even for harmonic oscillators that are bilinearly coupled via a purely dissipative interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' By appropriately tuned gain/loss stable dynamics may be achieved where for the cases studied in this work all oscillators are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' These findings are interpreted using the complex eigenvalues and eigenvectors of the non-Hermitian matrix describing the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' INTRODUCTION Synchronization is a fascinating phenomenon, which can be interpreted as a display of cooperative behavior appearing in many complex systems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Since the first observation by Huygens in the late 1600s [3], it has been studied in diverse communities, where it plays an im- portant role in our understanding for example in electric networks in engineering, circadian rhythms in biology, pattern formation in statistical mechanics, and chemical reactions in chemistry [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' By now, it is seen as a uni- versal phenomenon that is important both in fundamen- tal studies and in technical applications, ranging from laser networks [7], to phase-locked loops [8], Josephson junction arrays [9, 10], spin-torque resonators [11], and power grids [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Even today, the originally observed phenomenon of clock synchronization remains a crucial application for modern communication networks [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Typically synchronization is viewed in terms of the ad- justment of rhythms of autonomous oscillators, which at- tain stable periodic orbits without active regulation from the outside [15] and thus require nonlinearities in the governing equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Far less common is the investigation of synchronization in models that are lin- ear in both the oscillators and the couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Without dissipation, coupled harmonic oscillators form collective eigenmodes, where the individual oscillators perform mo- tion with a fixed phase relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, a system not initialized in an eigenmode usually stays in a superposi- tion of several eigenmodes with different eigenfrequencies resulting in a beating pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Moreover, if the number of coupled oscillators is large, the system dynamics does not need to exhibit perfect revivals in general and syn- chronized motion is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Hence in a closed system of oscillators, only for an eigenmode as initial condition one obtains a time-independent phase relation between the oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However if the system is not closed, but subject to gain and loss, the open system dynamics allow ∗ moreno@pks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='de † cwwaechtler@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='edu ‡ eisfeld@pks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='des for a situation where all eigenmodes but one are damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, synchronization is possible as long as the respec- tive eigenstate is present in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, in order to achieve a situation where all but one mode are damped, one needs to carefully balance gain and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In contrast to a self-sustained system where the non- linearity counteracts the dissipation (or gain) in order to stabilize periodic orbits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' a single linear harmonic oscilla- tor only exhibits the following dynamics in the absence of periodic driving: Either the dissipation exceeds the gain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' such that the amplitude of the dissipative systems shrinks and eventually reaches a single point in phase space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' or in other way around,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' where the gain exceeds the dissipation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' the oscillation amplitude infinitely grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the special case where both are equivalent the system is effectively described by closed system dynamics with infinitely many closed orbits in phase space depending on the initial energy of system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, when coupling between linear oscillators are introduced, many more so- lutions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Here, we investigate a network of linear harmonic os- cillators subject to gain and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Generally, one would consider each oscillator to couple to its own environment and direct coupling between two or more entities in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, a purely dissipative coupling leads to intriguing phenomena also for self-sustained oscilla- tors like for example oscillator death [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In our model of linear oscillators, it allows for the emergence of dissi- pation free subspaces in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Within these subspaces we find periodic motion of all oscillators in the network, that is starting from an (nearly) arbitrary initial state the system reaches a regime during time propaga- tion in which all oscillators exhibit synchronized motion for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' At this point, let us specify the notion of synchronization we use throughout this work: — With ’long time’ we mean times long compared to the eigenfrequencies of the individual oscillators and we fo- cus on the case where all oscillators have small deviations from a common ’mean frequency’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the ideal case they oscillate forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' — With ’synchronoized’ we mean that the oscillators have a fixed phase relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Ideally we want that all oscillators have the same amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' If this is the case, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='13614v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='CD] 30 Jan 2023 2 then we denote it as full synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' If the system is not in a fully synchronized state, we will characterize its degree of synchronization by a suitable measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' — With ’arbitrary’ initial state we mean that for most initial states synchronization is achieved, yet there exist some special initial conditions that do not lead to syn- chronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We note that within the above definitions for uncoupled oscillators one only finds synchronization, when there is no gain and loss and all oscillators have the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The remainder of the paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II A we summarize some general considerations of synchronization for linearly coupled harmonic oscillators important for our work, followed by the specific model under investigation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the subsequent Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III we discuss our results, which includes the special case of two coupled oscillators in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III A and the more general case of many oscillators in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Finally, we conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' MODEL AND BASIC FORMALISM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' General considerations of synchronization in linear oscillator models To introduce the basic concepts and notation, we con- sider N harmonic oscillators in a network, each labeled by a subscript n = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The motional state of each oscillator is characterized by a time dependent complex amplitude an(t) = ∣an(t)∣eiφn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' If all oscillators in the network oscillate with a common real frequency ωsyn while their relative amplitudes remain constant, we will refer to it as synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Using a vector notation ⃗a(t) = [a1(t),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',aN(t)]⊺, such synchronized motion may be expressed as ⃗a(t) = f(t)⃗asyne−iωsynt, (1) where f(t) is a real function that takes into account the possibility that the amplitudes decay (or grow) over time, which we will discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II B in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the case of f(t) = 1 the motion represents a periodic steady state, which we refer to as ideal synchronized motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The above notion is not sufficient to fully characterize synchronized motion as for example a single oscillatory site in the network (while all other oscillators are at rest) also fulfills Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' It is thus necessary to also quan- tify the degree of synchronization of a vector ⃗a, which we denote by S(⃗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' To this end, we use the inverse partici- pation ratio [16] S(⃗a) = 1 ∑N n=1 ∣an∣4 , (2) which takes values between 1 and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Here, a value of S = 1 corresponds to the aforementioned case of a single oscillator in motion, whereas a value of S = N indicates FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Illustration of potentially attainable synchronized motion in a network of N = 3 oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The inverse partici- pation ratio S(⃗a) increases from top to bottom in accordance with the transition from partially to fully synchronized mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' fully synchronized motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' all nodes have the same amplitude (without phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Values of S = ˜N < N cor- respond to partial synchronization of approximately ˜N oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 1, we illustrate different degrees of synchronization and their respective dynamics in a net- work of three oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The time evolution of a linearly coupled network of harmonic oscillators in the presence of gain and loss is generally expressed as d dt⃗a = −iW ⃗a, (3) where we assume the non-Hermitian matrix W to be time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, the state of the system at time t is simply given by ⃗a(t) = e−iW t⃗a(0), (4) where ⃗a(0) denotes the initial state at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, the dynamics of the network is fully characterized by the matrix W, in particular by its eigenvalues and eigenvec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Since W is (in general) non-Hermitian, there exist right and left eigenvectors defined via W ⃗cj =wj⃗cj and ⃗z† jW = ⃗z† jwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (5) 0003 Here, † indicates the complex conjugated and transpose, and the eigenvectors are normalized according to ⃗c† j⃗cj = 1 and ⃗z† j′⃗cj = δj′j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (6) Note, that in general ⃗c† j ≠ ⃗z† j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The matrix W can now be expressed as W = ∑j wj⃗cj ⃗z† j, such that the time evolution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (4) is conveniently given by ⃗a(t) = ∑ j ⃗cje−iwjt ⃗z† j⃗a(0), (7) where ⃗z† j⃗a(0) is the initial weight of the eigenstate j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' While the real part of the complex eigenvalue wj deter- mines the oscillation frequency of eigenmode j, the imag- inary part Im[wj] determines whether the oscillatory mo- tion is damped (Im[wj] < 0), growing (Im[wj] > 0) or oscillates forever (Im[wj] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In order to obtain a time evolution of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (1) with f(t) = 1 after some initial transient time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' dynamically reach the eigenstate with Im[wsync] = 0, the initial state needs to have non-vanishing overlap with the synchronized eigenstate [⃗z† synca(0) ≠ 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Furthermore, all other eigenstates present in the initial state need to have Im[wj] < 0, such that they are damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the following, we will therefore search for conditions and parameters under which one eigenstate fulfills Im[wsync] = 0 while all other eigenstates fulfill Im[wj] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Subsequently, we will characterize the degree of synchronization of the re- sulting state in terms of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Linear oscillators with purely dissipative coupling After the general considerations of the previous Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II A, let us now specify the network of interest throughout the remainder of this work: The individual oscillators have frequencies Ωn ∈ R and are arranged on a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Each oscillator is subject to gain/loss mediated via the rate γ ∈ R and interacts with its two nearest neighbors via a purely dissipative coupling v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For simplicity we assume that the coupling and dissipation is equal for all oscillators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' we are interested in the pos- sibility of synchronization when the frequency of each oscillator is different, which corresponds to the notion of synchronization as an adjustment of rhythms due to the presence of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The equation of motion of the n-th oscillator is then given by d dtan =(−iΩn − γ)an − v(an+1 + an−1), (8) with a0 ≡ aN and aN+1 ≡ a1 to fulfill periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note that positive values of γ represent loss whereas negative values correspond to gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' To simplify notation we express all energies in units of v and take v to be positive (the case of negative v will be discussed later), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' ωn = Ωn/v, g = γ/v and τ = tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Furthermore, we parameterize the frequencies as ωn = ¯ω + ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (8) becomes d dτ an =[−i(¯ω + ∆n) − g]an − (an+1 + an−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (9) Our goal in the following is to determine the values of g for a given set of frequency differences ∆n, such that the oscillators perform synchronized motion in the sense discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As the term (−i¯ω − g) is independent of the oscilla- tor index n, it only trivially contributes to the overall dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' specifically oscillations with frequency ¯ω and damping/growing with rate g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In matrix representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (9) can be written in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (3) with t → τ and W = (¯ω − ig)I + M, where M = ⎛ ⎜⎜⎜⎜⎜ ⎝ ∆1 −i 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' −i −i ∆2 −i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 0 0 −i ⋮ −i 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' −i ∆N ⎞ ⎟⎟⎟⎟⎟ ⎠ (10) Note, that the (left and right) eigensvectors of W and M are identical and their eigenvalues are simply shifted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=', if M⃗cj = λj⃗cj then W ⃗cj = wj⃗cj with wj = ¯ω + Re[λj] + i(−g + Im[λj]), v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (11) Moreover, as M only depends on ∆n, the eigenvectors and thus the degree of synchronization S(⃗c) is indepen- dent of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Let us summarize the general conditions of the pre- vious Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II A for synchronized motion tailored to the specifics of our system discussed here: (i) There exists a single eigenstate ⃗csync of W with purely real eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This corresponds to a state ⃗csync that fulfills −g+Im[λsync] = 0, where M⃗csync = λsync⃗csync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (ii) All other eigenstates of W have negative imaginary part for the set of parameters determined in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' That corresponds to −g+Im[λj] < 0 for all j ≠ sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (iii) The synchronization measure S(⃗csync) should be as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Ideally S(⃗csync) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' So far, we have taken v to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For negative values of v we define the scaled energies in terms of −v such that ωn = ¯ω + ∆n = −Ωn/v, g = −γn/v, and τ = −tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (9) becomes d dτ an = [−i(¯ω + ∆n) − g]an + (an+1 + an−1), (12) where the first term remains identical while the sign changes in front of the oscillator couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As a result, the eigenvalues of W [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (10) and (11)] are given by wj = ¯ω + Re[λj] + i(−g − Im[λj]), v < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (13) 4 Here, the real part of the eigenvalues (as well as the cor- responding eigenstates and thus the measure S) remains unchanged, while the imaginary part simply changes its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, eigenstates that are decaying for v > 0, are growing for v < 0 and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' RESULTS In the following we first discuss the case of N = 2 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III A, which provides a clear picture of the basic mechanism underlying synchronization of linear oscilla- tors interacting via dissipative couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Subsequently in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III B, we consider a ring of N > 2 oscillators and show that also in this case synchronized motion may be achieved and follows similar arguments as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Two coupled oscillators (N = 2) Without loss of generality, we may choose the scaled frequency differences of the two oscillators to be ∆1 = +∆ and ∆2 = −∆, such that matrix M governing the dynamics [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (10)] is given by M = (∆ −i −i −∆) (14) Here, we have chosen v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, from the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II B we know that a negative value of v simply results in a change of sign of the imaginary part of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The two eigenvalues and corresponding right eigenvectors of M are given by λ± = ± √ ∆2 − 1 (15) ⃗c± = 1 √ 1 + ∣∆ ± √ ∆2 − 1∣2 (i(∆ ± √ ∆2 − 1) 1 ) (16) If ∣∆∣ < 1 ( ∣∆∣ > 1) the eigenvalues λ± are both purely imaginary (real) and non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In contrast, for ∆ = ±1 not only are the eigenstates degenerate but also the corresponding eigenvectors coalesce, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=', these values of ∆ correspond to exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The impact of exceptional points on synchronization goes beyond the scope of the present work and we will focus in the follow- ing on the cases ∣∆∣ > 1 and ∣∆∣ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Overview: As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II B, the eigenen- ergies w± = ¯ω+Re[λ±]+i(−g+Im[λ±]) describe the overall possibility of long lasting synchronized motion in terms of oscillation frequency and damping, while S quantifies the degree of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Let us start by consid- ering the imaginary part of the eigenenergies w± given by Im[w±] = −g + Im[λ±], which determines the (expo- nential) damping or growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2 (a) and (b) we show Im(w−) and Im(w+), respectively, as a function of the frequency difference ∆ and the dissipation strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note, that ∆ as well as g can take on positive and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Top row: Density plots of the imaginary part Im(w±) as a function of the frequency difference ∆ and the dissipation strength g: (a) w− and (b) w+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Dissipation-free synchroniza- tion is found along the white line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Middle row: Correspond- ing real part (c) Re(w−) and (d) Re(w+) as a function of ∆, which corresponds to the oscillation frequency of the respec- tive eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Last row: Degree of synchronization S as function of ∆ of the eigenvalue (e) ⃗c− and (f) ⃗c+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The largest value is found for ∣∆∣ < 1 corresponding to fully synchronized motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The red areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(a) and (b) indi- cate positive values corresponding to amplitude growth whereas the blue areas indicate negative values and thus amplitude damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The two regions are separated by a white region, where amplitudes neither increase nor decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We discuss this most relevant region for dissi- pation free synchronization in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As expected from the discussion above, quite different behavior of Im[w±] is observed depending on whether ∣∆∣ > 1 or ∣∆∣ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Similarly, a pronounced difference is found in the behavior of the real part Re[w±] = ¯ω + Re[λ±], which describes the oscillation frequency of the eigenmodes and is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For ∣∆∣ < 1 the frequency remains unchanged and both eigenstates oscillate with the mean frequency ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, for ∣∆∣ > 1 the frequency of the − state [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(c)] is decreasing, while that of the + state [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(d)] is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Both follow the functional form of a square-root with opposite sign, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Lastly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(e) and (f) we show the degree of synchronization S as function of ∆, which state +state 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 (a) (b) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 Im(W=) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 2 (c) (d) 0 2 + t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 (e) (f) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 2 1 0 1 2 2 1 0 1 2 V V5 is given by [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (16)] S(⃗c±,∆) = { 2 , ∣∆∣ < 1 2 ∆2 2∆2−1 , ∣∆∣ > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (17) As expected, the maximum value lies within the range of ∣∆∣ < 1 and rapidly decreases as ∣∆∣ increases, indi- cating the absence of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' After this broad overview we will in the following discuss in more detail the potential of synchronized motion in the system of N = 2 oscillators, focusing on the three criteria (i)–(iii) formulated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Detailed discussion of the regime ∣∆∣ > 1: In this case, the eigenvalues λ± become purely real [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (15)], such that the eigenenergies take the simple form w± = (¯ω± √ ∆2 − 1)−ig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Most importantly, the imaginary part is solely given by −g for both states and is independent of ∆, which can also be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, both eigenstates show the same dynamical response to dissipation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=', either both are dissipation free (g = 0) or the amplitudes decay/increase with the same rate given by −g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Although there exists a dissipation free subspace for g = 0, and thus requirement (i) is fulfilled, require- ment (ii) cannot be fulfilled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The reasons is that both states have different oscillation frequencies ¯ω ± √ ∆2 − 1 and none of them is decaying, resulting in a beating pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We show an example of such a time evo- lution of the real amplitudes Re(an) governed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (9) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 3(a) for ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Detailed discussion of the regime ∣∆∣ < 1: Af- ter we have ruled out the possibility of synchroniza- tion [according to our conditions (i)–(iii)] in the previ- ous regime, we now discuss the case of ∣∆∣ < 1, where dissipation free synchronized motion is indeed possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For ∣∆∣ < 1 the eigenvalues λ± are purely imaginary [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (15)] and dissipation free states are determined by 0 = −g ± √ ∣1 − ∆2∣, such that condition (i) may be ful- filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In contrast to the previous case, we need to differ- entiate between the two states: Dissipation vanishes for the + state if g = g+ ≡ √ ∣1 − ∆2∣, and for the − state if g = g− ≡ − √ ∣1 − ∆2∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Each of these solutions describes a half circle with radius one, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We now examine whether condition (ii) is also ful- filled in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' When the − state is dissipation free, the amplitude of the + state is growing exponen- tially as Im[w+(g−)] = −g− + √ 1 − ∆2 = 2 √ 1 − ∆2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is also verified by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2: Along the white region in panel (a) within the regime ∣∆∣ < 1, the area in panel (b) is red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In contrast, along the white region in panel (b), the area in panel (a) is blue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' while the + state is dissipation free, the − state is damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Specifically, Im[w−(g+)] = −g+ − √ 1 − ∆2 = −2 √ 1 − ∆2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, synchronized motion for ∣∆∣ < 1 is found whenever the condition g = √ 1 − ∆2 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Moreover, this state has a degree of synchronization of S = 2 and is therefore fully synchronized for all ∣∆∣ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 3(b) we show the dynamics for the parameters ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='6 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='8 when starting in the initial state FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Examples of different dissipation free dynamics found for the case of N = 2 oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We plot the real amplitude Re(an(τ)) of the first oscillator in red (n = 1) and the second one in blue (n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (a) For ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1 and g = 0, the pres- ence of two oscillation frequencies within the dissipation free subspace leads to beating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (b) For ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='6 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='8, only a single eigenstate with its respective oscillation frequency is dissipation free, while the other is damped leading to a pe- riodic steady state of both oscillators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=', synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Parameters: ¯ω = 10, ⃗a(0) = (1, 0)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' These results are ob- tained by direct integration of the differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' It agrees perfectly with the results obtained via diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' ⃗a(0) = (1,0)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As discussed previously, we expect to find synchronized motion for these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Indeed, after a short transient time of τ ≳ 2 a stationary oscil- latory motion emerges where both oscillators have the same amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note the phase shift between the two oscillators, which may be understood as follows: Consid- ering the + state ⃗c+ [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (16)], the long time dynamics is given by ⃗async(t) = ⃗c+ exp[−iω+t];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, Re[⃗async(t)] =N (cos(ω+t + φ) cos(ω+t) ), (18) where the phase difference φ fulfills tan(φ) = − √ 1 − ∆2/∆ and N = (1 + ∣∆ + √ ∆2 − 1∣2)−1/2 is the normalization constant from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Many coupled oscillators on a ring In this section, we generalize our results from the pre- vious Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III A for the case of two coupled oscillators to large numbers of oscillators arranged on a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Also for the case of N oscillators, the dynamics is governed by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (9)–(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the following we will first discuss the case of equal frequencies of all oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Afterwards, we discuss the more relevant case of frequency differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 (a) Re(ai) Re(a2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 Re(ai) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='0 2 (b) Re(ai) Re(a2) 1 Re(ai) 188888888888888 0 1 2 0 2 4 6 8 10 26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Identical frequencies of all oscillators To gain a basic understanding of the eigenstates and eigenvector structure we now consider the case when all frequencies are identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' ∆n = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, the eigen- values and (right) eigenvectors of W are given by wj = (¯ω + ∆) − i(g ± 2cos(2πj N )), v ≷ 0, (19) ⃗cj = 1 √ N N ∑ n=1 ei 2π N jn⃗en, (20) where ⃗en is the nth unit-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As all eigenstates are independent of ∆ or g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' One sees that most eigenstates are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For even N only the eigenstates with j = N and j = N/2 are not degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' for odd N only the state with j = N is not degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Moreover, the real part of the eigenenergies wj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' the oscillation frequen- cies, is simply shifted by ∆ for all eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, the imaginary part of wj, which dictates the dissipation and more importantly the possibility of dissipation free dynamics, requires a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Positive v: The imaginary part of the jth eigen- value Im[wj] = 0 if g = gj ≡ −2cos(2πj/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, all other eigenvalues wj′ with j′ ≠ j have imaginary part given by Im[wj′(gj)] = 2cos(2πj N ) − 2cos(2πj′ N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (21) Furthermore, we need to distinguish the two cases of odd and even N: For an odd number of oscillators and j ≠ (N ± 1)/2 there is always at least one j′ with Im[wj′(gj)] > 0, and thus condition (ii) is not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' On the other hand, if j = (N ± 1)/2 all other eigenstates are damped except for j′ = j ∓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Yet, this state is also dissipation free and condition (ii) cannot be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For even N, however, there exists a non-degenerate eigen- state j = N/2 that fulfills (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, g = 2 and ⃗csyn ≡ ⃗cN/2 = 1 √ N (−1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',−1,1)⊺, which corresponds to anti-phase synchronization between nearest neighbors with the same frequency ¯ω + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Negative v: In contrast to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' the imaginary part of the jth eigenstate now is equal to zero if g = gj ≡ +2cos(2πj/N) and thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (21) becomes Im[wj′(gj)] = −2cos(2πj N ) + 2cos(2πj′ N ) (22) for all other eigenvalues wj′ with j′ ≠ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Here, only if j = N are all other states damped and conditions (i) and (ii) fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The corresponding eigenstate is ⃗csyn ≡ ⃗cN = 1 √ N (1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',1)⊺, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=', in-phase synchronization of all oscillators with frequency ¯ω + ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Oscillators with different frequencies In this section, we discuss the case of arbitrary fre- quency differences ∆n for each oscillator on the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In this case, the matrix M [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (10)] can no longer be diagonalized analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Therefore, we discuss the basic behavior along a few examples of ∆n and solve the eigenvalue problem numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Yet, these examples demonstrate that dissipation free synchronized motion also exists in such a general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' A convenient way to investigate how the properties of synchronization are affected by changes of ∆n, is to parametrize the frequency difference according to ∆n = sn∆, (23) and analyze the behavior of the eigenvalues and eigen- vectors of W as a function of ∆ for a given (and fixed) set of sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Furthermore, we choose v to be negative, such that for ∆ = 0 there exists a fully synchronized eigenstate if g = 2 (see the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III B 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note that a negative value of v implies gj = Im[λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the following we consider as example the case of N = 5 oscillators and show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4 the results of the nu- merical diagonalization of the matrix M for three differ- ent realizations of ⃗s = (s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',s5) (different columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We choose the largest difference between neighboring values of sn to be equal to one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' max[sn−sn+1] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, for ∆ < 1 all frequency differences between neighboring os- cillators are always smaller than the dissipative coupling between them (which has magnitude one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The case of N = 2 in our network of oscillators allows us to represent the full parameter space as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2 and identify the dissipation free subspaces and synchro- nization within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, for larger system sizes (as con- sidered now) a representation similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2 becomes very space consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Yet, a dissipation free subspace is always necessary for synchronization, which corresponds to the white lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, in or- der to determine whether conditions (i)–(iii) are fulfilled, it is sufficient to only search along the parameters for which each eigenstate becomes dissipation free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In par- ticular, the relevant information of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 2(a) and (b) may be conveniently combined to contain only g± = Im[λ±] as function of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Accordingly, the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4 shows the imaginary part of all eigenvalues Im[λj] as function of the parameter ∆ and the middle row shows the re- spective real parts Re[λj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Lastly, in the bottom row we plot the degree of synchronization S of each eigen- vector also as function of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The eigenvalues of M are sorted in descending order of their imaginary parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Im[λ1] > Im[λ2] > ⋅⋅⋅ > Im[λN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the following we discuss different regimes of ∆ and its impact on the possibility of synchronized motion in accordance with conditions (i)–(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We focus on the eigenstate ⃗c1 with largest imaginary part Im[λ1] (high- lighted as thick blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The reason is that for g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free while all other eigenstates are simultaneously damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In contrast, if we would choose g such that another eigen- state ⃗cj≠1 would become dissipation free, there is at least one eigenstate that is exponentially growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' It is thus 7 2 1 0 1 2 Im( i) 1 2 3 4 5 10 5 0 5 10 Re( i) 0 1 2 3 4 5 1 2 3 4 5 (ci) s=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='2, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='46, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1] 1 2 3 4 5 0 1 2 3 4 5 s=[-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='24, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='93, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='08, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='54] 1 2 3 4 5 0 1 2 3 4 5 s=[-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='22, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='22, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='77, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='63] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Examples of dissipation free and (fully) synchronized dynamics in a ring of N = 5 oscillators with random frequency disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The three different columns correspond to three different set of (scaled) frequency realizations ⃗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The value of v is taken to be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the top row we show the imaginary part Im[λj] of the eigenvalues λj of the matrix M as a function ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The middle row shows the corresponding real part Re[λj] and the bottom row the degree of synchronization S(⃗cj) of the corresponding eigenstates ⃗cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For all three considered realizations, there exists an eigenstate (blue) with the maximum value of S (bottom row) for small values of ∆ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This eigenstate also has the largest imaginary part of its associated eigenvalue (top row), which allows the tuning g in such a way that it becomes dissipation free while all other eigenstates are damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' sufficient to only analyze the possibility of synchroniza- tion of ⃗c1 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' No frequency difference (∆ = 0): This means that there are no variations in the oscillator frequen- cies and the situation is exactly the same as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III B 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Consequently, the eigenvalues of W are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' From the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III B 1b, we know that if g = 2 = Im[λsyn] there exists a dissi- pation free synchronized state ⃗csyn ≡ 1 √ 5(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=',1)⊺ with associated real eigenvalue wsyn = ¯ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' all oscillators are in phase and oscillate with frequency ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is exactly what we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4: the eigenvalue with largest imaginary part has imaginary part Im[λ1] = 2 (blue thick lines in the top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note that Im[λ2] = Im[λ3] and Im[λ4] = Im[λ5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Furthermore, Re[λj] = 0 (middle row) which implies an oscillation frequency of ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Small frequency differences (0 < ∆ < 1): In this regime, the disorder in the frequency differences between nearest neighboring oscillators always remains smaller than the coupling between them (which is 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We thus expect that the degree of synchronization also remains large [S(⃗c1) ≈ N], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' the full delocalization of the eigen- state ⃗c1 persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4 we observe exactly this behavior of the thick blue line correspond- ing to ⃗c1: For small values of ∆, S(⃗c1) is maximal and slowly decreases as ∆ approaches the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, the synchronized state remains close to be fully synchro- nized within this regime [condition (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note, that the values for which S(⃗c1) starts to decrease depends on the specific realization of disorder ⃗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The imaginary part of the corresponding eigenvalue (top row) continues to be the largest value of all eigen- values (blue thick line), Im[λ1] > Im[λj≠1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Thus, for g = Im[λ1] the eigenstate ⃗c1 becomes dissipation free while all other eigenstates are damped, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' conditions (i) and (ii) are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As ∆ increases, Im[λ1] decreases resulting from the larger amount of frequency disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Simultaneously, the real part Re[λ1] remains close to 0 such that the oscillation frequency of the synchronized state ⃗c1 also continues to be close ¯ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note, the value of Re[λ1] only affects the oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Large frequency differences (∆ ≥ 1): As ∆ is increased further, the frequency difference exceeds the nearest neighbor interaction such that – similar to (An- derson) localization in finite systems [17] – the degree of synchronization S(⃗c1) of the previously delocalized eigen- state ⃗c1 rapidly decreases as ∆ increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' see blue thick 8 lines in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Hence, only partial synchronization is possible in this regime and condition (iii) is not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' At the same time, the largest imaginary value Im[λ1] continues to decrease as function of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Yet, close to ∆ = 1 it remains well separated from the second largest imaginary value Im[λ2] such that a suitable choice of g still allows for dissipation free dynamics with a sinlge os- cillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, Im[λ1] may coalesce with Im[λ2] for larger values of ∆ depending on the specific realization of ⃗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' An example of such a degeneracy is ob- served for ∆ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='6 in the top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As a re- sult, both eigenstates would be dissipation free resulting in the beating pattern discussed previously in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, as mentioned above, only partial synchroniza- tion is possible in this regime anyways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Very large frequency differences (∆ ≫ 1): In the regime of very large frequency differences, we expect that the degree of synchronization takes its minimum value S(⃗cj) = 1 for all eigenstates j since the scaling follows ∆ ≫ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This implies that the values ∆n = ∆sn are much larger than the dissipative coupling strength v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then, M is approximately diagonal with eigenvectors ⃗cj nearly lo- calized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Note that in this limit there is no synchronized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We have checked numerically that for ∆ larger than the smallest difference between the sn the synchro- nization measure of all eigenstates approaches one, as expected (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Lastly, to demonstrate that the dynamics of the sys- tem of oscillators is consistent with our discussion of the different regimes above (obtained from analyzing the eigenvectors and eigenfrequencies), we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 5 ex- amples of Re[an(τ)] as a function of the scaled time τ for ⃗s = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='14,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='20,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='20,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='46,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1) (corresponding to the first column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4) for three different values of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In all cases, we choose the initial state ⃗a0 = (1,1,2,−1,−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Panel (a) corresponds to the case of vanishing fre- quency difference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We choose the dissipation g = 2 such that only the eigenstate with largest imagi- nary part is dissipation free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' As expected after a short transient time of τ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 all oscillators are in-phase syn- chronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In panel (b), we increase the frequency difference to be ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Hence, the synchronized state is dissipation free for g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Analogues to the previous case (a), all oscillators are synchronized after a transient time of τ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5, yet with a small phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Importantly, all oscillators have the same amplitude consistent with the finding of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 4 that the degree of synchronization is maximal [S(⃗c1) = 5 for this value of ∆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Contrarily, in panel (c) where ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1 (and g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='51 to match the condition of dissipation free dynamics) the amplitudes vary among the oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is in accor- dance with S(⃗c1) < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, still only a single oscil- lation frequency is present (after some transient time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is an example of partial synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Dynamical behavior of Re(ai(τ)) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (14) for different values of the scaling factor ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In all three cases the mean frequency of the oscillators is ¯ω = 10 and the disorder is the same of the first panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' (4), namely ⃗s = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='20, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='20, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='46, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1) The coupling strength v is taken to be negative and all frequencies are given in units of ∣v∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' The initial condition is ⃗a0 = (1, 1, 2, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Panels (a) and (b) show fully synchronized motion, while panel (c) is an example of partial synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' CONCLUSIONS In this work we have investigated the possibility of long-lived synchronized motion in networks of harmonic oscillators, which are subject to gain/loss and interact via nearest neighbor dissipative couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In this con- text, we refer to synchronization as the existence of a single eigenstate of the dynamical matrix, which is dis- sipation free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Furthermore, if it attains the maximum value of the (inverse) participation ratio we refer to it as ‘fully synchronized’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We find that in the case of only two coupled oscillators, synchronization may always be achieved by tuning the gain appropriately as long as the frequency difference between the two oscillators is smaller than their interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' A similar behavior may be observed in larger net- works, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' many oscillators arranged on a ring with near- est neighbor interactions, yet the possibility of synchro- nization then depends on the specifics of the system at hand: If all oscillators are identical, synchronized col- 1 2 3 4 5 2 (a) A=0 g=2 Re(ai) 2 (b) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='5 Re( 3 A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='1 g=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='51 2 Re(ai) 0 2 二 0 2 4 6 8 10 12 149 lective motion may be achieved for an even number of sites with repulsive dissipative couplings (v positive) or an odd number of sites with attractive dissipative in- teractions (v negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For small frequency differences compared to the coupling between the oscillators, this behavior remains, which we show specifically for the case of N = 5, yet it should also hold for larger networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, as the number of coupled oscillators increases, it becomes increasingly difficult to achieve full synchro- nization and may only be observed for very small fre- quency differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' For larger frequency differences, the (inverse) participation ratio decreases significantly such that only partial synchronization may be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is in accordance with Anderson localization, where on- site disorder results in localized eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' However, as the dynamical matrix in this work is non-Hermitian, Anderson localization is not directly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Here, future work is needed to study the interplay of synchro- nization and localization, in particular in the thermody- namic limit and arbitrary small frequency perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Synchronization as discussed in this work is intimately related to the existence of dissipation free dynamics and thus isolated points/submanifolds in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Hence, they require a very precise tuning of gain and loss in order to obtain periodic steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' This is however hard to achieve in any realistic experiment and the synchronized state will experience some gain or loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' We can relax the condition Im[wj] = 0 by solely requir- ing ∣Im[wj]∣ ≪ ∣Re[wj]∣, which means that the change of amplitude of oscillation is small over many oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In addition, we then require Im[wj] ≪ Im[wsync], which means that all other eigenstates decay much faster than the ’synchronized’ one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' In principle, one may relax the condition even further and demand that there exists only one state with Im[wj] > 0, while all other states fulfill Im[wi] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' Then the synchronized state would grow while all other states are exponentially damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' ACKNOWLEDGMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' acknowledges support from the Max-Planck Gesellschaft via the MPI-PKS Next Step fellowship and is financially supported by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) – Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' 496502542 (WA 5170/1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='4475 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} +page_content='dis-nn] (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFRT4oBgHgl3EQfpzcC/content/2301.13614v1.pdf'} diff --git a/CdFAT4oBgHgl3EQftB4M/content/tmp_files/2301.08661v1.pdf.txt b/CdFAT4oBgHgl3EQftB4M/content/tmp_files/2301.08661v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bcb20ef2f628001a61999fc673cc03685c713fa6 --- /dev/null +++ b/CdFAT4oBgHgl3EQftB4M/content/tmp_files/2301.08661v1.pdf.txt @@ -0,0 +1,1412 @@ +Emergence of complex network topologies from flow-weighted optimization of network +efficiency +Sebastiano Bontorin,1, 2 Giulia Cencetti,1 Riccardo Gallotti,1 Bruno Lepri,1 and Manlio De Domenico3, 4, ∗ +1Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (TN), Italy +2Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy +3University of Padua, Via Francesco Marzolo 8, 35131, Padua, Italy +4Padua Center for Network Medicine, University of Padua +(Dated: January 23, 2023) +Transportation and distribution networks are a class of spatial networks that have been of interest +in recent years. These networks are often characterized by the presence of complex structures such +as central loops paired with peripheral branches, which can appear both in natural and man-made +systems, such as subway and railway networks. +In this study, we investigate the conditions for +the emergence of these non-trivial topological structures in the context of human transportation +in cities. We propose a minimal model for spatial networks generation, where a network lattice +acts as a spatial substrate and edge velocities and distances define an effective temporal distance +which quantifies the efficiency in exploring the urban space. Complex network topologies can be +recovered from the optimization of joint network paths and we study how the interplay between a flow +probability between two nodes in space and the associated travel cost influences the resulting optimal +network. In the perspective of urban transportation we simulate these flows by means of human +mobility models to obtain Origin-Destination matrices. We find that when using simple lattices, the +obtained optimal topologies transition from tree-like structures to more regular networks, depending +on the spatial range of flows. Remarkably, we find that branches paired to large loops structures +appear as optimal structures when the network is optimized for an interplay between heterogeneous +mobility patterns of small range travels and longer range ones typical of commuting. Finally, we +show that our framework is able to recover the statistical spatial properties of the Greater London +Area subway network. +I. +INTRODUCTION +Cities represent one of the most fascinating man-made +complex systems, exhibiting complex features ranging +on different scales: from their structure and dynamical +behavior, up to the scaling of socio-economic factors +with their size [1–5]. These features represent a strong +hint +towards the existence of universal +underlying +mechanics behind apparently very different cities [6–8]. +Out of these structural properties, one of the most +relevant, as it plays a fundamental role mediating the +complex interplay between human dynamics [9, 10] and +mobility in urban context, are transportation networks +[11–15]. These networks are a class of spatial networks +whose properties have been investigated in the literature +during the last two decades [14, 16]. +In particular, +they have been studied under the lens of optimality +conditions and minimization of cost-based functionals +[16], in order to identify specific features behind efficient +networks. +The concept of optimal networks [2] and +energy-like minimization [17] has its natural under- +standing in the physics language. +States of a system +which minimize a functional defining trade-offs between +system’s observables (e.g., free energy) represent the +most likely to be observed states of many real world +systems. While in some complex systems, such as cities, +these physical variables can not be derived from first +∗ Corresponding author: manlio.dedomenico@unipd.it +principles, these analogies and concepts can still offer +a valid perspective and provide an embedding of these +systems in a space where the interplay between their +structure and dynamics can be unfolded and better +understood. Simple laws have been studied [16, 18] to +better understand the emergence of hierarchy and the +role of traffic in the network state. +Moreover, global +and local optimization criteria lie in the evolution of +man-made systems where policy makers and planners +can adopt some of these criteria in their plans [14]. +Transportation networks are often characterized by +the presence of complex structures [19–21] such as loops +paired with branches [22], which can appear both in +natural [23] and man-made systems [14], like railway +and subway networks. +These structures represent the +key topological elements behind efficient public trans- +portation systems [20]. In this study we investigate the +conditions for the emergence of these non-trivial struc- +tures [18, 24] in the context of human transportation in +cities. We aim to reconstruct these topologies by means +of an optimal configuration [25] of the network state. +Under the assumption of a fixed total cost and a limited +set of high-capacity connections (e.g., a constraint in +the expenditure available on infrastructure), the optimal +configuration is the assignation of connections’ velocities, +or edges’ weights, such that the joint amount of time +required to travel between two nodes is minimized for +all pairs of nodes. Moreover, as these networks represent +the arteries in urban exploration/navigation via public +arXiv:2301.08661v1 [physics.soc-ph] 20 Jan 2023 + +2 +transportation, we study the role of traffic [18, 26] in +weighting the importance of specific set of connected +edges (paths). +We model the urban morphological +structures which generate heterogeneous distributions of +human mobility in space, biasing these optimal networks +to converge towards specific topological features. +We +aim to explore the minimum requirements and the +conditions for these optimization processes to reproduce +the empirical structures aforementioned. +At variance with the recent works on network effi- +ciency, we adopt some fundamentally different model- +ing choices. We evaluate the efficiency in terms of time +necessary to explore the network, where edges’ weights +act as travel speeds. We also weight path efficiency by +the traffic probability between nodes. +The underlying +network lattice (as represented in its simplest form by +the triangular lattice in the next sections) acts as a sub- +strate that allows the network to evolve [27] and pos- +sibly exhibit the network topological features typical to +real world systems. +On this framework, we show how +introducing simple probabilities biasing the optimal effi- +ciency between points in space force a transition between +a tree-like topology and a network resembling a simple +lattice. +We show also that the modeling of traffic-like +flows forces the emergence of preferential shared paths in +space. The optimal configuration of these shared paths +leads to complex topologies, which ultimately shows fea- +tures seen in real systems. Features such as a bi-modality +in the edges’ velocity distribution, characteristic of multi- +layered transportation, and the central core with loops +paired with branches typical of subway systems [20, 22] +are recovered. +We finally show an application of the +model within the Greater London Area, finding similari- +ties of the optimal model with its London Underground +network. +II. +FRAMEWORK FOR URBAN SPATIAL +MORPHOLOGY +We introduce here a general framework for spatial net- +works with the aim of recovering a minimal model for ur- +ban morphologies that encode both transportation prop- +erties and urban features such as population and density +of points of interest (POIs). To this aim, we begin from +the definition of a network substrate which acts as an +effective discretization of the spatial dimension. Its sim- +plest form can be found in an hexagonal 2-dimensional +tiling [28] and its planar dual, the triangular lattice. +More formally, in this network substrate each tile in space +is represented by a node, connected to its set of neighbor- +ing nodes (see Fig. 1). The existence of a physical edge +between nodes/tiles i and j is encoded in the adjacency +matrix A where Aij = 1 if the regions are neighbors in the +lattice. Distances and metrics are therefore computed on +top of this network substrate and are biased by nodes’ +features. +Simulated +Annealing: + +minimization +E({we}) + {we}init : we = 1.0 ∀e + + +{we}optimal +we +dij +cΠ = ∑ +e∈Π +de +we +de +we +Euclidean distance +Network +distance +Urban +morphology +Urban street +network +Minimal +model +A +B +C +FIG. 1. +Spatial network model for urban mor- +phology: A) Mapping population distribution and urban +transportation network to a minimal spatial network where +nodes encode urban features. Example with hexagonal tiling +mapped to the triangular lattice. B) Network-based distance +cΠij versus euclidean distance dij; edges weights/velocity we +are depicted as widths. C) Edges weights of the lattice sub- +strate are optimized via simulated annealing to unravel spatial +features of the optimal transportation network. +Nodes of this network can encode spatial features +at the urban scale, such as population or amenities’ +density in a given node. We therefore have a minimal +representation +of +a +urban +morphological +structure +(see Fig. +1), and a network substrate that acts as a +transportation system and can be optimized to generate +optimal transportation networks [27]. +The path-based +temporal distance on top of the transportation network +acts as the fundamental metric we aim to minimize. +The +rationale +behind +a +network-based +distance +is +grounded on the assumption that in the context of +public transportation, urban systems are not navigated +by considering geographical distance but rather by eval- +uating the travel-time between departure and arrival. +More specifically, multi-layer transportation networks +[11, 21] are characterized by layers having a hierarchical +organization with different characteristic speeds [29]. + +3 +Spatial probability + of targets +from sample source s +psj(L, β) +β = 0.1 +β = 5.0 +β = 0.1 +β = 5.0 +source +Optimized Network +States + +G({we}) +A +B +pij ∝ e−β +rij +⟨r⟩ +spatial +network +(HEX) +pij ∝ e−β +Lij +⟨L⟩ +non-spatial +network +(ER) +0.0 +1.0 +psj +FIG. 2. +Optimization of synthetic networks: The role of β is studied for two network models: the triangular lattice +(HEX) and the non-spatial (ER) network. A) Heatmap of target nodes probabilities pij from source node (yellow) under two +different β values: as the penalty parameter grows, farther nodes are more penalized and flows tend to stay close to the source. +B) Samples of the associated optimized network states: when flows are not affected by distances (β = 0.1) source nodes target +all the other nodes in the network with approximately equal probability, the optimal network converges to a tree-like structure. +With larger β (β = 5.0), trip probabilities are more localized and the presence of loops appear in the optimal structure. +Thus, an effective temporal distance becomes fundamen- +tal in determining accessibility and efficiency in urban +space exploration. +In this model, we denote e as an edge in the network, +and we as the associated edge’s weight which can be seen +as a velocity of the edge in the transportation network. +de is the euclidean distance of edge e between the nodes +it is connecting; here edge weights are visually mapped +as widths of the links. Information about edges distance +in this framework can be relevant when generalizing to +the case of random spatial networks where edges have +different lengths. +In the case of a general non-spatial +network, where there is no notion of spatial distances, +the model can be adapted by fixing de = 1. Finally, we +define ΩΓij as the set of paths connecting the two nodes. +We then maximize the efficiency of this underlying sub- +strate. The transportation efficiency between two nodes +i−j is computed as a cost in terms of time [30], and we do +not account for congestion, which can be introduced in a +possible extension of this framework. We find the path +(a set of connected edges starting from source node i and +ending in destination node j) with the smallest cumula- +tive time, where the time delay introduced by choosing +en edge e is measured as a ratio between its euclidean dis- +tance and its weight, representing a proxy of speed. Refer +to Fig. 1 for a graphic depiction. Here G({we}) is used +to indicate the network configuration with the associated +set of edges weights {we}. We therefore aim to find the +assignment of weights {we} as a trade-off between net- +β = 0.01 +β = 1.5 +β = 10 +A +B +FIG. 3. +Loop Dimension vs β: Minimum cycle basis +is used as a network’s observable to study the appearance of +loops. For each point the median and its absolute deviation +are shown. A) The average size (number of edges) of the loops +that constitute the cycle basis. B) Cycle basis dimension as +number of loops. +The optimal network ranges from a tree +structure to a lattice-like with small loops, as the probability +of long range movement decreases (large β). A transition in +the cycle basis property is observed at β ∼ 1.0 for the trian- +gular lattice under study, where the optimal network results +in an intermediate state with large loops. + +Average Loop Size vs β +8 - +Size [# edges] +6 +2 +0 +10-3 +10-2 +10-1 +100 +101 +102 +Cycle Basis Dimension vs β +Dimension [# cycles] +10 +TI +8 +6 +4 +2 +0 +10-3 +10-2 +10-1 +100 +101 +102 +β4 +work efficiency in transportation, by minimizing the set +of costs {cij} of travelling between pair of nodes i − j +where each element cij is defined as: +cij({we}) = +min +Π∈ΩΓij +� +� � +e∈Πij +de +we +� +� , +(1) +and in absence of further information, the optimiza- +tion procedure is the equivalent of minimizing the travel +costs � +ij cij. Here we add a novel ingredient, in which +we couple the optimization of the network temporal dis- +tances with a traffic flow or probability between pairs +of nodes. Operationally, when dealing with real world +Origin-Destination (OD) matrices, this probability can +be then mapped to a traffic Tij between two points. Tij +represents the probability of a person from node i to +travel to node j, and a traffic can be recovered when in- +formation about populations in source and target nodes +is added. Tij effectively acts as a rank in the importance +of a specific path in the network. As paths connecting +different pairs of nodes may share common edges of the +network substrate, complex topologies emerge from the +shared paths jointly optimizing the network efficiency. +The flow-weighted transportation efficiency therefore be- +comes: +E(G({we})) = +1 +N(N − 1) +N +� +i +N +� +j̸=i +Tij · cij({we}) +(2) +We also require that the total network infrastructure +cost, defined as the cumulative sum of edges weights +per unit length, multiplied by edge distance CG += +� +e∈G dewe is conserved. +This is a generalization of a +standard optimization process, in the sense that when +Tij = 1, ∀(i, j), the efficiency is optimized for all possible +trip pairs (i, j) with equal importance, where the Mini- +mum Spanning Tree often represents the optimal solution +[14]. +Before tackling the problem of traffic-like (OD) flows, +we study a simpler definition of Tij, which allows to un- +derstand the role of distance in the optimization process, +in absence of other nodes’ features: +Tij ∝ e−βdij. +(3) +The coefficient β appearing in Eq. 3 is introduced as +a penalizing parameter and determines how relevant is +the pair-wise distance dij when computing probabilities. +We can understand it as the inverse of a characteristic +traveling distance for an agent on the network β ∼ +1 +d0 . +While several alternatives on the integration of distance +in spatial-dependent probabilities (such as power-laws +Tij ∼ d−γ +ij ) can be employed, we focus on the exponential +dependence as it represents the foundational result from +the maximum entropy derivation of gravity flows [31]. +The introduction of gravity-like flows will be discussed +in Section IV. +In the following, we introduce the application of the +model on simple substrates to explore the role of β in +absence of spatial urban features. +III. +OPTIMIZATION OF SIMPLE NETWORK +SUBSTRATES +In order to asses the role of the characteristic distance +parameter β in the emergence of specific topologies, we +compute networks statistics on a set of generative models +for both spatial and non spatial networks. As hexagonal +tiling of space is preferable when an analysis includes as- +pects of connectivity [28], the first model we study is a +triangular lattice. The reason behind this choice is that +it represents the planar dual [14, 32] of the hexagonal lat- +tice. Therefore, as space is discretized in hexagonal tiles, +the spatial network connecting its centers is the triangu- +lar lattice, which is isotropic and presents less equivalent +degenerate paths of a rectangular lattice. +As a direct +reference to hexagonal tiling, we refer to this network +as HEX (see Fig. 2). We also extend the analysis also +to the case of a random network model where nodes are +not embedded in a metric space. Specifically, we study +an Erd˝os-R´enyi (ER) network topology, where the def- +inition of distance between nodes Lij can be defined in +terms of topological shortest path distance [33]. +As a first benchmark we simplify flows as a spatial +probability Tij = pij that decays exponentially with dis- +tance and does not consider nodes features, the resulting +equation for pij is: +pij = +e−βdij/⟨d⟩ +�N−1 +k̸=i e−βdik/⟨d⟩ +(4) +Where ⟨d⟩ = +1 +N(N−1) +� +i̸=j dij is the average distance +of points in the network and acts as a normalization fac- +tor (euclidean distance ⟨r⟩ in case of a spatial network or +topological ⟨L⟩ for the ER network). +Therefore pij encodes how much of the nearby space is +explored by a single source node. An illustration of the +spatial dependence of target probabilities and samples of +the resulting optimal topologies are presented in Fig. 2. +For a range of β values the optimization process is +performed on an ensemble of these models. To assess the +emergence of complex structures we observe the number +of loops that emerge in the optimal state. +This mea- +sure is relevant in the context of spatial networks, where +loops break the symmetry introduced by optimal struc- +tures such as trees. We compute the minimum cycle ba- +sis set as a metric to observe the emergence of loops [36]: +i.e. the minimum set of loops (where a single loop is en- +coded in a set of edges that defines a closed path in the +graph) such that any other closed path in the network can + +5 +Optimal networks and edges weights distribution P(we) +Exponential +Distribution +Morphology +A +B +β = 3.0 +3-Points +Steiner Tree +Morphology +Steiner node +10.0 +0.0 +Wj +β = 4.0 +β = 0.1 +G-KDE +FIG. 4. +Minimal models of urban morphology and attractiveness distributions under study (3-Points and +Exponential decay): Morphology of POIs, where attractiveness Wj is mapped with color intensity (yellow being higher). +Optimized edges weights distributions P({we}) are characterized by the bi-modal nature that reveals the multi-layered structure +of the optimal transportation networks when close-range flows are paired with long-range traffic typical of commuting towards +city centers (Insets P(Tij) with peaks on large-flows due to POIs). Gaussian KDE is shown in orange as a visual aid. A) +3-points polycentric distribution of POIs, resembling the euclidean Steiner Tree problem [34, 35] for three points. The network +is optimized with β = 0.1 and β = 4.0, and shows the appearance of branches connecting the POIs paired to large loops in +the periphery. B) Optimal state and distribution of speeds with exponential decay of wj from the center and an exemplifying +result with β = 3.0. The optimal topology is characterized by a central loop paired with branches. +be reconstructed via combination of this cycle basis [36]. +Specifically, we investigate the cycle basis dimension (the +number of loops that constitute this set) and the average +loop size, against a range of β values. This metric allows +to quantify the emergence of spatial topological features +that differentiate the optimal state from a tree structure. +Results for these synthetic systems are presented in Fig. +3. Additional boxplots are shown in SM Figures 1-2. A +tree-like topology is recovered when the flows probabil- +ities are distributed uniformly across all nodes in space +(when β → 0 and distance is therefore not a penalizing +variable in Eq. 4), while loops emerge when farther tar- +gets become less likely to be explored and the network +is globally optimized for close-range trips. Notably, in +Fig. 3 around β ≈ 1.0, we observe a sharp transition +in the average loop size in the HEX lattice under analy- +sis: connections appear between neighboring nodes which +are far from the tree-root as it becomes more efficient to +have a direct link. In this β regime the tree topology +does not guarantee the most efficient configuration for +peripheral nodes, which have their high probability tar- +gets in their direct neighborhood (see Eq. 4). Thus in the +optimization process edges appear between leaves nodes +which are in separated branches: this ultimately breaks +the tree structure and leads to the emergence of large- +scale loops. Eventually the optimal network converges +to a simpler structure with small loops as the network is +optimized for nodes to target only direct neighbors in the +lattice. Finally, in SM Section 2 we show an application +on the case of a single target node in the perimeter of +the lattice, where the model reproduces leaves venation +patterns [14, 37]. +IV. +SPATIAL ATTRACTIVENESS AND +TRAFFIC-LIKE FLOWS +In the context of urban systems, optimal transporta- +tion networks need to be devised to accommodate traffic +flows [26] towards specific areas of interest, e.g. due to +high commercial and business land use density. Hence we +extend the efficiency optimization framework in the case +where we have more realistic traffic on top of the urban +networks, as the presence of nodes with high attractive- +ness (POIs) biases the flows towards them. In urban sce- +narios we adopt spatial-interaction models to mimic more +traffic-like flows. In these models, flows are obtained via a +gravity-like equation: Tij ∝ pipj exp (−βdij) [10] which +can be derived from first principles via entropy maxi- +mization, thus representing the most likely set of flows +to be observed. In the context of urban exploration, the +gravity equation can be mapped to a model for spatial in- +teraction [31, 38] where nodes with a given attractiveness +Wj compete as possible targets for traffic: +Tij ∝ 1 +Z PiWj exp (−βdij) +(5) +Normalization Z accounts for all possible trip al- +ternatives � +k Wk exp (−βdik). +Pi is the population +density in node i and Wj encodes a suitable definition +of benefit/attractiveness of node j as a possible target +[38]. +Tij is therefore the fraction of population in + +β: 0.1 +102 +("1) +P +101 +0.0 +0.1 +Tijbeta is:4.0 +100 +Gaussian KDE +P(We) +10-1 +2 +3 +4 +Weβ: 4.0 +101 +P(Tij) +100 +10-1 +10-2 +0.0 +0.2 +0.4 +Tij.beta is:0.01 +100 +Gaussian KDE +P(We) +10-1 +10-2 +2 +4 +6 +8 +We6 +node i commuting/travelling on average to node j. To +better understand the role of nodes’ attractiveness, we +start with the simplest assumption of equal population +distribution on all nodes: Pi = 1.0 ∀i; we will introduce +more realistic population distribution in the next section +with the London case study. +We apply these models on the triangular lattice to +unravel the optimal topologies that emerge when traf- +fic probabilities are biased towards some nodes having +high attractiveness (simulating POIs) and we study two +spatial configurations for nodes’ Wj. In the first configu- +ration high Wj is assigned to three nodes (POIs) placed +at the vertices of an equilateral triangle. We study the 3- +points distribution as it mimics a prototypical polycentric +distribution of city-centers, and it can be linked to the +solution of the euclidean Steiner Tree problem [34, 35]. +The Steiner Tree is a class of problems where given a set +of N points in a plane the goal is to find the set of lines +connecting the points with minimum cumulative length. +In our case, the solution would lie in the central node of +the lattice being the Fermat point [35] and the Steiner +node, which connects the three vertices of the high Wj +triangle, as illustrated in Fig. 4 panel A. The second case +is a distribution of Wj that decays exponentially from the +center, mimicking a more realistic morphology for a ur- +ban monocentric structure. +The two morphologies are +depicted in Fig. 4. +We find that due to nodes in the network biasing the +traffic flows, as it can be seen in the insets of Fig. 4 A, +the traffic flows get divided into two types: a close range +paired to a long range set of flows, due to POI polariza- +tion. We show in Fig. 4 optimal solutions for values of +β = 0.1, 4.0. Interestingly, optimal solutions are char- +acterized by three central lines branching from the cen- +ter (which therefore acts as Steiner node) and connecting +the three nodes with high attractiveness, therefore resem- +bling the solution of the Steiner tree problem. Moreover, +in the case of more localized flows (β = 4.0) these lines +are also paired with large scale loops connecting farther +nodes. We also find that the heterogeneity of traffic flows +forces the appearance of a second mode in the distribu- +tion of speeds we (see Fig. 4). The two peaks in the +optimal P(we) can be interpreted as two different lev- +els of speed, which suggests that the entire process can +be decomposed in two distinct mechanisms which can +be mapped as a bi-layer network: one layer at high ca- +pacity with long-range/commuting trajectories and the +other one at low velocity with short-range paths. These +two layers can be ideally separated, hinting towards a +possible extension of the model to multilayer networks. +V. +GREATER LONDON AREA: GENERATIVE +MODEL FOR THE SUBWAY SYSTEM +We extend in this section the application of the model +by integrating data from a real urban structure. Specifi- +cally, we model the morphology of Greater London Area +(GLA) on top of our framework and apply the efficiency +optimization process with the aim of understanding if +the temporal efficiency optimization of the spatial sub- +strate paired with realistic flows is sufficient to yield a +transportation network with similar topological features +(such as a central core paired with peripheral branches +[22]) as the London subway system. To extend the model +to real urban scenarios, we first obtain the distribution +of amenities [39] from OpenStreetMap [40] and we use +this density of points in space as a proxy to estimate +the attractiveness Wj of a tile. Census data for Greater +London Area yards from 2014 is used to recover popu- +lation density Pi. These densities are then mapped to +Uber’s H3 tiling to recover the spatial discretization in +hexagonal tiles, such that we can have direct mapping to +the nodes on a triangular lattice, as in the examples dis- +cussed in previous sections. We thus have the ingredients +to finally simulate the spatial interaction flows Tij in Eq. +5. In Fig. 5 the integration of urban data describing the +London’s morphology in the model is explained and we +provide a depiction of the OD matrix that arises from +the spatial interaction model. +With the aim of repro- +ducing real features, we impose an upper limit on edge +weight, so that the distribution of weights is bounded +during the optimization process: we ∈ (0, w∗). This bet- +ter simulates the upper bound in speed of real multilayer +systems. Further explanation of data recovery and inte- +gration in the model is provided in SM Section 3. We +find (see SM Figure 6) that {we} distribution displays a +bi-modal shape, and this allows the analysis of the gen- +erated network in a sub-graph defined by the set of high +speed edges. In Fig. 5, panel C, we show a sample re- +sult for β = 0.35 of this sub-graph. The characterization +of the network into a central core paired with peripheral +branches as the optimal state can be visually observed. +The model’s subgraph of high speed edges is compared +to the real tube network in the Greater London Area +[21] to assess the similarities between the optimal struc- +ture and the real subway system. We quantify this sim- +ilarity by means of spatial scaling laws [22], these are +convenient to highlight the recovery of the central core +structure characterized by loops, paired with quasi mono- +dimensional lines branching from the core. We investi- +gate the distribution of nodes stations using the profile +function N(r) that quantifies the total number of stations +at a distance r from the network barycenter, computed +as the average location of all station nodes [22]. Results +of this scaling analysis for the real and simulated net- +works are presented in Fig. 6. The two scaling regimes +indicate the separation of core and branches: the scaling +of r2 in the core center and a second trend due to mono- +dimensional branches for r > rC, where rC is the radius +of the core structure. The second trend can be computed +analytically via an integral curve for N(r > rC) which +can be approximated by a power law rγ (γ = 1.25±0.02, +see SM Section 4), as in Ref. [22]. The curve of N(r) +is consistent with the real network and confirms scaling + +7 +Efficiency optimized +Network state on GLA +Morphology +Pi +Wj +POI Distribution +(OSM) +Census Data +Tij ∝ PiWα +j exp (−βdij) +OD fluxes +Network optimization +under realistic fluxes +Greater +London Area ++ +Metro network +A +B +C +FIG. 5. +Optimal network model for Greater London Area subway system: Application of the efficiency optimization +with realistic flows on the urban structure of the Greater London Area. A) Urban morphology data is recovered from Census +and OSM and population and POI densities are mapped to the H3 tiling. B) Data is mapped to the triangular lattice, with +nodes having features which allow the calculation of traffic-like flows, a sample OD matrix is shown where Tij are computed +with β = 0.35. C) Optimal network state for the London model, where only edges and nodes corresponding to the second +mode are shown (see SM Section 3). Central core structure with loops paired with peripheral branches can be visually seen. +laws prediction from [22]. +core +branches +FIG. 6. +Scaling properties of GLA Tube stations: +Profile of the number of stations (nodes in the optimal dis- +cretized network (see SM Section 3) reproducing GLA under- +ground) versus the distance from the barycenter. The scaling +of N(r) profile of the model is compared with the real net- +work system. Scaling properties predicted in [22] are verified, +finding the two different scaling regimes separated at +r +rC ∼ 1 +for core paired with branches systems, where rC is the core +radius (characterized by r2 scaling), and NC is the number of +stations in the core. The scaling exponent γ = 1.25 ± 0.02 is +obtained as a linear fit of the integral curve [22] for r > rC +(see SM Section 4 for more details). +VI. +DISCUSSION +Starting from simple conditions on temporal efficiency +on a spatial network substrate, we show that network op- +timization paired with traffic-like flows weighting the im- +portance of specific connections in space can reproduce +complex networks features from man-made transporta- +tion networks. +Specifically, we devise a framework for +spatial networks where nodes can encode features of ur- +ban systems and can ultimately lead to the study of opti- +mal topologies in real scenarios. A key novelty lies in the +optimization process happening on a spatial substrate, +such that edges of the resulting optimal network are op- +timized to improve the efficiency of the shared space by +all nodes in the network. We show how the probabili- +ties of moving from one point to another in space force a +transition between a tree-like and a lattice-like topology +in the optimal network. Fixing certain target points in +space with a higher attractiveness for flows can repro- +duce theoretical results such as the Steiner tree solution +or leaves venation patterns. We also show that extend- +ing these probabilities using urban spatial information +and traffic-like flows modeling forces the emergence of +shared preferential paths that are organized as complex +topologies, resulting from traffic weighted optimization +of network time efficiency, which ultimately exhibits the +characteristics seen in real systems. We recover features +such as a bi-modality in the speed distribution of the +edges of the network, characteristic of multilayer trans- +portation. Or the appearance of a central core with loops +coupled to branches typical of underground systems, as +in the case of the London underground system. We find +that branches paired to large loops structures appear as +optimal structures when the network is optimized for an + +Tiling (HEX3-Res6) and POI Density +OSM amenities distribution +Mapping to the hex lattice model +W; extraction - here log(density) +London Greater Area +52.0 +51.8 +Lat +51. +51.2 +51.0 +0.2 +0.0 +0.2 +Lon51.70 +51.65 +51.60 +51.55 +51.5 +51.45 +51.40 +51.35 +51.30 +0.4 +-0.2 +0.D +0.21.0 +0.9 +M +0.8 +0.7 +Traffic Density +0.6 +0.5 +0.4 +0.3 +0.21.0 +0.9 +0.8 +0.7 +Traffic Density +0.6 +0.5 +0.4 +0.3 +0.2Rescaled number of stations at distance r from barycenter +Power law r2 +101 +Model +LGA Tube +Powerlaw ry +100 +10-1 +.OL +10-1 +1008 +interplay of traffic flows mixed between small range trav- +els and longer range ones typical of commuting. +This +novel framework for the optimization of spatial networks +in urban contexts may show further improvements and +extensions to better accommodate the concepts of multi- +layer and shared space. It could be extended also to the +case of inter-cities transportation, where specific nodes in +the network substrate represent cities. To conclude, in +this work the problem is addressed in a theoretical way +with the aim of reproducing and understanding some fea- +tures observed in real spatial networks, but future works +can exploit this framework as a basis to understand how +to generate optimal transportation networks in a urban +planning scenario. +Competing Financial Interests +The authors declare no competing financial interests +Data Availability +The data used in this work are publicly available from +the original references +Code Availability +The code to perform the analysis will be available upon +request. +[1] M. Batty, Science 319, 769 (2008). +[2] M. Barthelemy, Nature Reviews Physics 1, 406 (2019). +[3] L. Bettencourt and G. West, Nature 467, 912 (2010). +[4] W. +Pan, +G. +Ghoshal, +C. +Krumme, +M. +Cebrian, +and A. Pentland, Nature Communications 4 (2013), +10.1038/ncomms2961. +[5] E. Arcaute, E. Hatna, P. Ferguson, H. Youn, A. Johans- +son, and M. Batty, Journal of The Royal Society Inter- +face 12, 20140745 (2015). +[6] L. M. A. Bettencourt, Science 340, 1438 (2013). +[7] A. Bassolas, H. Barbosa-Filho, B. Dickinson, X. Doti- +walla, P. Eastham, R. Gallotti, G. Ghoshal, B. Gip- +son, S. A. Hazarie, H. Kautz, O. Kucuktunc, A. Lieber, +A. Sadilek, and J. J. Ramasco, Nature Communications +10 (2019), 10.1038/s41467-019-12809-y. +[8] L. M. A. Bettencourt, Science Advances 6 (2020), +10.1126/sciadv.aat8812. +[9] M. Schl¨apfer, L. Dong, K. O’Keeffe, P. Santi, M. Szell, +H. Salat, S. Anklesaria, M. Vazifeh, C. Ratti, and G. B. +West, Nature 593, 522 (2021). +[10] H. Barbosa, M. Barthelemy, G. Ghoshal, C. R. James, +M. Lenormand, T. Louail, R. Menezes, J. J. Ramasco, +F. Simini, +and M. Tomasini, Physics Reports 734, 1 +(2018). +[11] L. Alessandretti, L. G. N. Orozco, M. Saberi, M. Szell, +and F. Battiston, Environment and Planning B: Urban +Analytics and City Science , 239980832211081 (2022). +[12] M. Lee, H. Barbosa, H. Youn, P. Holme, and G. Ghoshal, +Nature Communications 8 (2017), 10.1038/s41467-017- +02374-7. +[13] R. Gallotti, P. Sacco, and M. D. Domenico, Complexity +2021, 1 (2021). +[14] M. Barth´elemy, Physics Reports 499, 1 (2011). +[15] R. G. Morris and M. Barthelemy, Physical Review Let- +ters 109 (2012), 10.1103/physrevlett.109.128703. +[16] M. T. Gastner and M. E. J. Newman, Physical Review +E 74 (2006), 10.1103/physreve.74.016117. +[17] M. Barth´elemy and A. Flammini, Journal of Statisti- +cal Mechanics: Theory and Experiment 2006, L07002 +(2006). +[18] R. Louf, P. Jensen, and M. Barthelemy, Proceedings of +the National Academy of Sciences 110, 8824 (2013). +[19] J. R. Banavar, A. Maritan, and A. Rinaldo, Nature 399, +130 (1999). +[20] A. Pei, F. Xiao, S. Yu, and L. Li, Scientific Reports 12 +(2022), 10.1038/s41598-022-12053-3. +[21] R. Gallotti and M. Barthelemy, Scientific Data 2 (2015), +10.1038/sdata.2014.56. +[22] C. Roth, S. M. Kang, M. Batty, +and M. Barthelemy, +Journal of The Royal Society Interface 9, 2540 (2012). +[23] A. Tero, S. Takagi, T. Saigusa, K. Ito, D. P. Bebber, +M. D. Fricker, K. Yumiki, R. Kobayashi, and T. Naka- +gaki, Science 327, 439 (2010). +[24] R. Louf and M. Barthelemy, Scientific Reports 4 (2014), +10.1038/srep05561. +[25] A. A. Ibrahim, A. Lonardi, and C. D. Bacco, Algorithms +14, 189 (2021). +[26] X. Zhang, +A. Adamatzky, +F. T. Chan, +Y. Deng, +H. Yang, X.-S. Yang, M.-A. I. Tsompanas, G. C. Sir- +akoulis, and S. Mahadevan, Scientific Reports 5 (2015), +10.1038/srep10794. +[27] M. Szell, S. Mimar, T. Perlman, G. Ghoshal, +and +R. Sinatra, Scientific Reports 12 (2022), 10.1038/s41598- +022-10783-y. +[28] C. P. Birch, S. P. Oom, and J. A. Beecham, Ecological +Modelling 206, 347 (2007). +[29] R. +Gallotti, +A. +Bazzani, +S. +Rambaldi, +and +M. +Barthelemy, +Nature +Communications +7 +(2016), +10.1038/ncomms12600. +[30] Y. Ren, M. Ercsey-Ravasz, P. Wang, M. C. Gonz´alez, +and Z. Toroczkai, Nature Communications 5 (2014), +10.1038/ncomms6347. +[31] A. Wilson, Transportation Research 9, 167 (1975). +[32] M. P. Viana, E. Strano, P. Bordin, and M. Barthelemy, +Scientific Reports 3 (2013), 10.1038/srep03495. +[33] M. Newman, Networks (Oxford University Press, 2010). +[34] S. E. Dreyfus and R. A. Wagner, Networks 1, 195 (1971). +[35] M. Brazil, R. L. Graham, D. A. Thomas, and M. Zachari- +asen, Archive for History of Exact Sciences 68, 327 +(2013). +[36] T. Kavitha, K. Mehlhorn, D. Michail, and K. E. Paluch, +Algorithmica 52, 333 (2007). +[37] E. Katifori, G. J. Sz¨oll˝osi, +and M. O. Magnasco, +Physical Review Letters 104 (2010), 10.1103/phys- +revlett.104.048704. + +9 +[38] D. Piovani, C. Molinero, and A. Wilson, PLOS ONE 12, +e0185787 (2017). +[39] C. A. Hidalgo, E. Casta˜ner, +and A. Sevtsuk, Habitat +International 106, 102205 (2020). +[40] OpenStreetMap +contributors, +“Planet +dump +re- +trieved +from +https://planet.osm.org +,” +https: +//www.openstreetmap.org (2017). + +10 +SUPPLEMENTARY MATERIAL FOR “EMERGENCE OF COMPLEX NETWORKS TOPOLOGIES +FROM FLOW-WEIGHTED OPTIMIZATION OF NETWORK EFFICIENCY” +S1. +APPLICATION ON SIMPLE NETWORK SUBSTRATES +In this Section we report more detailed results for the optimization of simple network substrates that was discussed +in the Main text, specifically for the triangular (HEX) lattice (planar dual of the hexagonal tiling of space) and +Erd˝os-R´enyi (ER) network. The aim is to explore the resulting topologies that emerge both in spatial and non spatial +networks when simple probabilities are taken into consideration (see Main text). +S1.1. +HEX Lattice and ER Network +Results of model application on the triangular (HEX) lattice and results of model application on an Erd˝os-R´enyi +(ER) non-spatial network. For the non-spatial generative model, an ensemble of 20 networks with N = 30 nodes and +edge probability ρ = 0.2 is generated. For the spatial case, the optimization process is repeated 20 times for each +value of β on the triangular lattice with N = 37 nodes. +SM Fig. 1 and SM Fig. 2 show boxplots for the distributions of metrics computed on the cycle basis in panels A +on both figures. In panels B, samples of the optimized network states are shown for different values of β. + +11 +Hex Lattice +N = 37 +A +B +Supplementary Figure S1. +Cycle basis properties and samples - HEX lattice. Boxplot statistics for the cycle basis +dimension and optimal network samples for triangular lattice across different β values. + +Average Loop Size vs β +10 +8 - +edges] +6 +# +Size +4 : +2 +0 +0.0001 0.001 +0.01 +0.1 +0.5 +1.0 +1.5 +2.0 +3.0 +4.0 +5.0 +10.0 +15.0100.0 +Cycle Basis Dimension vs β +Dimension [# cycles] +10.0 +7.5 +5.0 +2.5 +0.0 +0.0001 0.001 +0.01 +0.1 +0.5 +1.0 +1.5 +2.0 +3.0 +4.0 +5.0 +10.0 +15.0 +100.0 +βB: 10.0 +Yβ: 0.001 +7β: 1.0 +YYβ: 4.0β: 0.1β: 2.0 +蒸β: 5.0 +茶β: 0.5β: 3.012 +ER Network +N = 30, ρ = 0.2 +A +B +Supplementary Figure S2. +Cycle basis properties and samples - ER network. Boxplot statistics for the cycle basis +dimension and optimal network samples for an Erdos-Renyi (ER) network across different β values. + +Average Loop Size vs β +8 +6 +5 +#1 +4 +Size +S +2 +1 +0 +0.0001 0.001 +0.01 +0.1 +0.5 +1.0 +1.5 +2.0 +3.0 +4.0 +5.0 +10.0 +15.0 +100.0 +Cycle Basis Dimension vs β +Dimension [# cycles] +15 +10 +5 +0 +0.0001 0.001 +0.01 +0.1 +0.5 +1.0 +1.5 +2.0 +3.0 +4.0 +5.0 +10.0 +15.0 +100.0 +ββ: 100.0B: 0.0001β: 1.5β: 5.0β: 0.01β: 3.0β: 10.0β: 0.5β: 4.013 +S2. +LEAVES PATTERNS +Here we show an application of the model to reproduce leaf venation patterns [37]. A single attracting node (single +target or sink) is considered at one of the perimeter nodes of the lattice, and the substrate is optimized using all the +other nodes as sources. +Supplementary Figure S3. +Optimal networks resembling leaves’ veins patterns. Optimal state when a single target +(orange node) in a single spatial extremity is considered. Efficiency is optimized for all nodes in space to reach the target. The +resulting optimal state resembles tree-like patterns found in leaves (A), while the distribution of edges weights is shown in (B). + +B +beta is:1.0 +100 +10-1 +P(We) +102. +100 +101 +We14 +S3. +APPLICATION ON LONDON TUBE NETWORK +In this section we discuss more in detail the application of the model on the Greater London Area urban morphology. +We show that the London subway network spatial properties can be recovered by means of the optimal network state. +In particular, to simulate a real transportation system, an upper bound on edges travel velocity is imposed. +S3.1. +Data Integration: Census and OpenStreetMap Data +To extend the model to real urban scenarios, we gather data regarding the urban morphological structure from +OpenStreetMap (OSM) [40] and Census. +1. +OSM Data +To model the attractiveness of nodes in the lattice, we use the density of Points Of Interest in the urban space. We +use amenities [39] points in OSM as a proxy for Points Of Interest (POIs), and a node j attractiveness (Wj) therefore +encodes the density of amenities in space. +The bounding box for Greater London Area (GLA) is obtained via OSM (https://wiki.openstreetmap.org/ +wiki/Bounding_Box) and POI densities are recovered inside this box. Specifically we use the following amenities +sub-categories: +’cafe’,’college’,’library’,’school’,’university’,’kindergarten’,’restaurant’,’pub’, ’fast +food’,’bar’,’bank’,’dentist’,’pharmacy’,’hospital’,’clinic’,’doctor’,’arts +centre’,’cinema’,’community centre’,’police’,’post office’,’marketplace’ +In Fig. S4 the amenities points recovered from OSM are plotted in the bounding box. +Lat +Lon +51.72 +51.25 +-0.60 +0.35 +Map background is provided by OSM under Open Database License +Supplementary Figure S4. +OSM amenities query for GLA bounding box. +Amenities retrieved from OSM +(https://openstreetmap.org/copyright) for the bounding box defined by the following longitude [−0.6, 0.35] and latitude +interval [51.25, 51.75]. + +tMisisenden +Chippin +AONB +Amersham +Ingatestone +ghwycombe +Billericay +consfielo +wWickto +Basildon +arlow +Maidenhead +Stanford-le +endon +M4 +Windsol +Bracknel +M2 +ton +M20 +ghtwater +Snodland +Biggir +LHET +WestMalling +Ma +nam +eafletLDataby@OpenStreetMapunderODbl15 +2. +Census Data +London Wards - Census Data 2014 +A +B +C +Supplementary Figure S5. +Census data retrieval. A) London’s wards data from Census 2014 (https://data.london. +gov.uk/dataset/ward-profiles-and-atlas). +B) Points are generated in space following wards polygons with a density +proportional to Census data. C) Points are mapped to HEX3 tiles with associated densities of points. Finally, a restriction to +a disc is used to enforce symmetry in the distribution of nodes and to ease the computational load. +Points are mapped to density in space by using HEX3 tiling (https://eng.uber.com/h3/) with spatial resolution +RES = 8. This spatial discretization process via tiling is of particular relevance to map this information to the +HEX triangular lattice model which was described in the Main text. Both Fig. S5 and Fig. S4 show how model +information Pi and Wj are recovered from data and then mapped to tiles as in shown in panel C in Fig. S5. Tiles +covering the urban area are then mapped to the spatial network model as nodes in the triangular lattice (see Fig. 5 +in Main text). +For computational simplicity and to preserve the isotropy of the lattice substrate from its central point, we restrict +our analysis to a disc centered in the London’s region with highest attractiveness, that lies approximately in the City +Of London district. Results are robust against integration of the remaining GLA region. As the optimal edges weights +are influenced by the traffic on the substrate, the discarded regions in the Greater London Area do not add relevant +contributions when compared to more central regions with higher population density. +S3.2. +Results with distributions for traffic and bounded weights distribution +To simulate a real system we limit upper edges weights to a fixed value w∗ = 7 ∗ winit where winit is the initial +edges weight assigned to the network state (winit = 1.0) before optimization process, such that � +e∈G dewe,init = C. +In this simulation we work with β = 0.35. + +'LS +51.6 +51.5 +51.4 +51.3 +.4 +0.2 +0.D +0.216 +A +B +Pi +Wj +Urban morphology mapped to +The hexagonal lattice +Optimal network and +edges weights distribution +Supplementary Figure S6. +Mapping the data to Hex lattice + GLA optimized network. A) Urban morphology +data mapped to the HEX tiling and then mapped to the triangular (HEX) lattice. At this step, OD matrix is generated and +the network is then optimized. B) Resulting optimal network state with its distribution of edges weights showing a bi-modal +shape. Only fast edges having weight larger than a threshold (we > 5) are kept to isolate the sub-graph constituted by a high +velocity set of edges, such as a subway system. +A +London Metro network +(Restricted to disc) +Model output (pruned) +B +Supplementary Figure S7. +Pruned network vs real tube. A) London subway network, restricted to the disc under +study. B) GLA model output, limited to the nodes and edges which have a weight we > 5.0, where this value was chosen +from the previously plotted {we} distribution as a threshold to separate the second “mode” with fast edges. This “discretized” +network is the model on which statistical measures are performed. + +Model network0.5 . +0.4 +P(we) +0.3 - +0.2 +0.1 +0.0 +0 +1 +2 +4 +5 +6 +7 +WeCG Real network +Model pruned (2nd mode)17 +S4. +SCALING OF NETWORK STATIONS +The spatial organization of the transportation network can be inspected by taking into consideration the number +of nodes’ stations N(r) up to a distance r from the barycenter of stations. In the Main text we analyze the scaling +regimes for the simulated and real London Tube network [21] following the analysis employed in [22]. Specifically, +they obtain functional forms for the scaling properties of N(r) for different distances from the barycenter. We report +here the scalings obtained in [22]: +N(r) ∼ +� +� +� +� +� +ρCπr2 +for r < rC +ρCπr2 +C + NB +� r +rC +dr +∆(r) +for +rC < r < rmax +N +for r > rmax +(S1) +Specifically, they show that in the large distance regime (r > rC and r < rmax ) the number of stations can be +approximated by adding the integral curve NB +� r +rC +dr +∆(r) to a constant term. In [22] it is also reported that the large +distance behavior can be also, in general, approximated by a scaling law. Therefore we plot here the computation of +the integral curve against rescaled values of r, and show that in that regime it can be approximated by a power law, +and its exponent can be obtained via a linear fit. The N(r) curve in the Main text was computed on the real London +Tube restricted to the disc area as presented in SM Fig. 7. The associated exponent γ = 1.25 ± 0.02 was used in +the Main text to highlight the secondary scaling for r > rC. As discussed in Ref. [22], due to ∆(r) in SM Eq. S1 +being often noisy, this scaling property is not often well reproduced in empirical networks, and it is often restricted +to a small region of r values. In Fig. 6 of the Main text, we see that the secondary scaling with the exponent γ +approximates N(r) in a limited interval for r > rC. The value of the exponent in the branches region is expected to +be γ < 2.0 [22] and our result is consistent with this. +Supplementary Figure S8. +Scaling of the integral curve. Log-Log plot of the integral curve and its linear fit. With +the associated value of the γ exponent approximated, to highlight a secondary scaling region as mentioned in [22]. + +Scaling of the integral curve - fit with y = 1.253 +Power law ry +Integral Curve +102 +6 × 101 +4 × 101 +3 × 101 +100 +2 × 100 +3 × 100 +r +Tc18 +[1] M. Batty, Science 319, 769 (2008). +[2] M. Barthelemy, Nature Reviews Physics 1, 406 (2019). +[3] L. Bettencourt and G. West, Nature 467, 912 (2010). +[4] W. Pan, G. Ghoshal, C. Krumme, M. 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Barthelemy, Scientific Data 2 (2015), 10.1038/sdata.2014.56. +[22] C. Roth, S. M. Kang, M. Batty, and M. Barthelemy, Journal of The Royal Society Interface 9, 2540 (2012). +[23] A. Tero, S. Takagi, T. Saigusa, K. Ito, D. P. Bebber, M. D. Fricker, K. Yumiki, R. Kobayashi, and T. Nakagaki, Science +327, 439 (2010). +[24] R. Louf and M. Barthelemy, Scientific Reports 4 (2014), 10.1038/srep05561. +[25] A. A. Ibrahim, A. Lonardi, and C. D. Bacco, Algorithms 14, 189 (2021). +[26] X. Zhang, A. Adamatzky, F. T. Chan, Y. Deng, H. Yang, X.-S. Yang, M.-A. I. Tsompanas, G. C. Sirakoulis, +and +S. Mahadevan, Scientific Reports 5 (2015), 10.1038/srep10794. +[27] M. Szell, S. Mimar, T. Perlman, G. Ghoshal, and R. Sinatra, Scientific Reports 12 (2022), 10.1038/s41598-022-10783-y. +[28] C. P. Birch, S. P. Oom, and J. A. Beecham, Ecological Modelling 206, 347 (2007). +[29] R. Gallotti, A. Bazzani, S. Rambaldi, and M. Barthelemy, Nature Communications 7 (2016), 10.1038/ncomms12600. +[30] Y. Ren, M. Ercsey-Ravasz, +P. Wang, +M. C. Gonz´alez, +and Z. Toroczkai, Nature Communications 5 (2014), +10.1038/ncomms6347. +[31] A. Wilson, Transportation Research 9, 167 (1975). +[32] M. P. Viana, E. Strano, P. Bordin, and M. Barthelemy, Scientific Reports 3 (2013), 10.1038/srep03495. +[33] M. Newman, Networks (Oxford University Press, 2010). +[34] S. E. Dreyfus and R. A. Wagner, Networks 1, 195 (1971). +[35] M. Brazil, R. L. Graham, D. A. Thomas, and M. Zachariasen, Archive for History of Exact Sciences 68, 327 (2013). +[36] T. Kavitha, K. Mehlhorn, D. Michail, and K. E. Paluch, Algorithmica 52, 333 (2007). +[37] E. Katifori, G. J. Sz¨oll˝osi, and M. O. Magnasco, Physical Review Letters 104 (2010), 10.1103/physrevlett.104.048704. +[38] D. Piovani, C. Molinero, and A. Wilson, PLOS ONE 12, e0185787 (2017). +[39] C. A. Hidalgo, E. Casta˜ner, and A. Sevtsuk, Habitat International 106, 102205 (2020). +[40] OpenStreetMap contributors, “Planet dump retrieved from https://planet.osm.org ,” https://www.openstreetmap.org +(2017). + diff --git a/CdFAT4oBgHgl3EQftB4M/content/tmp_files/load_file.txt b/CdFAT4oBgHgl3EQftB4M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d05797216585546333bc84f4768f3f569c2c8048 --- /dev/null +++ b/CdFAT4oBgHgl3EQftB4M/content/tmp_files/load_file.txt @@ -0,0 +1,1076 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf,len=1075 +page_content='Emergence of complex network topologies from flow-weighted optimization of network efficiency Sebastiano Bontorin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2 Giulia Cencetti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 Riccardo Gallotti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 Bruno Lepri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 and Manlio De Domenico3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' ∗ 1Fondazione Bruno Kessler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Via Sommarive 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 38123 Povo (TN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Italy 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' University of Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Via Sommarive 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 38123 Povo (TN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Italy 3University of Padua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Via Francesco Marzolo 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 35131,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Padua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Italy 4Padua Center for Network Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' University of Padua (Dated: January 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2023) Transportation and distribution networks are a class of spatial networks that have been of interest in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These networks are often characterized by the presence of complex structures such as central loops paired with peripheral branches, which can appear both in natural and man-made systems, such as subway and railway networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In this study, we investigate the conditions for the emergence of these non-trivial topological structures in the context of human transportation in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We propose a minimal model for spatial networks generation, where a network lattice acts as a spatial substrate and edge velocities and distances define an effective temporal distance which quantifies the efficiency in exploring the urban space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Complex network topologies can be recovered from the optimization of joint network paths and we study how the interplay between a flow probability between two nodes in space and the associated travel cost influences the resulting optimal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the perspective of urban transportation we simulate these flows by means of human mobility models to obtain Origin-Destination matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We find that when using simple lattices, the obtained optimal topologies transition from tree-like structures to more regular networks, depending on the spatial range of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Remarkably, we find that branches paired to large loops structures appear as optimal structures when the network is optimized for an interplay between heterogeneous mobility patterns of small range travels and longer range ones typical of commuting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Finally, we show that our framework is able to recover the statistical spatial properties of the Greater London Area subway network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' INTRODUCTION Cities represent one of the most fascinating man-made complex systems, exhibiting complex features ranging on different scales: from their structure and dynamical behavior, up to the scaling of socio-economic factors with their size [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These features represent a strong hint towards the existence of universal underlying mechanics behind apparently very different cities [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Out of these structural properties, one of the most relevant, as it plays a fundamental role mediating the complex interplay between human dynamics [9, 10] and mobility in urban context, are transportation networks [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These networks are a class of spatial networks whose properties have been investigated in the literature during the last two decades [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In particular, they have been studied under the lens of optimality conditions and minimization of cost-based functionals [16], in order to identify specific features behind efficient networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The concept of optimal networks [2] and energy-like minimization [17] has its natural under- standing in the physics language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' States of a system which minimize a functional defining trade-offs between system’s observables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=', free energy) represent the most likely to be observed states of many real world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' While in some complex systems, such as cities, these physical variables can not be derived from first ∗ Corresponding author: manlio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='dedomenico@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='it principles, these analogies and concepts can still offer a valid perspective and provide an embedding of these systems in a space where the interplay between their structure and dynamics can be unfolded and better understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Simple laws have been studied [16, 18] to better understand the emergence of hierarchy and the role of traffic in the network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Moreover, global and local optimization criteria lie in the evolution of man-made systems where policy makers and planners can adopt some of these criteria in their plans [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Transportation networks are often characterized by the presence of complex structures [19–21] such as loops paired with branches [22], which can appear both in natural [23] and man-made systems [14], like railway and subway networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These structures represent the key topological elements behind efficient public trans- portation systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In this study we investigate the conditions for the emergence of these non-trivial struc- tures [18, 24] in the context of human transportation in cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We aim to reconstruct these topologies by means of an optimal configuration [25] of the network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Under the assumption of a fixed total cost and a limited set of high-capacity connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=', a constraint in the expenditure available on infrastructure), the optimal configuration is the assignation of connections’ velocities, or edges’ weights, such that the joint amount of time required to travel between two nodes is minimized for all pairs of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Moreover, as these networks represent the arteries in urban exploration/navigation via public arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='08661v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='soc-ph] 20 Jan 2023 2 transportation, we study the role of traffic [18, 26] in weighting the importance of specific set of connected edges (paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We model the urban morphological structures which generate heterogeneous distributions of human mobility in space, biasing these optimal networks to converge towards specific topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We aim to explore the minimum requirements and the conditions for these optimization processes to reproduce the empirical structures aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' At variance with the recent works on network effi- ciency, we adopt some fundamentally different model- ing choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We evaluate the efficiency in terms of time necessary to explore the network, where edges’ weights act as travel speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We also weight path efficiency by the traffic probability between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The underlying network lattice (as represented in its simplest form by the triangular lattice in the next sections) acts as a sub- strate that allows the network to evolve [27] and pos- sibly exhibit the network topological features typical to real world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' On this framework, we show how introducing simple probabilities biasing the optimal effi- ciency between points in space force a transition between a tree-like topology and a network resembling a simple lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We show also that the modeling of traffic-like flows forces the emergence of preferential shared paths in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The optimal configuration of these shared paths leads to complex topologies, which ultimately shows fea- tures seen in real systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Features such as a bi-modality in the edges’ velocity distribution, characteristic of multi- layered transportation, and the central core with loops paired with branches typical of subway systems [20, 22] are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We finally show an application of the model within the Greater London Area, finding similari- ties of the optimal model with its London Underground network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' FRAMEWORK FOR URBAN SPATIAL MORPHOLOGY We introduce here a general framework for spatial net- works with the aim of recovering a minimal model for ur- ban morphologies that encode both transportation prop- erties and urban features such as population and density of points of interest (POIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' To this aim, we begin from the definition of a network substrate which acts as an effective discretization of the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Its sim- plest form can be found in an hexagonal 2-dimensional tiling [28] and its planar dual, the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' More formally, in this network substrate each tile in space is represented by a node, connected to its set of neighbor- ing nodes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The existence of a physical edge between nodes/tiles i and j is encoded in the adjacency matrix A where Aij = 1 if the regions are neighbors in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Distances and metrics are therefore computed on top of this network substrate and are biased by nodes’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Simulated Annealing: minimization E({we}) {we}init : we = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 ∀e {we}optimal we dij cΠ = ∑ e∈Π de we de we Euclidean distance Network distance Urban morphology Urban street network Minimal model A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Spatial network model for urban mor- phology: A) Mapping population distribution and urban transportation network to a minimal spatial network where nodes encode urban features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Example with hexagonal tiling mapped to the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Network-based distance cΠij versus euclidean distance dij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' edges weights/velocity we are depicted as widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' C) Edges weights of the lattice sub- strate are optimized via simulated annealing to unravel spatial features of the optimal transportation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Nodes of this network can encode spatial features at the urban scale, such as population or amenities’ density in a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We therefore have a minimal representation of a urban morphological structure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1), and a network substrate that acts as a transportation system and can be optimized to generate optimal transportation networks [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The path-based temporal distance on top of the transportation network acts as the fundamental metric we aim to minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The rationale behind a network-based distance is grounded on the assumption that in the context of public transportation, urban systems are not navigated by considering geographical distance but rather by eval- uating the travel-time between departure and arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' More specifically, multi-layer transportation networks [11, 21] are characterized by layers having a hierarchical organization with different characteristic speeds [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 3 Spatial probability of targets from sample source s psj(L, β) β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 source Optimized Network States G({we}) A B pij ∝ e−β rij ⟨r⟩ spatial network (HEX) pij ∝ e−β Lij ⟨L⟩ non-spatial network (ER) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 psj FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Optimization of synthetic networks: The role of β is studied for two network models: the triangular lattice (HEX) and the non-spatial (ER) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) Heatmap of target nodes probabilities pij from source node (yellow) under two different β values: as the penalty parameter grows, farther nodes are more penalized and flows tend to stay close to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Samples of the associated optimized network states: when flows are not affected by distances (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1) source nodes target all the other nodes in the network with approximately equal probability, the optimal network converges to a tree-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' With larger β (β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0), trip probabilities are more localized and the presence of loops appear in the optimal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Thus, an effective temporal distance becomes fundamen- tal in determining accessibility and efficiency in urban space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In this model, we denote e as an edge in the network, and we as the associated edge’s weight which can be seen as a velocity of the edge in the transportation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' de is the euclidean distance of edge e between the nodes it is connecting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' here edge weights are visually mapped as widths of the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Information about edges distance in this framework can be relevant when generalizing to the case of random spatial networks where edges have different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the case of a general non-spatial network, where there is no notion of spatial distances, the model can be adapted by fixing de = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Finally, we define ΩΓij as the set of paths connecting the two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We then maximize the efficiency of this underlying sub- strate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The transportation efficiency between two nodes i−j is computed as a cost in terms of time [30], and we do not account for congestion, which can be introduced in a possible extension of this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We find the path (a set of connected edges starting from source node i and ending in destination node j) with the smallest cumula- tive time, where the time delay introduced by choosing en edge e is measured as a ratio between its euclidean dis- tance and its weight, representing a proxy of speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1 for a graphic depiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Here G({we}) is used to indicate the network configuration with the associated set of edges weights {we}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We therefore aim to find the assignment of weights {we} as a trade-off between net- β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 β = 10 A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Loop Dimension vs β: Minimum cycle basis is used as a network’s observable to study the appearance of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' For each point the median and its absolute deviation are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) The average size (number of edges) of the loops that constitute the cycle basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Cycle basis dimension as number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The optimal network ranges from a tree structure to a lattice-like with small loops, as the probability of long range movement decreases (large β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A transition in the cycle basis property is observed at β ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 for the trian- gular lattice under study, where the optimal network results in an intermediate state with large loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Average Loop Size vs β 8 - Size [# edges] 6 2 0 10-3 10-2 10-1 100 101 102 Cycle Basis Dimension vs β Dimension [# cycles] 10 TI 8 6 4 2 0 10-3 10-2 10-1 100 101 102 β4 work efficiency in transportation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' by minimizing the set of costs {cij} of travelling between pair of nodes i − j where each element cij is defined as: cij({we}) = min Π∈ΩΓij � � � e∈Πij de we � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' (1) and in absence of further information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' the optimiza- tion procedure is the equivalent of minimizing the travel costs � ij cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Here we add a novel ingredient, in which we couple the optimization of the network temporal dis- tances with a traffic flow or probability between pairs of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Operationally, when dealing with real world Origin-Destination (OD) matrices, this probability can be then mapped to a traffic Tij between two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Tij represents the probability of a person from node i to travel to node j, and a traffic can be recovered when in- formation about populations in source and target nodes is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Tij effectively acts as a rank in the importance of a specific path in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As paths connecting different pairs of nodes may share common edges of the network substrate, complex topologies emerge from the shared paths jointly optimizing the network efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The flow-weighted transportation efficiency therefore be- comes: E(G({we})) = 1 N(N − 1) N � i N � j̸=i Tij · cij({we}) (2) We also require that the total network infrastructure cost, defined as the cumulative sum of edges weights per unit length, multiplied by edge distance CG = � e∈G dewe is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This is a generalization of a standard optimization process, in the sense that when Tij = 1, ∀(i, j), the efficiency is optimized for all possible trip pairs (i, j) with equal importance, where the Mini- mum Spanning Tree often represents the optimal solution [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Before tackling the problem of traffic-like (OD) flows, we study a simpler definition of Tij, which allows to un- derstand the role of distance in the optimization process, in absence of other nodes’ features: Tij ∝ e−βdij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' (3) The coefficient β appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 3 is introduced as a penalizing parameter and determines how relevant is the pair-wise distance dij when computing probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We can understand it as the inverse of a characteristic traveling distance for an agent on the network β ∼ 1 d0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' While several alternatives on the integration of distance in spatial-dependent probabilities (such as power-laws Tij ∼ d−γ ij ) can be employed, we focus on the exponential dependence as it represents the foundational result from the maximum entropy derivation of gravity flows [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The introduction of gravity-like flows will be discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the following, we introduce the application of the model on simple substrates to explore the role of β in absence of spatial urban features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' OPTIMIZATION OF SIMPLE NETWORK SUBSTRATES In order to asses the role of the characteristic distance parameter β in the emergence of specific topologies, we compute networks statistics on a set of generative models for both spatial and non spatial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As hexagonal tiling of space is preferable when an analysis includes as- pects of connectivity [28], the first model we study is a triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The reason behind this choice is that it represents the planar dual [14, 32] of the hexagonal lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Therefore, as space is discretized in hexagonal tiles, the spatial network connecting its centers is the triangu- lar lattice, which is isotropic and presents less equivalent degenerate paths of a rectangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As a direct reference to hexagonal tiling, we refer to this network as HEX (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We also extend the analysis also to the case of a random network model where nodes are not embedded in a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifically, we study an Erd˝os-R´enyi (ER) network topology, where the def- inition of distance between nodes Lij can be defined in terms of topological shortest path distance [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As a first benchmark we simplify flows as a spatial probability Tij = pij that decays exponentially with dis- tance and does not consider nodes features, the resulting equation for pij is: pij = e−βdij/⟨d⟩ �N−1 k̸=i e−βdik/⟨d⟩ (4) Where ⟨d⟩ = 1 N(N−1) � i̸=j dij is the average distance of points in the network and acts as a normalization fac- tor (euclidean distance ⟨r⟩ in case of a spatial network or topological ⟨L⟩ for the ER network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Therefore pij encodes how much of the nearby space is explored by a single source node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' An illustration of the spatial dependence of target probabilities and samples of the resulting optimal topologies are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' For a range of β values the optimization process is performed on an ensemble of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' To assess the emergence of complex structures we observe the number of loops that emerge in the optimal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This mea- sure is relevant in the context of spatial networks, where loops break the symmetry introduced by optimal struc- tures such as trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We compute the minimum cycle ba- sis set as a metric to observe the emergence of loops [36]: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' the minimum set of loops (where a single loop is en- coded in a set of edges that defines a closed path in the graph) such that any other closed path in the network can 5 Optimal networks and edges weights distribution P(we) Exponential Distribution Morphology A B β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 3-Points Steiner Tree Morphology Steiner node 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 Wj β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 G-KDE FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Minimal models of urban morphology and attractiveness distributions under study (3-Points and Exponential decay): Morphology of POIs, where attractiveness Wj is mapped with color intensity (yellow being higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Optimized edges weights distributions P({we}) are characterized by the bi-modal nature that reveals the multi-layered structure of the optimal transportation networks when close-range flows are paired with long-range traffic typical of commuting towards city centers (Insets P(Tij) with peaks on large-flows due to POIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Gaussian KDE is shown in orange as a visual aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) 3-points polycentric distribution of POIs, resembling the euclidean Steiner Tree problem [34, 35] for three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The network is optimized with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 and β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0, and shows the appearance of branches connecting the POIs paired to large loops in the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Optimal state and distribution of speeds with exponential decay of wj from the center and an exemplifying result with β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The optimal topology is characterized by a central loop paired with branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' be reconstructed via combination of this cycle basis [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifically, we investigate the cycle basis dimension (the number of loops that constitute this set) and the average loop size, against a range of β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This metric allows to quantify the emergence of spatial topological features that differentiate the optimal state from a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Results for these synthetic systems are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Additional boxplots are shown in SM Figures 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A tree-like topology is recovered when the flows probabil- ities are distributed uniformly across all nodes in space (when β → 0 and distance is therefore not a penalizing variable in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4), while loops emerge when farther tar- gets become less likely to be explored and the network is globally optimized for close-range trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Notably, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 3 around β ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0, we observe a sharp transition in the average loop size in the HEX lattice under analy- sis: connections appear between neighboring nodes which are far from the tree-root as it becomes more efficient to have a direct link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In this β regime the tree topology does not guarantee the most efficient configuration for peripheral nodes, which have their high probability tar- gets in their direct neighborhood (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Thus in the optimization process edges appear between leaves nodes which are in separated branches: this ultimately breaks the tree structure and leads to the emergence of large- scale loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Eventually the optimal network converges to a simpler structure with small loops as the network is optimized for nodes to target only direct neighbors in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Finally, in SM Section 2 we show an application on the case of a single target node in the perimeter of the lattice, where the model reproduces leaves venation patterns [14, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' SPATIAL ATTRACTIVENESS AND TRAFFIC-LIKE FLOWS In the context of urban systems, optimal transporta- tion networks need to be devised to accommodate traffic flows [26] towards specific areas of interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' due to high commercial and business land use density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Hence we extend the efficiency optimization framework in the case where we have more realistic traffic on top of the urban networks, as the presence of nodes with high attractive- ness (POIs) biases the flows towards them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In urban sce- narios we adopt spatial-interaction models to mimic more traffic-like flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In these models, flows are obtained via a gravity-like equation: Tij ∝ pipj exp (−βdij) [10] which can be derived from first principles via entropy maxi- mization, thus representing the most likely set of flows to be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the context of urban exploration, the gravity equation can be mapped to a model for spatial in- teraction [31, 38] where nodes with a given attractiveness Wj compete as possible targets for traffic: Tij ∝ 1 Z PiWj exp (−βdij) (5) Normalization Z accounts for all possible trip al- ternatives � k Wk exp (−βdik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Pi is the population density in node i and Wj encodes a suitable definition of benefit/attractiveness of node j as a possible target [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Tij is therefore the fraction of population in β: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 102 ("1) P 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 Tijbeta is:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 100 Gaussian KDE P(We) 10-1 2 3 4 Weβ: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 101 P(Tij) 100 10-1 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='beta is:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 100 Gaussian KDE P(We) 10-1 10-2 2 4 6 8 We6 node i commuting/travelling on average to node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' To better understand the role of nodes’ attractiveness, we start with the simplest assumption of equal population distribution on all nodes: Pi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 ∀i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' we will introduce more realistic population distribution in the next section with the London case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We apply these models on the triangular lattice to unravel the optimal topologies that emerge when traf- fic probabilities are biased towards some nodes having high attractiveness (simulating POIs) and we study two spatial configurations for nodes’ Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the first configu- ration high Wj is assigned to three nodes (POIs) placed at the vertices of an equilateral triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We study the 3- points distribution as it mimics a prototypical polycentric distribution of city-centers, and it can be linked to the solution of the euclidean Steiner Tree problem [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The Steiner Tree is a class of problems where given a set of N points in a plane the goal is to find the set of lines connecting the points with minimum cumulative length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In our case, the solution would lie in the central node of the lattice being the Fermat point [35] and the Steiner node, which connects the three vertices of the high Wj triangle, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4 panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The second case is a distribution of Wj that decays exponentially from the center, mimicking a more realistic morphology for a ur- ban monocentric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The two morphologies are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We find that due to nodes in the network biasing the traffic flows, as it can be seen in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4 A, the traffic flows get divided into two types: a close range paired to a long range set of flows, due to POI polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4 optimal solutions for values of β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Interestingly, optimal solutions are char- acterized by three central lines branching from the cen- ter (which therefore acts as Steiner node) and connecting the three nodes with high attractiveness, therefore resem- bling the solution of the Steiner tree problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Moreover, in the case of more localized flows (β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0) these lines are also paired with large scale loops connecting farther nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We also find that the heterogeneity of traffic flows forces the appearance of a second mode in the distribu- tion of speeds we (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The two peaks in the optimal P(we) can be interpreted as two different lev- els of speed, which suggests that the entire process can be decomposed in two distinct mechanisms which can be mapped as a bi-layer network: one layer at high ca- pacity with long-range/commuting trajectories and the other one at low velocity with short-range paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These two layers can be ideally separated, hinting towards a possible extension of the model to multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' GREATER LONDON AREA: GENERATIVE MODEL FOR THE SUBWAY SYSTEM We extend in this section the application of the model by integrating data from a real urban structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifi- cally, we model the morphology of Greater London Area (GLA) on top of our framework and apply the efficiency optimization process with the aim of understanding if the temporal efficiency optimization of the spatial sub- strate paired with realistic flows is sufficient to yield a transportation network with similar topological features (such as a central core paired with peripheral branches [22]) as the London subway system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' To extend the model to real urban scenarios, we first obtain the distribution of amenities [39] from OpenStreetMap [40] and we use this density of points in space as a proxy to estimate the attractiveness Wj of a tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Census data for Greater London Area yards from 2014 is used to recover popu- lation density Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' These densities are then mapped to Uber’s H3 tiling to recover the spatial discretization in hexagonal tiles, such that we can have direct mapping to the nodes on a triangular lattice, as in the examples dis- cussed in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We thus have the ingredients to finally simulate the spatial interaction flows Tij in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 5 the integration of urban data describing the London’s morphology in the model is explained and we provide a depiction of the OD matrix that arises from the spatial interaction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' With the aim of repro- ducing real features, we impose an upper limit on edge weight, so that the distribution of weights is bounded during the optimization process: we ∈ (0, w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This bet- ter simulates the upper bound in speed of real multilayer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Further explanation of data recovery and inte- gration in the model is provided in SM Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We find (see SM Figure 6) that {we} distribution displays a bi-modal shape, and this allows the analysis of the gen- erated network in a sub-graph defined by the set of high speed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 5, panel C, we show a sample re- sult for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='35 of this sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The characterization of the network into a central core paired with peripheral branches as the optimal state can be visually observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The model’s subgraph of high speed edges is compared to the real tube network in the Greater London Area [21] to assess the similarities between the optimal struc- ture and the real subway system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We quantify this sim- ilarity by means of spatial scaling laws [22], these are convenient to highlight the recovery of the central core structure characterized by loops, paired with quasi mono- dimensional lines branching from the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We investi- gate the distribution of nodes stations using the profile function N(r) that quantifies the total number of stations at a distance r from the network barycenter, computed as the average location of all station nodes [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Results of this scaling analysis for the real and simulated net- works are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The two scaling regimes indicate the separation of core and branches: the scaling of r2 in the core center and a second trend due to mono- dimensional branches for r > rC, where rC is the radius of the core structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The second trend can be computed analytically via an integral curve for N(r > rC) which can be approximated by a power law rγ (γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='02, see SM Section 4), as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The curve of N(r) is consistent with the real network and confirms scaling 7 Efficiency optimized Network state on GLA Morphology Pi Wj POI Distribution (OSM) Census Data Tij ∝ PiWα j exp (−βdij) OD fluxes Network optimization under realistic fluxes Greater London Area + Metro network A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Optimal network model for Greater London Area subway system: Application of the efficiency optimization with realistic flows on the urban structure of the Greater London Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) Urban morphology data is recovered from Census and OSM and population and POI densities are mapped to the H3 tiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Data is mapped to the triangular lattice, with nodes having features which allow the calculation of traffic-like flows, a sample OD matrix is shown where Tij are computed with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' C) Optimal network state for the London model, where only edges and nodes corresponding to the second mode are shown (see SM Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Central core structure with loops paired with peripheral branches can be visually seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' laws prediction from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' core branches FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Scaling properties of GLA Tube stations: Profile of the number of stations (nodes in the optimal dis- cretized network (see SM Section 3) reproducing GLA under- ground) versus the distance from the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The scaling of N(r) profile of the model is compared with the real net- work system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Scaling properties predicted in [22] are verified, finding the two different scaling regimes separated at r rC ∼ 1 for core paired with branches systems, where rC is the core radius (characterized by r2 scaling), and NC is the number of stations in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The scaling exponent γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='02 is obtained as a linear fit of the integral curve [22] for r > rC (see SM Section 4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' DISCUSSION Starting from simple conditions on temporal efficiency on a spatial network substrate, we show that network op- timization paired with traffic-like flows weighting the im- portance of specific connections in space can reproduce complex networks features from man-made transporta- tion networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifically, we devise a framework for spatial networks where nodes can encode features of ur- ban systems and can ultimately lead to the study of opti- mal topologies in real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A key novelty lies in the optimization process happening on a spatial substrate, such that edges of the resulting optimal network are op- timized to improve the efficiency of the shared space by all nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We show how the probabili- ties of moving from one point to another in space force a transition between a tree-like and a lattice-like topology in the optimal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Fixing certain target points in space with a higher attractiveness for flows can repro- duce theoretical results such as the Steiner tree solution or leaves venation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We also show that extend- ing these probabilities using urban spatial information and traffic-like flows modeling forces the emergence of shared preferential paths that are organized as complex topologies, resulting from traffic weighted optimization of network time efficiency, which ultimately exhibits the characteristics seen in real systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We recover features such as a bi-modality in the speed distribution of the edges of the network, characteristic of multilayer trans- portation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Or the appearance of a central core with loops coupled to branches typical of underground systems, as in the case of the London underground system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We find that branches paired to large loops structures appear as optimal structures when the network is optimized for an Tiling (HEX3-Res6) and POI Density OSM amenities distribution Mapping to the hex lattice model W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' extraction - here log(density) London Greater Area 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='8 Lat 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 51.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='9 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='7 Traffic Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='7 Traffic Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2Rescaled number of stations at distance r from barycenter Power law r2 101 Model LGA Tube Powerlaw ry 100 10-1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='OL 10-1 1008 interplay of traffic flows mixed between small range trav- els and longer range ones typical of commuting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This novel framework for the optimization of spatial networks in urban contexts may show further improvements and extensions to better accommodate the concepts of multi- layer and shared space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' It could be extended also to the case of inter-cities transportation, where specific nodes in the network substrate represent cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' To conclude, in this work the problem is addressed in a theoretical way with the aim of reproducing and understanding some fea- tures observed in real spatial networks, but future works can exploit this framework as a basis to understand how to generate optimal transportation networks in a urban planning scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Competing Financial Interests The authors declare no competing financial interests Data Availability The data used in this work are publicly available from the original references Code Availability The code to perform the analysis will be available upon request.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Hidalgo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Casta˜ner, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Sevtsuk, Habitat International 106, 102205 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' [40] OpenStreetMap contributors, “Planet dump re- trieved from https://planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='osm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='org ,” https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='org (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 10 SUPPLEMENTARY MATERIAL FOR “EMERGENCE OF COMPLEX NETWORKS TOPOLOGIES FROM FLOW-WEIGHTED OPTIMIZATION OF NETWORK EFFICIENCY” S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' APPLICATION ON SIMPLE NETWORK SUBSTRATES In this Section we report more detailed results for the optimization of simple network substrates that was discussed in the Main text, specifically for the triangular (HEX) lattice (planar dual of the hexagonal tiling of space) and Erd˝os-R´enyi (ER) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The aim is to explore the resulting topologies that emerge both in spatial and non spatial networks when simple probabilities are taken into consideration (see Main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' HEX Lattice and ER Network Results of model application on the triangular (HEX) lattice and results of model application on an Erd˝os-R´enyi (ER) non-spatial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' For the non-spatial generative model, an ensemble of 20 networks with N = 30 nodes and edge probability ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' For the spatial case, the optimization process is repeated 20 times for each value of β on the triangular lattice with N = 37 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1 and SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 2 show boxplots for the distributions of metrics computed on the cycle basis in panels A on both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In panels B, samples of the optimized network states are shown for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 11 Hex Lattice N = 37 A B Supplementary Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Cycle basis properties and samples - HEX lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Boxplot statistics for the cycle basis dimension and optimal network samples for triangular lattice across different β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Average Loop Size vs β 10 8 - edges] 6 # Size 4 : 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 Cycle Basis Dimension vs β Dimension [# cycles] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 βB: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 Yβ: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='001 7β: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 YYβ: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0β: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1β: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 蒸β: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 茶β: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5β: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='012 ER Network N = 30, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 A B Supplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Cycle basis properties and samples - ER network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Boxplot statistics for the cycle basis dimension and optimal network samples for an Erdos-Renyi (ER) network across different β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Average Loop Size vs β 8 6 5 #1 4 Size S 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 Cycle Basis Dimension vs β Dimension [# cycles] 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 ββ: 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0B: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0001β: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5β: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0β: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='01β: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0β: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0β: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5β: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='013 S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' LEAVES PATTERNS Here we show an application of the model to reproduce leaf venation patterns [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A single attracting node (single target or sink) is considered at one of the perimeter nodes of the lattice, and the substrate is optimized using all the other nodes as sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Supplementary Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Optimal networks resembling leaves’ veins patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Optimal state when a single target (orange node) in a single spatial extremity is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Efficiency is optimized for all nodes in space to reach the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The resulting optimal state resembles tree-like patterns found in leaves (A), while the distribution of edges weights is shown in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B beta is:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 100 10-1 P(We) 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 100 101 We14 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' APPLICATION ON LONDON TUBE NETWORK In this section we discuss more in detail the application of the model on the Greater London Area urban morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We show that the London subway network spatial properties can be recovered by means of the optimal network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In particular, to simulate a real transportation system, an upper bound on edges travel velocity is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Data Integration: Census and OpenStreetMap Data To extend the model to real urban scenarios, we gather data regarding the urban morphological structure from OpenStreetMap (OSM) [40] and Census.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' OSM Data To model the attractiveness of nodes in the lattice, we use the density of Points Of Interest in the urban space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We use amenities [39] points in OSM as a proxy for Points Of Interest (POIs), and a node j attractiveness (Wj) therefore encodes the density of amenities in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The bounding box for Greater London Area (GLA) is obtained via OSM (https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='org/ wiki/Bounding_Box) and POI densities are recovered inside this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifically we use the following amenities sub-categories: ’cafe’,’college’,’library’,’school’,’university’,’kindergarten’,’restaurant’,’pub’, ’fast food’,’bar’,’bank’,’dentist’,’pharmacy’,’hospital’,’clinic’,’doctor’,’arts centre’,’cinema’,’community centre’,’police’,’post office’,’marketplace’ In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S4 the amenities points recovered from OSM are plotted in the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Lat Lon 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='72 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='35 Map background is provided by OSM under Open Database License Supplementary Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' OSM amenities query for GLA bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Amenities retrieved from OSM (https://openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='org/copyright) for the bounding box defined by the following longitude [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='35] and latitude interval [51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='25, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' tMisisenden Chippin AONB Amersham Ingatestone ghwycombe Billericay consfielo wWickto Basildon arlow Maidenhead Stanford-le endon M4 Windsol Bracknel M2 ton M20 ghtwater Snodland Biggir LHET WestMalling Ma nam eafletLDataby@OpenStreetMapunderODbl15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Census Data London Wards - Census Data 2014 A B C Supplementary Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Census data retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) London’s wards data from Census 2014 (https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='london.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='uk/dataset/ward-profiles-and-atlas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Points are generated in space following wards polygons with a density proportional to Census data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' C) Points are mapped to HEX3 tiles with associated densities of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Finally, a restriction to a disc is used to enforce symmetry in the distribution of nodes and to ease the computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Points are mapped to density in space by using HEX3 tiling (https://eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='uber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='com/h3/) with spatial resolution RES = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This spatial discretization process via tiling is of particular relevance to map this information to the HEX triangular lattice model which was described in the Main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S4 show how model information Pi and Wj are recovered from data and then mapped to tiles as in shown in panel C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Tiles covering the urban area are then mapped to the spatial network model as nodes in the triangular lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 5 in Main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' For computational simplicity and to preserve the isotropy of the lattice substrate from its central point, we restrict our analysis to a disc centered in the London’s region with highest attractiveness, that lies approximately in the City Of London district.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Results are robust against integration of the remaining GLA region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As the optimal edges weights are influenced by the traffic on the substrate, the discarded regions in the Greater London Area do not add relevant contributions when compared to more central regions with higher population density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Results with distributions for traffic and bounded weights distribution To simulate a real system we limit upper edges weights to a fixed value w∗ = 7 ∗ winit where winit is the initial edges weight assigned to the network state (winit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0) before optimization process, such that � e∈G dewe,init = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In this simulation we work with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=" 'LS 51." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='216 A B Pi Wj Urban morphology mapped to The hexagonal lattice Optimal network and edges weights distribution Supplementary Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Mapping the data to Hex lattice + GLA optimized network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) Urban morphology data mapped to the HEX tiling and then mapped to the triangular (HEX) lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' At this step, OD matrix is generated and the network is then optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) Resulting optimal network state with its distribution of edges weights showing a bi-modal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Only fast edges having weight larger than a threshold (we > 5) are kept to isolate the sub-graph constituted by a high velocity set of edges, such as a subway system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A London Metro network (Restricted to disc) Model output (pruned) B Supplementary Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Pruned network vs real tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' A) London subway network, restricted to the disc under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' B) GLA model output, limited to the nodes and edges which have a weight we > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0, where this value was chosen from the previously plotted {we} distribution as a threshold to separate the second “mode” with fast edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' This “discretized” network is the model on which statistical measures are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Model network0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='4 P(we) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 0 1 2 4 5 6 7 WeCG Real network Model pruned (2nd mode)17 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' SCALING OF NETWORK STATIONS The spatial organization of the transportation network can be inspected by taking into consideration the number of nodes’ stations N(r) up to a distance r from the barycenter of stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In the Main text we analyze the scaling regimes for the simulated and real London Tube network [21] following the analysis employed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Specifically, they obtain functional forms for the scaling properties of N(r) for different distances from the barycenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' We report here the scalings obtained in [22]: N(r) ∼ � � � � � ρCπr2 for r < rC ρCπr2 C + NB � r rC dr ∆(r) for rC < r < rmax N for r > rmax (S1) Specifically, they show that in the large distance regime (r > rC and r < rmax ) the number of stations can be approximated by adding the integral curve NB � r rC dr ∆(r) to a constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In [22] it is also reported that the large distance behavior can be also, in general, approximated by a scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Therefore we plot here the computation of the integral curve against rescaled values of r, and show that in that regime it can be approximated by a power law, and its exponent can be obtained via a linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The N(r) curve in the Main text was computed on the real London Tube restricted to the disc area as presented in SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The associated exponent γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='02 was used in the Main text to highlight the secondary scaling for r > rC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' As discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' [22], due to ∆(r) in SM Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' S1 being often noisy, this scaling property is not often well reproduced in empirical networks, and it is often restricted to a small region of r values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' 6 of the Main text, we see that the secondary scaling with the exponent γ approximates N(r) in a limited interval for r > rC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' The value of the exponent in the branches region is expected to be γ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='0 [22] and our result is consistent with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content=' Supplementary Figure S8.' metadata={'source': 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https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='openstreetmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} +page_content='org (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFAT4oBgHgl3EQftB4M/content/2301.08661v1.pdf'} diff --git a/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/2301.03561v1.pdf.txt b/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/2301.03561v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b094ca71ce0fa074b3931af91395659ea2726091 --- /dev/null +++ b/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/2301.03561v1.pdf.txt @@ -0,0 +1,2536 @@ +IEEE INTERNET OF THINGS JOURNAL +1 +Ancilia: Scalable Intelligent Video Surveillance for +the Artificial Intelligence of Things +Armin Danesh Pazho∗, Student Member, IEEE, Christopher Neff∗, Student Member, IEEE, Ghazal Alinezhad +Noghre, Student Member, IEEE, Babak Rahimi Ardabili, Student Member, IEEE, Shanle Yao, Mohammadreza +Baharani, Member, IEEE, Hamed Tabkhi, Member, IEEE +Abstract—With the advancement of vision-based artificial in- +telligence, the proliferation of the Internet of Things connected +cameras, and the increasing societal need for rapid and eq- +uitable security, the demand for accurate real-time intelligent +surveillance has never been higher. This article presents Ancilia, +an end-to-end scalable, intelligent video surveillance system for +the Artificial Intelligence of Things. Ancilia brings state-of-the- +art artificial intelligence to real-world surveillance applications +while respecting ethical concerns and performing high-level +cognitive tasks in real-time. Ancilia aims to revolutionize the +surveillance landscape, to bring more effective, intelligent, and +equitable security to the field, resulting in safer and more secure +communities without requiring people to compromise their right +to privacy. +Index Terms—Surveillance, artificial intelligence, IoT, com- +puter vision, application, real-world, real-time, edge, anomaly. +I. INTRODUCTION +There is a growing need for effective and efficient surveil- +lance technologies that can be deployed to protect our cities, +people, and infrastructure. For example, in Itaewon, South +Korea, a holiday celebration left over 150 dead due to severe +overcrowding, with many blaming the tragedy on careless +government oversight [1]. In Moore County, North Carolina, +directed attacks against two power substations left over 45,000 +residents without power for days as technicians rushed to +restore power and authorities struggled to find the source of +the attacks [2]. With enough forewarning through smart video +surveillance, they could have been prevented. +With the recent emergence of the Artificial Intelligence +of Things (AIoT), some surveillance solution providers have +started adding basic forms of artificial intelligence to their +systems. However, their methods are still naive and unable +to enhance security in a truly meaningful way [3]. This is +because, while a lot of research is conducted on tasks that +would benefit surveillance systems, most works focus on +algorithmic improvements in a lab environment instead of +paying attention to factors that are prevalent in real-world +scenarios [4], [5]. Most research focuses on a single algorithm +and how to tweak it to get the best possible results on readily +available datasets that often do not reflect a real surveillance +environment. Few works explore how different algorithms af- +fect the performance of other downstream algorithms in multi- +The authors are with the Electrical and Computer Engineering Department, +The University of North Carolina at Charlotte, Charlotte, NC, 28223 USA. +{adaneshp, cneff1, galinezh, brahimia, mbaharan, htabkhiv}@uncc.edu +∗ Corresponding authors have equal contribution. +algorithm systems. Few still explore the effects of noise (both +data derived and the system produced) in end-to-end accuracy. +Beyond this, real-world intelligent surveillance necessitates +real-time performance. The cognitive abilities of advanced +artificial intelligence are only helpful if they can be provided +to security personnel quickly enough to take appropriate action +before it is too late. +EdgeN-1 +Camera(s) +C0 +C1 +CN-1 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Global +Node +EdgeN-1 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +EdgeN-1 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +EdgeN-1 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local Node(s) +L0 +L1 +LN-1 +Global Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Global +Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Edge0 +Camera(s) +C0 +C1 +CN-1 +Local +Node(s) +L0 +L1 +LN-1 +Global +Node +Cloud +Service(s) +User +Device(s) +D0 +D1 +DN-1 +User +Device(s) +D0 +D1 +DN-1 +Fig. 1. Conceptual overview of Ancilia. +In this article, we present Ancilia, the first end-to-end +scalable, intelligent video surveillance system able to perform +high-level cognitive tasks in real-time while achieving state- +of-the-art results. Ancilia takes advantage of the prevalence +of cameras in the Internet of Things (IoT) and uses localized +servers and existing cameras, facilitating processing on the +edge without the need for additional infrastructure upgrades. +Shown in Fig. 1, Ancilia exists within three logical and +physical segments: the edge, the cloud, and user devices. +The edge uses a plethora of advanced artificial intelligence +algorithms processing data received from cameras to facilitate +intelligent security. Using a single workstation to perform edge +processing, Ancilia can monitor up to 4 cameras in real-time at +30 FPS, or up to 8 cameras at 15 FPS, in scenarios with both +medium and heavy crowd density. Ancilia performs high-level +cognitive tasks (i.e. action recognition, anomaly detection) +with ∼ 1% deviation in accuracy from current SotA. +Ancilia is designed from the ground up to respect the pri- +vacy of the people and communities being surveilled. Ancilia +does not store any personally identifiable information in any +databases and does not make use of invasive artificial intelli- +gence techniques such as facial recognition or gait detection. +Ancilia strictly provides a pose and locational information for +high-level tasks (i.e. action recognition, anomaly detection), +as opposed to identity information, which is common. Ancilla +arXiv:2301.03561v1 [cs.CV] 9 Jan 2023 + +0IEEE INTERNET OF THINGS JOURNAL +2 +looks at what a person is doing, not who they are. This allows +Ancilia to act as a buffer to help remove biases based on +race, ethnicity, gender, age, and socio-economic factors, which +can lead to a reduction in the unnecessary conflict between +authorities and marginalized communities that has become +increasingly problematic. After data is processed on edge and +sent to the cloud for communication and service management +with user devices. A mobile app allows user devices to receive +data from the cloud, including alerts when potential security +concerns arise. +In summary, this article has the following contributions: +• We present Ancilia, the first end-to-end scalable real- +world intelligent video surveillance system capable of +performing high-level cognitive tasks in real-time while +achieving SotA accuracy. +• We analyze the ethical concerns of intelligent video +surveillance, both from a privacy and fairness perspective, +and illustrate how Ancilia’s design is purpose-built to +address them. +• We perform an end-to-end empirical evaluation of Ancilia +using two high-level cognitive tasks directly related to +intelligent surveillance, action recognition, and anomaly +detection, investigating the trade-off in accuracy required +to achieve real-time performance. +• We perform an exhaustive system-level evaluation of +Ancilia’s real-time performance and scalability across +different classes of hardware and increasing scenario +intensities, displaying how Ancilia is able to meet real- +time intelligent security needs in different contexts. +II. RELATED WORK +There has been a plethora of research regarding the use +of artificial intelligence for video surveillance [4], [6]–[8]. +[9] proposes the use of region proposal based optical flow to +suppress background noise and a bidirectional Bayesian state +transition strategy to model motion uncertainty to enhance +spatio-temporal feature representations for the detection of +salient objects in surveillance videos. [10] proposes the use +of a person detector, tracking algorithm, and mask classifier +for tracking pedestrians through surveillance video streams. +In [4], it is determined that in order to address the latency +concerns of real-time video surveillance, a shift towards edge +computing is needed. Nikouei et al. [11]–[13] explore the +feasibility of using low-power edge devices to perform object +detection and tracking in surveillance scenarios. They argue +that in worst case 5 FPS is high enough throughput for tracking +humans in surveillance applications, and as such computation +can be pushed to the edge. However, their results show that +even light weight convolutional neural networks can prove +problematic for low-power devices, often reducing throughput +below the 5 FPS threshold. [14] proposes a system using low- +power embedded GPUs to perform detection, tracking, path +prediction, pose estimation, and multi-camera re-identification +in a surveillance environment, while placing a focus on real- +time execution and the privacy of tracked pedestrians. [15] +proposes a similar system, focusing solely on object detec- +tion, tracking, and multi-camera re-identification to increase +throughput. [16] proposes using a combination of lightweight +object detection models on the edge and more computation- +ally expensive models in the cloud, splitting computation +between the two to provide real-time video surveillance in +a construction site environment. [17] proposes the use of +background detection, vehicle detection, and kalman filter [18] +based tracking for parking lot surveillance and determining lot +occupancy. [19] proposes a system that uses object detection, +person tracking, scene segmentation, and joint trajectory and +activity prediction for pedestrians in a surveillance setting. +The future of intelligent surveillance is heading towards +systems able to perform high-level cognitive tasks. A recent +survey focusing on real-world video surveillance [4] asserts +that while the domain of video surveillance is comprised +of understanding stationary object, vehicles, individuals, and +crowds, the ability to determine when anomalous events oc- +cur is paramount for intelligent surveillance systems. Other +research has supported this assertion [6]. [20] utilizes the +Infinite Hidden Markov Model and Bayesian Nonparametric +Factor Analysis to find patterns in video streams and detect +abnormal events. [21] proposes active learning and fuzzy +aggregation to learn what constitutes an anomaly continually +over time, adapting the scenarios not seen in standard datasets. +[22] proposes a system to detect suspicious behaviors in a +mall surveillance setting, using lightweight algorithms such +as segmentation, blob fusion, and kalman filter based tracking +[18]. AnomalyNet [23] is a recently proposed recurrent neutral +network with adaptive iterative hard-thresholding and long +short-term memory that works directly off pixel information to +eliminate background noise, capture motion, and learn sparse +representation and dictionary to perform anomaly detection in +video surveillance. +III. ETHICAL CONCERNS +Video surveillance has always been associated with social +and ethical concerns, whether in traditional form or more +recent intelligent formats. Respecting citizens’ privacy and +autonomy while improving public safety and security are the +most well-known and enduring ethical issues in this context +[24]–[27]. Developing a successful smart video surveillance +solution that addresses the public safety problem and engages +the community up to a certain level is only possible by +considering these concerns. +There is rising attention among scholars to the issue of +incorporating privacy concerns at the design level, referred to +as ”privacy by design” [28]. The source of discrimination and +privacy violation in many data-driven and AI-based systems, +such as Smart video surveillance technology, is using Personal +Identifiable Information (PII) [29], [30]. Using PII, such as +actual footage of people’s daily activities at any stage of the +technology, can increase the risk of privacy violation. There is +a long-lasting debate on the ethical challenges of using facial +recognition technologies in different sectors and how using +this technology can result in privacy violation [31]–[34]. +Avoiding facial recognition technologies does not guarantee +the system is entirely privacy persevering. Storing images of +pedestrians is another source of ethical violation. From the + +IEEE INTERNET OF THINGS JOURNAL +3 +Neural +Network +Filter +Match and +Combine +Algorithm +SQL +Database +Node +Boundary +Cloud +Cloud +(C) Cloud +Node +SA +Flow +within a +Node +Flow +Between +Nodes +Communication +AR +R +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node N-1 +F +PB +OB +T +P +PT +C +PT +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node N-1 +F +PB +OB +T +P +PT +C +PT +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +Database +Global +Tracker +Statistical +Analysis +(B) Global Node +FL +FD +IDG +FL +D +I +SA +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +High-level +Tasks +Crop +Selection +(A) Local Node 0 +BBP +BBO +IDL +P +P,IDL +C +P,IDL +BBP +FL +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +High-level +Tasks +Crop +Selection +(A) Local Node 0 +BBP +BBO +IDL +P +P,IDL +C +P,IDL +BBP +FL +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node N-1 +F +PB +OB +T +P +PT +C +PT +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Video +Stream +Tracker +Pose +Estimator +Feature +Extractor +Downstream +Tasks +Crop +Selection +(A) Local Node 0 +F +PB +OB +T +P +PT +C +PT +PB +E +D +Database +Global +Tracker +Statistical +Analysis +(B) Global Node +FL +FD +IDG +FL +D +I +SA +IoU +Confidence +Confidence +Object +Detector +Pedestrian +Tracker +Pose +Estimator +Feature +Extractor +High-level +Tasks +Crop +Selection +(A) Local Node 0 +BBP +BBO +IDL +P +P,IDL +C +P,IDL +BBP +FL +D +Edge +Boundary +Video +Stream +from +Camera +(D) User +Device(s) +AR +R +User +Device +Fig. 2. Ancilia algorithmic details. N local nodes are connected to a single global node on the edge. The final analyses are transferred to the cloud node to +feed the application on the user device. Multiple edges may be connected to the could, though this figure only shows one edge for clarity. BBP , BBO, IDL, +P, C, FL, D, FD, IDG, I, SA, R, and AR refer to bounding boxes for pedestrians, bounding boxes of objects, local identities, poses, person crops that +passed selection, features from the local node, data from the downstream tasks, features from the database, global identities, information from the database, +completed statistical analysis, requests from users, and requested attributes respectively. +discrimination perspective, using any form of PII can con- +tribute to the issue of marginalization in policing systems [35]. +Therefore, an essential step in designing a non-discriminatory +system is to ensure the system is not dependent on PII. +This requires a specific approach toward the design of such +technology in the choice of algorithm, the type of data used,, +and the storing of such data. +Ancilia addresses this by not storing any PII or sending any +PII across the network. Such data is destroyed after it is used. +Ancilia utilizes pose-based methods for all high-level cognitive +tasks, ensuring no PII is ever used in such algorithms. This +allows for such processing to occur without any potential +for gender, ethnicity, or class-based discrimination. As such, +Ancilia is able to address the ethical challenge of privacy in +smart video surveillance systems while also addressing the +ethical issue of discrimination. +IV. ANCILIA ALGORITHMIC FRAMEWORK +The algorithmic core of Ancilia is separated into two +conceptual systems: the local nodes containing the algorithmic +pipeline of each camera and the global node that handles +all processing that requires understanding of multiple camera +perspectives. These two systems make up the algorithmic core +of Ancilia and are the basis on which all higher understanding +is achieved. A visual representation of this algorithmic core +can be seen in Fig. 2. +A. Single Camera Vision Pipeline +As seen in Fig. 2, the local algorithmic pipeline starts when +an image is extracted from the camera. The image is first run +through an object detector to locate people, vehicles, animals, +and other important objects in the scene. This is important not +only because it acts as the basis for the rest of the algorithmic +pipeline but also because it can be used for basic situational +awareness. Sometimes, just the presence of a certain object +in a scene is noteworthy, like a person in an unauthorized +location, a bag left unattended, or the presence of a firearm. +Ancilia uses YOLOv5 [36] for this purpose (however, it can be +any detector). The locational coordinates of persons are sent to +a tracker, where tracklets are created, matching each person +with their previous detections in prior images. Ancilia uses +ByteTrack [37]. The tracking allows for understanding how +a person moves throughout a scene, which is vital for many +surveillance applications. It also allows Ancilia to understand +which poses belong to which persons over time, which is +vital for many high-level tasks that provide much-needed +situational awareness. Image crops of the people detected in +the image are also sent to a human pose estimator, where two- +dimensional pose skeletons are created. Ancilia uses HRNet +[38] for extracting 2D skeletons. Using pose data for higher- +level tasks has two major benefits over simply using raw pixel +data. First, pose data is of much lower dimensionality than +pixel data, making it much less computationally expensive and +allowing the Ancilia to function in real-time. Second, pose +data contains absolutely zero identifiable information, making +it impossible for high-level tasks to form unintended biases +based on ethnicity, gender, age, or other identity-based metrics. +B. Multi-Camera Person Re-identification +While the tracker tracks people within a single camera, +locational information cannot accurately re-identify a person +across multiple cameras. For this, the same person crops that +were sent to the human pose estimator are also sent to a person +re-identification feature extractor, where an abstract feature +representation is created for each person. Only one feature +representation is created for each person over a period of 30 +frames, and only when the quality of the representation can +be assured, as poor quality representations are detrimental to +accurate multi-camera person re-identification. Ancilia uses a +feature representation filtering algorithm to verify two qualities + +IEEE INTERNET OF THINGS JOURNAL +4 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β2 +β1 +β1 +β1 +β1 +Local Node 0 +β3 +β3 +β2 +β2 +β3 +β3 +β1 +β1 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β1 +β1 +Local Node 0 +β3 +β2 +β3 +β1 +β3 +β3 +β3 +β3 +δ0 +δ1 +δN-1 +δ0 +δ1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β2 +β1 +β1 +β1 +β1 +Local Node 0 +β3 +β3 +β2 +β2 +β3 +β3 +β1 +β1 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β1 +β1 +Local Node 0 +β3 +β2 +β3 +β1 +β3 +β3 +β3 +β3 +δ0 +δ1 +δN-1 +δ0 +δ1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β2 +β1 +β1 +β1 +β1 +Local Node 0 +β3 +β3 +β2 +β2 +β3 +β3 +β1 +β1 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +β3 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β1 +β1 +Local Node 0 +β3 +β2 +β3 +β1 +β3 +β3 +β3 +β3 +δ0 +δ1 +δN-1 +δ0 +δ1 +δN-1 +Camera +Neural +Network +Algorithm +Data +Batched +Data +Sequential +Data +Frame +Batching +Object +Batching +Comm. +Process +IoT +Flow within +a Node +Transfer to +Global Node +Frame +Unbatching +De-identified +Data +Queue +Comm. +Process +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β2 +β1 +β1 +β1 +β1 +Local Node 0 +β3 +β3 +β2 +β2 +β3 +β3 +β1 +β1 +β3 +β3 +β1 +β1 +β1 +β1 +β1 +β1 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +δ0 +δ0 +δ1 +δ1 +δN-1 +δN-1 +Pre- +Processor +Object +Detector +Tracker +A +Pose +Estimator +Task 0 +Task 1 +Task N-1 +Feature +Extractor +Crop +Selection +A +Transfer +FL +D0 +D1 +DN-1 +∞ +∞ +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β1 +β2 +β1 +β1 +Local Node 0 +β3 +β2 +β3 +β1 +β3 +β1 +β1 +β1 +δ0 +δ1 +δN-1 +δ0 +δ1 +δN-1 +Fig. 3. A detailed view of system design in Ancilia’s local nodes. β and δ refer to different batch sizes. FL and D represent local features and data received +from downstream tasks respectively. +for person crops. First, a person crop must contain a high- +quality view of the person. To this end, the filter algorithm uses +the 2D pose skeleton and verifies that at least 9 keypoints were +detected with at least 60% confidence. The filter algorithm +looks at the overlap (i.e. Intersection of Union) of the bounding +boxes generated by the object detector. An individual’s bound +box must have an Intersection over Union (IoU) of no more +than 0.1 with any other person. If those two conditions are met, +the person crop is determined to be of high enough quality to +produce an adequate feature representation. If more than one +crop is deemed suitable for a single person during a 30 frame +window, the one with the most confident pose is selected. The +features created by the feature extractor are sent to the global +node for multi-camera person re-identification. Ancilia uses +OSNet [39] to extract feature representations. +C. higher Level Tasks +High-level tasks are executed on the local node, and have +access to the object, tracking, and pose data generated in the +previous steps. Since the decision of which high-level tasks are +needed is highly application dependent, we do not consider +these tasks to be part of the Ancilia algorithmic core, but +instead an extension to be customized based on intended use. +In this paper, we use action recognition and anomaly detection +as two common examples of high-level tasks that are highly +relevant to intelligent surveillance. For action recognition, we +choose PoseConv3D [40] and CTR-CGN [41], two state-of- +the-art networks that can utilize the 2D human pose skeletons +provided by Ancilia. For anomaly detection, we use GEPC +[42] and MPED-RNN [43], which are based on 2D human +pose skeletons. +V. SYSTEM DESIGN +Beyond the algorithmic design, Ancilia can be analyzed +from a system-level design and implementation perspective. +The local node in particular has a complex system design, as +seen in Fig. 3. The global node and cloud are much simpler, +as shown in Fig. 2. +A. Parallelism +A key design objective of Ancilia is to achieve higher +efficiency by balancing throughput and latency. Ancilia uses +pipelining to take advantage of process parallelism. Each +major task is implemented as a separate process, which +executes concurrently with other processes. These processes +communicate with each other using queues to utilize memory +resources better and enable fast inter-process communication. +While pipelining is a well-known technique for optimization, +the overhead associated with its implementation means a +balance needs to be found. Figure 3 shows a detailed view of +the system design on the local node. Each pipeline stage is sep- +arated by a queue with a size limit of λ1 elements, preventing +any potential overflow from uneven execution speed between +pipeline stages. By default, Ancilia uses a λ1 value of 4. As +is common, Ancilia offloads highly parallel tasks that rely on +neural networks (i.e. object detection, pose estimation, feature +extraction, and many high-level tasks) to Graphics Processing +Units (GPUs) for execution. +B. Data Batching +Batching is another technique Ancilia implement to better +utilize hardware resources. Generally, batching is able to +greatly increase the throughput of a system at the cost of end- +to-end latency. However, many high-level tasks (e.g. action + +IEEE INTERNET OF THINGS JOURNAL +5 +recognition, anomaly detection) require multiple video frames +worth of input data (often called a window) before the can start +processing, so the latency that would be incurred by batching +input frames is already inherent in these high-level task, as +long as the frame batch and high-level task window are of +the same size. Further, as frame batching ultimately increases +the throughput, the end-to-end latency is decreased when +compared processing each frame sequentially. While object +detection works on entire frames, all other neural networks in +Ancilia work off individual objects. These objects are batched +together before being input to the network, greatly increasing +hardware utilization. There can be multiple object batches +within a single frame batch, based on how many of the relevant +objects are detected in the video. +C. Local Node +Once the local node receives the video stream from the +camera, the proprocessor is responsible for all basic image +processing necessary before sending the frames through the +algorithmic core. After preprocessing, frames are batched in +sequential segements of size β1. Ancilia sets β1 = 30 to match +the window size of high-level tasks. This is also convenient +as most modern security and IoT cameras record video at +either 30 or 60 FPS. The batched frames are sent to the object +detector, which outputs a list of objects with class labels and +bounding box coordinates. Bounding boxes for pedestrians are +sent to the tracker, while bounding boxes for other objects +are passed through the system for use in high-level tasks +and statistical analysis. A crop of each pedestrian from the +original frame is passed through to the pose estimator. At +the tracker, bounding boxes for pedestrians are unbatched +to fit the tracker’s sequential operation. The tracker groups +the pedestrians and either matches them with previously seen +pedestrians or assigns them a unique local ID. Afterwards, +the pedestrians are once again batched by frame and sent to +the pose estimator. At the pose estimator, the object batching +is performed on the person crops, with a batch size of +β2 = 32. These batches are fed to the pose estimator, which +outputs human pose skeletons for each person crop. Then the +pedestrian bounding boxes, person crops, local IDs, and human +pose skeletons are once again batched by frame and combined +with the object bounding boxes from the object detector. +The pedestrian bounding boxes, person crops, local IDs, and +pose skeletons are sent to crop selection, while the pedestrian +bounding boxes, object bounding boxes, local IDs, and pose +skeletons are sent to each high-level task as necessary. While +the person crops are necessary for the feature extraction that +enables multi-camera re-identification, no identifiable data is +sent to any of the high-level tasks, keeping in line with the +ethical concerns mentioned in Sec. III. +Crop selection filters out low-quality person crops based on +bounding box overlap and keypoint confidence, as described +in Sec. IV. The remaining crops are then batched, with size β3 +being dynamic based on the number of persons in the scene, +and sent to the feature extractor. Once features are extracted, +the are sent for transfer to the server. Each high-level task +receives data at the granularity of a frame batch, and sends +data to the server at whatever granularity that task requires. +Each high-level task has its own process and works in parallel +with other tasks as well as with crop selection and feature +extraction. Communication is completely decoupled from the +pipeline, so once the data is sent the local node pipeline +continues to function as normal. Importantly, no identifiable +information is ever sent to the global node, keeping in line +with the privacy and ethical concerns mentioned in Sec. III +D. Global Node +All received data is stored in a relational database on the +global node. The matching algorithm described in Sec. IV +compares the received features with existing features in the +database over the period λ5 and assigns a global ID based on +the results. The default value for lλ5 is set to 1 hour, but this +should be changed to suit the needs of the application. An +assortment of algorithms performs statistical analysis using +the relational database, as detailed in Sec. IV. The analysis +is transmitted to the cloud node using APIs provided by the +cloud service provider. By default, Ancilia uses Amazon Web +Services, but this can be altered based on user/application +needs. The cloud (e.g. Amazon Web Services (AWS)) receives +analyzed data from the global node. +VI. EXPERIMENTAL RESULTS +A. Algorithmic Core +The algorithmic core of Ancilia consist of multiple algo- +rithms, each of which works off of data generated by the +previous algorithms. As these algorithms leverage imperfect +neural networks, they generate noise that accumulates through +the system. To understand the source of this noise, we must +first look at the accuracy of each of these core algorithms +in isolation. Table I shows the accuracies of the algorithmic +core’s four main tasks: object detection, pedestrian tracking, +human pose estimation, and person re-identification. The table +also shows the accuracies of the top SotA models in each +task. These SotA methods are not suitable for intelligent +surveillance applications, as their excessive computation and +vast parameters make real-time execution impossible, but the +comparison allows us to see the maximum potential allowable +by current research and the accuracy loss incurred to keep +Ancilia performing in real-time. +TABLE I +ACCURACY OF ANCILIA’S ALGORITHMIC CORE NETWORKS IN +ISOLATION. ACCURACIES OF OBJECT DETECTION AND POSE ESTIMATION +ARE USING THE COCO DATASET [44], TRACKING USING MOT20 [45], +AND PERSON REID USING DUKEMTMC [46]. SOTA REPRESENT THE +HIGHEST ACCURACIES CURRENTLY ACHIEVABLE WITH +STATE-OF-THE-ART METHODS WHEN COMPUTATION AND LATENCY ARE +NOT A CONCERN. +Task +Method +Metric +Accuracy +SotA +Object Detection +YOLOv5 [36] +mAP +49.0 +65.0 [47] +Tracking +ByteTrack [37] +MOTA +77.8 +77.9 [48] +Pose Estimation +HRNet [38] +AP +75.1 +81.1 [49] +Person ReID +OSNet [39] +Top-1 +88.6 +95.6 [50] +Object detection sees the biggest hit to accuracy, with a 16% +drop from SotA. This is intuitive, as YOLOv5 [36] is not only + +IEEE INTERNET OF THINGS JOURNAL +6 +the largest model in the algorithmic core, but also the only +one that operates on the raw camera stream. So while larger +models are available and would be able to produce higher +accuracy, even a slight increase in model size or computation +would result in a noticeable decrease in throughput. Human +pose estimation sees a decrease in accuracy for a similar +reason, though much smaller in scale at only 6%. While +HRNet [38] is not run on the raw camera stream, it is run +individually for each person detected by the object detector. +As such, maintaining a small model size is preferable. Person +re-identification sees a slightly larger drop in accuracy than +human pose estimation at 7%. While this is partly due to +using a lightweight model, OSNet [39], the SotA model for +person reID is also lightweight. However, the SotA uses a +centroid based retrieval method not suitable for pen-set reID, +of which most surveillance scenarios are. Pedestrian tracking +sees almost no drop in accuracy, approximately 0.1%. This +stems from the comparative ease of tracking pedestrians in +a single camera, where a simple, lightweight algorithm like +ByteTrack [37] see almost no performance difference from +the top of the line SotA approaches. +B. High-level Tasks +To better understand how the noise generated by the al- +gorithmic core effects overall performance, and thus how +well Ancilia performs in the realm of real-world intelligent +surveillance, we examine the performance of two high-level +cognitive surveillance tasks when running on Ancilia. For +Ancilia to be a benefit to intelligent surveillance tasks, we +must ensure that excess false alarms or missed positive events +do not occur. To assess this, we choose action recognition +and anomaly detection, as these tasks can utilize the human +pose information generated by the algorithmic core, resulting +in faster and less biased inference. Since both these methods +utilize temporal batches of human poses for each individual, +these experiments will directly reflect the quality of the object +detection, tracking, re-identification, and pose estimation data +generated by Ancilia. +1) High-level Task - Action Recognition: We select two +state-of-the-art action recognition models, PoseConv3d [40] +and CTR-GCN [41], and train them using data generated with +Ancilia. For each model, we train and test with full (30 FPS) +and half (15 FPS) throughput on NTU60-XSub [51]. Both +models use a window size of 30 and are trained for 24 epochs +using Stochastic Gradient Descent (SGD) with a momentum of +0.9 and Cosine Annealing scheduling. PoseConv3d and CTR- +GCN have weight decay of 3e−4 and 5e−4 and an initial +learning rate of 0.4 and 0.2, respectively. +The results of these experiments can be seen in Tab. II. +We report the Top-1 and Top-5 accuracy and compare the +results using data generated by Ancilia to the original data +available through the PYSKL toolbox [52]. We can see that +Ancilia is able to provide data of comparable quality to the +original; action recognition as a high-level task in Ancilia sees +around 1% drop in accuracy compared to the original data +using PoseConv3D [40] at full throughput, and around 3% at +half throughput. Using CTR-GCN [41], Ancilia sees a 2.5% +TABLE II +TOP-1 AND TOP-5 ACCURACIES ON NTU60-XSUB [51] IN FULL AND +HALF THROUGHPUT MODES FOR POSECONV3D [40] AND CTR-GCN +[41]. +Model +Data +FPS +Top-1 (%) +Top-5 (%) +PoseConv3D [40] +[52] +15 +91.96 +99.47 +30 +92.76 +99.57 +Ours +15 +88.79 +98.82 +30 +91.99 +99.28 +CTR-GCN [41] +[52] +15 +86.36 +98.46 +30 +83.07 +98.26 +Ours +15 +81.58 +97.52 +30 +80.44 +97.2 +drop at full throughput and a 4.8% drop at half throughput, +compared to the original data. From this we can infer that +PoseConv3D is more robust to noise than CTR-GCN, however +both performed reasonably well with data generated from +Ancilia, demonstrating its efficacy for intelligent surveillance +applications. +Another interesting observation is that CTR-GCN [41] +actually performed noticeably better at half throughput than +at full throughput. This means that CTR-GCN is more suited +to taking advantage of the higher temporal window allowed +when using half throughput. This is something to consider +when choosing an action recognition model when a real-time +throughput of 30 FPS cannot be guaranteed. +2) High-level Task - Anomaly Detection: Using the Shang- +haiTech dataset [53] we train two state-of-the-art anomaly +detection models, GEPC [42] and MPED-RNN [43], using +both data generated by Ancilia and the data provided by the +original authors. The same training strategy from Sec. VI-B1 +is used, with both models trained in full (20 FPS) and half (10 +FPS) modes. GEPC is trained for 25 epochs with a window +size of 30 and stride of 20 using Adam optimizer with a +learning rate of 1e-4, weight decay of 1e-5, and batch size +of 512. MPED-RNN is trained with an input window size of +30, a reconstruction window of 12, and a prediction window of +6. The model is trained for 5 epochs using the Adam optimizer +with a learning rate of 1e−3 and a batch size of 265. +TABLE III +AUC ROC, AUC PR, AND EER ON SHANGHAITECH DATASET [53] IN +FULL AND HALF THROUGHPUT MODES FOR GEPC [42] AND MPED-RNN +[43]. +Model +Data +FPS +AUC ROC +AUC PR +EER +GEPC [42] +[42] +10 +0.6906 +0.5951 +0.35 +20 +0.7372 +0.6427 +0.31 +Ours +10 +0.6888 +0.5905 +0.35 +20 +0.7223 +0.6023 +0.32 +MPED-RNN [43] +[43] +10 +0.6645 +0.5733 +0.37 +20 +0.7023 +0.5869 +0.36 +Ours +10 +0.6685 +0.5661 +0.37 +20 +0.6679 +0.5487 +0.37 +The results of this experiment can be seen in Tab. III. +In line with current practices, we report Area Under the +Receiver Operating Characteristic Cure (AUC ROC), Area + +IEEE INTERNET OF THINGS JOURNAL +7 +1 +2 +3 +4 +5 +6 +7 +8 +0 +10 +20 +30 +40 +50 +60 +Normal +Heavy +Extreme +Nodes +Throughput (FPS) +(a) Server A +1 +2 +3 +4 +5 +6 +7 +8 +15 +20 +25 +30 +Normal +Heavy +Extreme +Nodes +Throughput (FPS) +(b) Server B +1 +2 +3 +4 +5 +6 +7 +8 +0 +10 +20 +30 +40 +50 +60 +Normal +Heavy +Extreme +Nodes +Throughput (FPS) +(c) Workstation +Fig. 4. Throughput of Ancilia across different crowd densities. Hardware details can be seen in Tab. IV. +TABLE IV +SYSTEM CONFIGURATIONS. STATS ARE PER CPU/GPU OF THE LISTED TYPE. +Processor +GPU +Name +Model +Cores +Clock Speed +Model +CUDA Cores +VRAM +Server A +2× EPYC 7513 +32 +2.6 GHz +4× V100 +5120 +32 GB +Server B +2× Xeon E5-2640 v4 +10 +2.4 GHz +2× Titan V +5120 +12 GB +Workstation +Threadripper Pro 3975WX +32 +3.50 GHz +3× A6000 +10752 +48 GB +Under the Precision-recall Curve (AUC PR), and the Equal +Error Rate (EER). With GEPC, we can see that Ancilia more +than measures up to the task, with only a 1.5% drop in AUC +ROC at full throughput and less than a 0.2% drop in AUC ROC +at half throughput. AUC PR shows a more substantial drop +of 4% at full throughput, but goes down to less than 0.5% at +half throughput. Equal Error Rates are almost identical, seeing +almost no change (less than 0.01) when using Ancilia. MPED- +RNN, which displayed lower overall accuracy in all regards to +begin with, sees a more significant drop in AUC ROC at full +throughput, losing 3.5%. However, at half throughput the AUC +ROC actually increases when using Ancilia, though only by +0.5%. The AUC PR results mirror that of GEPC, dropping +3.8% at full throughput and 0.7% at half throughput. The +Equal Error Rates are once again nearly identical. Being able +to perform a high-level task such as anomaly detection while +maintaining accuracies so close to current SotA in research, +demonstrates Ancilia’s ability to produce quality data, suitable +for intelligent surveillance applications. +C. Real-time System Performance +Algorithmic accuracy is vital for ensuring the information +provided by high-level cognitive tasks is beneficial for surveil- +lance applications. However, Ancilia’s ability to perform in +real-time is equally important. We conduct a series of ex- +periments, evaluating the runtime performance of Ancilia on +different hardware, with different scenario intensities, and for +increasing number of local nodes per hardware device. We +focus on the performance of the local node, as the global node +is completely decoupled from the algorithmic pipeline and has +no noticeable effect on throughput or latency. +We choose three different hardware configurations for these +experiments: a high-end server, a lower-end server, and a high- +end workstation, as seen in Tab. IV. For our scenarios, we +use the DukeMTMC-video dataset [46] and pick three scenes +with different crowd densities: normal density, heavy density, +and extreme density. The distribution of detection density +in each scenario can be seen in Fig. 5. Note that what is +considered ”normal density” will change based on application +environment, which is why we report on such a wide range. +Each video lasts for +32k frames, with +7k frames warm-up +and cool-down. We test using 1, 2, 4, 6, and 8 local nodes on +a single system, showing how throughput and latency scale +in such cases. Each experiment is conducted three times, the +throughput and latency averaged across runs. The results of +these experiments can be seen in Tab. V and Fig. 4. The +distribution of throughput in these scenes can be seen in Fig. 5. +Under normal crowd density, Server A and Workstation +are both able to achieve over 50 FPS with a single local +node, with an end-to-end latency of 5.39 and 4.90 seconds +respectively. This is well above the 30 and 20 FPS required +by action recognition and anomaly detection algorithms at +full throughput, and the latency is low enough to be suitable +for most surveillance applications where the main concern is +notify authorities in time for appropriate response. Server A +is able to handle 6 local nodes in the normal scenario while +maintaining above 30 FPS, while Workstation can do so with +4 nodes. Server A is able to maintain above 26 FPS while +running all 8 local nodes, while Workstation drops to just +below 18 FPS at 8 local nodes. Server B falls just short of +30 FPS even with a single node in normal crowd density. +However, it is able to maintain above 20 FPS while handling +two nodes simultaneously. Due to having only a single GPU +and limited VRAM, Server B was unable to run 4 or more +nodes concurrently. +Heavy crowd density proves more challenging, with both +Server A and Workstation only able to achieve above 30 +FPS with up to 4 nodes. The end-to-end latency is also +longer than it was under normal crowd density, with Server +A seeing almost double the latency and Workstation seeing +around a 20% to 80% increase in most cases. Server A and +Workstation are able to mainatin above 20 FPS at 8 and 6 + +IEEE INTERNET OF THINGS JOURNAL +8 +TABLE V +AVERAGE THROUGHPUT AND LATENCY WHEN RUNNING MULTIPLE LOCAL NODES ON A SINGLE SERVER. +Server A +Server B +Workstation +Crowd Density +Nodes +FPS +Latency (s) +FPS +Latency (s) +FPS +Latency (s) +Normal +(70 detections +per second) +1 +52.94 +5.39 +29.76 +9.73 +54.91 +4.90 +2 +48.99 +5.97 +22.07 +12.7 +45.06 +5.97 +4 +38.91 +7.29 +- +- +31.67 +8.53 +6 +31.35 +12.55 +- +- +23.10 +13.61 +8 +26.51 +17.94 +- +- +17.98 +24.95 +Heavy +(216 detections +per second) +1 +40.16 +15.66 +26.48 +11.93 +48.52 +5.95 +2 +41.22 +10.87 +19.55 +14.51 +41.45 +7.10 +4 +34.54 +14.48 +- +- +30.01 +11.20 +6 +27.05 +22.42 +- +- +20.99 +30.65 +8 +20.02 +34.68 +- +- +15.77 +46.28 +Extreme +(744 detections +per second) +1 +17.80 +36.04 +14.50 +43.81 +25.68 +24.78 +2 +20.90 +30.73 +13.40 +47.57 +23.52 +26.97 +4 +17.27 +38.15 +- +- +17.56 +39.43 +6 +9.56 +76.71 +- +- +8.94 +94.49 +8 +6.49 +130.08 +- +- +6.31 +134.82 +nodes respectively, while Workstation drops to just above 15 +FPS at 8 nodes. Interestingly, with heavy crowd density we +start to see unusual behavior with Server A having worse +performance with a singe node than it does with 2 nodes. +This is caused by the abundance of CPU and GPU resources +available to the server and a single node being unable to +fully utilize them. As such, the behavior of Server A in the +heavy and extreme crowd density scenarios does not start to +match the expected behavior and mimic the other systems until +multiple nodes are being run simultaneously. This behavior +is not too concerning, considering it does not make sense +to purchase such a high-end server class machine for only +running a single local node, when a more latency focused +workstation would be both cheaper and more effective. Server +B behaves similarly to how it did with normal crowd density, +except that it falls slightly below 20 FPS when running two +nodes. Assuming only half throughput was needed for high- +level tasks, Server B would still be suitable for running up to +two nodes. +With the extreme crowd density scenario, Ancilia begins +to struggle. None of the systems are able to achieve above +30 FPS even with a single camera, putting full throughput +action recognition out of reach. Server A is able to achieve +above 20 FPS with 2 nodes (but notably not with 1) and +Workstation is able to do so with 1 or 2 nodes. Both Server +A and Workstation can maintain above 15 FPS at 4 nodes, +but both drop to around 9 and 6 FPS at 6 and 8 nodes, +respectively. [54] argues that 5 FPS is suitable for tacking +pedestrians, and while that is true, high-level tasks that rely +on detailed human motion, such as action recognition and +anomaly detection, often struggle for accuracy when running +below 10 FPS. Another issue is with the increased latency. +Running only 1 node, Server A and Workstation have latencies +of 36 seconds and 25 seconds respectively, which is suitable +for many surveillance applications, but might be too much +for those that require sharper response times. The latency +increase to over 2 minutes for both systems with 8 nodes. +Combined with the low throughput, it becomes difficult to +recommend running more than 4 nodes on a single system with +Ancilia when operating under extreme crowd density, expect +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0 +5 +10 +15 +20 +Extreme +Heavy +Normal +Average Number of Detections/Batch +Probability +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +20 +30 +40 +50 +60 +70 +80 +90 +Extreme +Heavy +Normal +Throughput (FPS) +Probability +Fig. 5. +The performance of Ancilia in terms of throughput and detection +distribution. +for applications where low throughput and high latency are +not as much of a concern. Server B is unable to achieve 15 +FPS, but does stay above 10 FPS for both 1 and 2 nodes, +making it suitable for half throughput in anomaly detection. +However, the latencies of 44 and 48 seconds might be too +much for some applications. This is the extreme scenario, and +it understandably provides quite the challenge for real-time +execution. +Overall, Ancilia is able to meet the needs of high-level cog- +nitive tasks while still achieving performance suitable for real- + +IEEE INTERNET OF THINGS JOURNAL +9 +time intelligent surveillance applications. Exact performance is +dependent on both the hardware used and the intensity of the +scene, but these results show that even for the most extreme of +scenarios, Ancilia can be used to provide intelligent assistance +to surveillance applications. +VII. CONCLUSION +In this article we presented Ancilia, an end-to-end scal- +able intelligent video surveillance system for the Artificial +Intelligence of Things. Through empirical evaluation, Ancilia +has demonstrated its ability to bring state-of-the-art artificial +intelligence to real-world surveillance applications. Ancilia +performs high-level cognitive tasks (i.e. action recognition and +anomaly detection) in real-time, all while respecting ethical +and privacy concerns common to surveillance applications. +ACKNOWLEDGMENTS +This research is supported by the National Science Foun- +dation (NSF) under Award No. 1831795 and NSF Graduate +Research Fellowship Award No. 1848727. +REFERENCES +[1] C. +Alexandre, +“The +public +safety +implications +of +the +itaewon +tragedy,” Dec 2022. [Online]. Available: https://thediplomat.com/2022/ +12/the-public-safety-implications-of-the-itaewon-tragedy/ +[2] N. Salahieh, J. Miller, and H. Yan, “As north carolinians regain power, +investigators probe terrorism and threats against power substations +across +the +us. +one +expert +explains +what +needs +to +be +done,” +Dec 2022. [Online]. 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Lin, “Pyskl: Towards good +practices for skeleton action recognition,” 2022. [Online]. Available: +https://arxiv.org/abs/2205.09443 +[53] W. Liu, D. L. W. Luo, and S. Gao, “Future frame prediction for anomaly +detection – a new baseline,” in 2018 IEEE Conference on Computer +Vision and Pattern Recognition (CVPR), 2018. +[54] L. Wang, D. Q. Huynh, and P. Koniusz, “A comparative review of +recent kinect-based action recognition algorithms,” IEEE Transactions +on Image Processing, vol. 29, pp. 15–28, 2020. +BIOGRAPHY +Armin Danesh Pazho (S’22) is currently a Ph.D. +student at the University of North Carolina at Char- +lotte, NC, United States. With a focus on Artificial +Intelligence, Computer Vision, and Deep Learning, +his research delves into the realm of developing AI +for practical, real-world applications and addressing +the challenges and requirements inherent in these +fields. Specifically, his research covers action recog- +nition, anomaly detection, person re-identification, +human pose estimation, and path prediction. +Christopher Neff (S’18) is a National Science +Foundation Graduate Research Fellow and Doctoral +Candidate at the University of North Carolina at +Charlotte. His dissertation focus is on tackling the +challenges of bringing human-centric computer vi- +sion to real-world applications. His previous work +focuses on person re-identification, human pose es- +timation, action recognition, real-time system devel- +opment, lightweight algorithms, noisy data, domain +shift, and real-world applications. +Ghazal Alinezhad Noghre (S’22) is currently pur- +suing her Ph.D. in Electrical and Computer Engi- +neering at the University of North Carolina at Char- +lotte, NC, United States. Her research concentrates +on Artificial Intelligence, Machine Learning, and +Computer Vision. She is particularly interested in the +applications of anomaly detection, action recogni- +tion, and path prediction in real-world environments, +and the challenges associated with these fields. +Babak Rahimi Ardabili is a Ph.D. student in the +Public Policy Analysis program at the University +of North Carolina at Charlotte, United States. His +main research area is emerging technologies policy +making. He mainly focuses on the intersection of +Artificial Intelligence and policy from a privacy +perspective and the challenges of bringing the tech- +nology to the community. +Shanle Yao is an Electrical Engineering Graduate +student from the University of North Carolina at +Charlotte. His dissertation focus is on optimization +and application of Computer Vision pipeline perfor- +mance and throughput. His areas of interest include +object detection, multiple objects tracking, human +pose estimation, semantic segmentation and real- +world applications. +Mohammedreza Baharani is an ML researcher and +edge system deployment engineer at ForesightCares. +He received his Ph.D. in computer engineering in +2021 from the University of North Carolina at +Charlotte, USA, and was a postdoctoral researcher +at the TeCSAR Lab. His research focuses on the +intersection of computer architecture engineering +and machine learning, with the goal of enabling AI +algorithms on edge devices to have a positive impact +in fields such as healthcare. +Hamed Tabkhi (S’07–M’14) is an Associate Pro- +fessor in the Department of Electrical and Com- +puter Engineering, University of North Carolina at +Charlotte, USA. He was a post-doctoral research +associate at Northeastern University. Hamed Tabkhi +received his Ph.D. degree in 2014 from Northeast- +ern University under the direction of Prof. Gunar +Schirner. Overall, his research focuses on transfor- +mative computer systems and architecture for cyber- +physical, real-time streaming and emerging machine +learning applications. + +Award1831795 +d PI), Shannon Reid, Dougl. +er,Robert Phocas, Arun Ravi +leta,ChristopherNeff,James +&Integrative +Communit +oroach +ervice of Public +t"policing +edge +tiple +co \ No newline at end of file diff --git a/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/load_file.txt b/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e0726c9c2eab9e85162f351756d3f829cab49ff --- /dev/null +++ b/CtE1T4oBgHgl3EQf9wbT/content/tmp_files/load_file.txt @@ -0,0 +1,2143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf,len=2142 +page_content='IEEE INTERNET OF THINGS JOURNAL 1 Ancilia: Scalable Intelligent Video Surveillance for the Artificial Intelligence of Things Armin Danesh Pazho∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Christopher Neff∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ghazal Alinezhad Noghre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Babak Rahimi Ardabili,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Shanle Yao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Mohammadreza Baharani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Hamed Tabkhi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IEEE Abstract—With the advancement of vision-based artificial in- telligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' the proliferation of the Internet of Things connected cameras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' and the increasing societal need for rapid and eq- uitable security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' the demand for accurate real-time intelligent surveillance has never been higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This article presents Ancilia, an end-to-end scalable, intelligent video surveillance system for the Artificial Intelligence of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia brings state-of-the- art artificial intelligence to real-world surveillance applications while respecting ethical concerns and performing high-level cognitive tasks in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia aims to revolutionize the surveillance landscape, to bring more effective, intelligent, and equitable security to the field, resulting in safer and more secure communities without requiring people to compromise their right to privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Index Terms—Surveillance, artificial intelligence, IoT, com- puter vision, application, real-world, real-time, edge, anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' INTRODUCTION There is a growing need for effective and efficient surveil- lance technologies that can be deployed to protect our cities, people, and infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For example, in Itaewon, South Korea, a holiday celebration left over 150 dead due to severe overcrowding, with many blaming the tragedy on careless government oversight [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In Moore County, North Carolina, directed attacks against two power substations left over 45,000 residents without power for days as technicians rushed to restore power and authorities struggled to find the source of the attacks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' With enough forewarning through smart video surveillance, they could have been prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' With the recent emergence of the Artificial Intelligence of Things (AIoT), some surveillance solution providers have started adding basic forms of artificial intelligence to their systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, their methods are still naive and unable to enhance security in a truly meaningful way [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is because, while a lot of research is conducted on tasks that would benefit surveillance systems, most works focus on algorithmic improvements in a lab environment instead of paying attention to factors that are prevalent in real-world scenarios [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Most research focuses on a single algorithm and how to tweak it to get the best possible results on readily available datasets that often do not reflect a real surveillance environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Few works explore how different algorithms af- fect the performance of other downstream algorithms in multi- The authors are with the Electrical and Computer Engineering Department, The University of North Carolina at Charlotte, Charlotte, NC, 28223 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' {adaneshp, cneff1, galinezh, brahimia, mbaharan, htabkhiv}@uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='edu ∗ Corresponding authors have equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' algorithm systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Few still explore the effects of noise (both data derived and the system produced) in end-to-end accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Beyond this, real-world intelligent surveillance necessitates real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The cognitive abilities of advanced artificial intelligence are only helpful if they can 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='DN-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='Device(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='DN-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Conceptual overview of Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In this article, we present Ancilia, the first end-to-end scalable, intelligent video surveillance system able to perform high-level cognitive tasks in real-time while achieving state- of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia takes advantage of the prevalence of cameras in the Internet of Things (IoT) and uses localized servers and existing cameras, facilitating processing on the edge without the need for additional infrastructure upgrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 1, Ancilia exists within three logical and physical segments: the edge, the cloud, and user devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The edge uses a plethora of advanced artificial intelligence algorithms processing data received from cameras to facilitate intelligent security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Using a single workstation to perform edge processing, Ancilia can monitor up to 4 cameras in real-time at 30 FPS, or up to 8 cameras at 15 FPS, in scenarios with both medium and heavy crowd density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia performs high-level cognitive tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' action recognition, anomaly detection) with ∼ 1% deviation in accuracy from current SotA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia is designed from the ground up to respect the pri- vacy of the people and communities being surveilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia does not store any personally identifiable information in any databases and does not make use of invasive artificial intelli- gence techniques such as facial recognition or gait detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia strictly provides a pose and locational information for high-level tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' action recognition, anomaly detection), as opposed to identity information, which is common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilla arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='03561v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='CV] 9 Jan 2023 0IEEE INTERNET OF THINGS JOURNAL 2 looks at what a person is doing, not who they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This allows Ancilia to act as a buffer to help remove biases based on race, ethnicity, gender, age, and socio-economic factors, which can lead to a reduction in the unnecessary conflict between authorities and marginalized communities that has become increasingly problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' After data is processed on edge and sent to the cloud for communication and service management with user devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A mobile app allows user devices to receive data from the cloud, including alerts when potential security concerns arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In summary, this article has the following contributions: We present Ancilia, the first end-to-end scalable real- world intelligent video surveillance system capable of performing high-level cognitive tasks in real-time while achieving SotA accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We analyze the ethical concerns of intelligent video surveillance, both from a privacy and fairness perspective, and illustrate how Ancilia’s design is purpose-built to address them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We perform an end-to-end empirical evaluation of Ancilia using two high-level cognitive tasks directly related to intelligent surveillance, action recognition, and anomaly detection, investigating the trade-off in accuracy required to achieve real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We perform an exhaustive system-level evaluation of Ancilia’s real-time performance and scalability across different classes of hardware and increasing scenario intensities, displaying how Ancilia is able to meet real- time intelligent security needs in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' RELATED WORK There has been a plethora of research regarding the use of artificial intelligence for video surveillance [4], [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [9] proposes the use of region proposal based optical flow to suppress background noise and a bidirectional Bayesian state transition strategy to model motion uncertainty to enhance spatio-temporal feature representations for the detection of salient objects in surveillance videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [10] proposes the use of a person detector, tracking algorithm, and mask classifier for tracking pedestrians through surveillance video streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In [4], it is determined that in order to address the latency concerns of real-time video surveillance, a shift towards edge computing is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Nikouei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [11]–[13] explore the feasibility of using low-power edge devices to perform object detection and tracking in surveillance scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' They argue that in worst case 5 FPS is high enough throughput for tracking humans in surveillance applications, and as such computation can be pushed to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, their results show that even light weight convolutional neural networks can prove problematic for low-power devices, often reducing throughput below the 5 FPS threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [14] proposes a system using low- power embedded GPUs to perform detection, tracking, path prediction, pose estimation, and multi-camera re-identification in a surveillance environment, while placing a focus on real- time execution and the privacy of tracked pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [15] proposes a similar system, focusing solely on object detec- tion, tracking, and multi-camera re-identification to increase throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [16] proposes using a combination of lightweight object detection models on the edge and more computation- ally expensive models in the cloud, splitting computation between the two to provide real-time video surveillance in a construction site environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [17] proposes the use of background detection, vehicle detection, and kalman filter [18] based tracking for parking lot surveillance and determining lot occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [19] proposes a system that uses object detection, person tracking, scene segmentation, and joint trajectory and activity prediction for pedestrians in a surveillance setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The future of intelligent surveillance is heading towards systems able to perform high-level cognitive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A recent survey focusing on real-world video surveillance [4] asserts that while the domain of video surveillance is comprised of understanding stationary object, vehicles, individuals, and crowds, the ability to determine when anomalous events oc- cur is paramount for intelligent surveillance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Other research has supported this assertion [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [20] utilizes the Infinite Hidden Markov Model and Bayesian Nonparametric Factor Analysis to find patterns in video streams and detect abnormal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [21] proposes active learning and fuzzy aggregation to learn what constitutes an anomaly continually over time, adapting the scenarios not seen in standard datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [22] proposes a system to detect suspicious behaviors in a mall surveillance setting, using lightweight algorithms such as segmentation, blob fusion, and kalman filter based tracking [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' AnomalyNet [23] is a recently proposed recurrent neutral network with adaptive iterative hard-thresholding and long short-term memory that works directly off pixel information to eliminate background noise, capture motion, and learn sparse representation and dictionary to perform anomaly detection in video surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' ETHICAL CONCERNS Video surveillance has always been associated with social and ethical concerns, whether in traditional form or more recent intelligent formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Respecting citizens’ privacy and autonomy while improving public safety and security are the most well-known and enduring ethical issues in this context [24]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Developing a successful smart video surveillance solution that addresses the public safety problem and engages the community up to a certain level is only possible by considering these concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' There is rising attention among scholars to the issue of incorporating privacy concerns at the design level, referred to as ”privacy by design” [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The source of discrimination and privacy violation in many data-driven and AI-based systems, such as Smart video surveillance technology, is using Personal Identifiable Information (PII) [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Using PII, such as actual footage of people’s daily activities at any stage of the technology, can increase the risk of privacy violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' There is a long-lasting debate on the ethical challenges of using facial recognition technologies in different sectors and how using this technology can result in privacy violation [31]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Avoiding facial recognition technologies does not guarantee the system is entirely privacy persevering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Storing images of pedestrians is another source of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='IDL BBP FL D Edge Boundary Video Stream from Camera (D) User Device(s) AR R User Device Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia algorithmic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' N local nodes are connected to a single global node on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The final analyses are transferred to the cloud node to feed the application on the user device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Multiple edges may be connected to the could, though this figure only shows one edge for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' BBP , BBO, IDL, P, C, FL, D, FD, IDG, I, SA, R, and AR refer to bounding boxes for pedestrians, bounding boxes of objects, local identities, poses, person crops that passed selection, features from the local node, data from the downstream tasks, features from the database, global identities, information from the database, completed statistical analysis, requests from users, and requested attributes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' discrimination perspective, using any form of PII can con- tribute to the issue of marginalization in policing systems [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Therefore, an essential step in designing a non-discriminatory system is to ensure the system is not dependent on PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This requires a specific approach toward the design of such technology in the choice of algorithm, the type of data used,, and the storing of such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia addresses this by not storing any PII or sending any PII across the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Such data is destroyed after it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia utilizes pose-based methods for all high-level cognitive tasks, ensuring no PII is ever used in such algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This allows for such processing to occur without any potential for gender, ethnicity, or class-based discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' As such, Ancilia is able to address the ethical challenge of privacy in smart video surveillance systems while also addressing the ethical issue of discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' ANCILIA ALGORITHMIC FRAMEWORK The algorithmic core of Ancilia is separated into two conceptual systems: the local nodes containing the algorithmic pipeline of each camera and the global node that handles all processing that requires understanding of multiple camera perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' These two systems make up the algorithmic core of Ancilia and are the basis on which all higher understanding is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A visual representation of this algorithmic core can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Single Camera Vision Pipeline As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 2, the local algorithmic pipeline starts when an image is extracted from the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The image is first run through an object detector to locate people, vehicles, animals, and other important objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is important not only because it acts as the basis for the rest of the algorithmic pipeline but also because it can be used for basic situational awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Sometimes, just the presence of a certain object in a scene is noteworthy, like a person in an unauthorized location, a bag left unattended, or the presence of a firearm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses YOLOv5 [36] for this purpose (however, it can be any detector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The locational coordinates of persons are sent to a tracker, where tracklets are created, matching each person with their previous detections in prior images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses ByteTrack [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The tracking allows for understanding how a person moves throughout a scene, which is vital for many surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' It also allows Ancilia to understand which poses belong to which persons over time, which is vital for many high-level tasks that provide much-needed situational awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Image crops of the people detected in the image are also sent to a human pose estimator, where two- dimensional pose skeletons are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses HRNet [38] for extracting 2D skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Using pose data for higher- level tasks has two major benefits over simply using raw pixel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' First, pose data is of much lower dimensionality than pixel data, making it much less computationally expensive and allowing the Ancilia to function in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Second, pose data contains absolutely zero identifiable information, making it impossible for high-level tasks to form unintended biases based on ethnicity, gender, age, or other identity-based metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Multi-Camera Person Re-identification While the tracker tracks people within a single camera, locational information cannot accurately re-identify a person across multiple cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For this, the same person crops that were sent to the human pose estimator are also sent to a person re-identification feature extractor, where an abstract feature representation is created for each person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Only one feature representation is created for each person over a period of 30 frames, and only when the quality of the representation can be assured, as poor quality representations are detrimental to accurate multi-camera person re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='feature representation filtering algorithm to verify two qualities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='IEEE INTERNET OF THINGS JOURNAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A detailed view of system design in Ancilia’s local nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' β and δ refer to different batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' FL and D represent local features and data received from downstream tasks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' for person crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' First, a person crop must contain a high- quality view of the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' To this end, the filter algorithm uses the 2D pose skeleton and verifies that at least 9 keypoints were detected with at least 60% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The filter algorithm looks at the overlap (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Intersection of Union) of the bounding boxes generated by the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' An individual’s bound box must have an Intersection over Union (IoU) of no more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1 with any other person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' If those two conditions are met, the person crop is determined to be of high enough quality to produce an adequate feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' If more than one crop is deemed suitable for a single person during a 30 frame window, the one with the most confident pose is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The features created by the feature extractor are sent to the global node for multi-camera person re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses OSNet [39] to extract feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' higher Level Tasks High-level tasks are executed on the local node, and have access to the object, tracking, and pose data generated in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Since the decision of which high-level tasks are needed is highly application dependent, we do not consider these tasks to be part of the Ancilia algorithmic core, but instead an extension to be customized based on intended use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In this paper, we use action recognition and anomaly detection as two common examples of high-level tasks that are highly relevant to intelligent surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For action recognition, we choose PoseConv3D [40] and CTR-CGN [41], two state-of- the-art networks that can utilize the 2D human pose skeletons provided by Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For anomaly detection, we use GEPC [42] and MPED-RNN [43], which are based on 2D human pose skeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' SYSTEM DESIGN Beyond the algorithmic design, Ancilia can be analyzed from a system-level design and implementation perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The local node in particular has a complex system design, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The global node and cloud are much simpler, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Parallelism A key design objective of Ancilia is to achieve higher efficiency by balancing throughput and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia uses pipelining to take advantage of process parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each major task is implemented as a separate process, which executes concurrently with other processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' These processes communicate with each other using queues to utilize memory resources better and enable fast inter-process communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' While pipelining is a well-known technique for optimization, the overhead associated with its implementation means a balance needs to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Figure 3 shows a detailed view of the system design on the local node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each pipeline stage is sep- arated by a queue with a size limit of λ1 elements, preventing any potential overflow from uneven execution speed between pipeline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' By default, Ancilia uses a λ1 value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' As is common, Ancilia offloads highly parallel tasks that rely on neural networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' object detection, pose estimation, feature extraction, and many high-level tasks) to Graphics Processing Units (GPUs) for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Data Batching Batching is another technique Ancilia implement to better utilize hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Generally, batching is able to greatly increase the throughput of a system at the cost of end- to-end latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, many high-level tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' action IEEE INTERNET OF THINGS JOURNAL 5 recognition, anomaly detection) require multiple video frames worth of input data (often called a window) before the can start processing, so the latency that would be incurred by batching input frames is already inherent in these high-level task, as long as the frame batch and high-level task window are of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Further, as frame batching ultimately increases the throughput, the end-to-end latency is decreased when compared processing each frame sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' While object detection works on entire frames, all other neural networks in Ancilia work off individual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' These objects are batched together before being input to the network, greatly increasing hardware utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' There can be multiple object batches within a single frame batch, based on how many of the relevant objects are detected in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Local Node Once the local node receives the video stream from the camera, the proprocessor is responsible for all basic image processing necessary before sending the frames through the algorithmic core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' After preprocessing, frames are batched in sequential segements of size β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia sets β1 = 30 to match the window size of high-level tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is also convenient as most modern security and IoT cameras record video at either 30 or 60 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The batched frames are sent to the object detector, which outputs a list of objects with class labels and bounding box coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Bounding boxes for pedestrians are sent to the tracker, while bounding boxes for other objects are passed through the system for use in high-level tasks and statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' A crop of each pedestrian from the original frame is passed through to the pose estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' At the tracker, bounding boxes for pedestrians are unbatched to fit the tracker’s sequential operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The tracker groups the pedestrians and either matches them with previously seen pedestrians or assigns them a unique local ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Afterwards, the pedestrians are once again batched by frame and sent to the pose estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' At the pose estimator, the object batching is performed on the person crops, with a batch size of β2 = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' These batches are fed to the pose estimator, which outputs human pose skeletons for each person crop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Then the pedestrian bounding boxes, person crops, local IDs, and human pose skeletons are once again batched by frame and combined with the object bounding boxes from the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The pedestrian bounding boxes, person crops, local IDs, and pose skeletons are sent to crop selection, while the pedestrian bounding boxes, object bounding boxes, local IDs, and pose skeletons are sent to each high-level task as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' While the person crops are necessary for the feature extraction that enables multi-camera re-identification, no identifiable data is sent to any of the high-level tasks, keeping in line with the ethical concerns mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Crop selection filters out low-quality person crops based on bounding box overlap and keypoint confidence, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The remaining crops are then batched, with size β3 being dynamic based on the number of persons in the scene, and sent to the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Once features are extracted, the are sent for transfer to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each high-level task receives data at the granularity of a frame batch, and sends data to the server at whatever granularity that task requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each high-level task has its own process and works in parallel with other tasks as well as with crop selection and feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Communication is completely decoupled from the pipeline, so once the data is sent the local node pipeline continues to function as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Importantly, no identifiable information is ever sent to the global node, keeping in line with the privacy and ethical concerns mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' III D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Global Node All received data is stored in a relational database on the global node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The matching algorithm described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV compares the received features with existing features in the database over the period λ5 and assigns a global ID based on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The default value for lλ5 is set to 1 hour, but this should be changed to suit the needs of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' An assortment of algorithms performs statistical analysis using the relational database, as detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The analysis is transmitted to the cloud node using APIs provided by the cloud service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' By default, Ancilia uses Amazon Web Services, but this can be altered based on user/application needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The cloud (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Amazon Web Services (AWS)) receives analyzed data from the global node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' EXPERIMENTAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Algorithmic Core The algorithmic core of Ancilia consist of multiple algo- rithms, each of which works off of data generated by the previous algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' As these algorithms leverage imperfect neural networks, they generate noise that accumulates through the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' To understand the source of this noise, we must first look at the accuracy of each of these core algorithms in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Table I shows the accuracies of the algorithmic core’s four main tasks: object detection, pedestrian tracking, human pose estimation, and person re-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The table also shows the accuracies of the top SotA models in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' These SotA methods are not suitable for intelligent surveillance applications, as their excessive computation and vast parameters make real-time execution impossible, but the comparison allows us to see the maximum potential allowable by current research and the accuracy loss incurred to keep Ancilia performing in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' TABLE I ACCURACY OF ANCILIA’S ALGORITHMIC CORE NETWORKS IN ISOLATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' ACCURACIES OF OBJECT DETECTION AND POSE ESTIMATION ARE USING THE COCO DATASET [44], TRACKING USING MOT20 [45], AND PERSON REID USING DUKEMTMC [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' SOTA REPRESENT THE HIGHEST ACCURACIES CURRENTLY ACHIEVABLE WITH STATE-OF-THE-ART METHODS WHEN COMPUTATION AND LATENCY ARE NOT A CONCERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Task Method Metric Accuracy SotA Object Detection YOLOv5 [36] mAP 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='0 [47] Tracking ByteTrack [37] MOTA 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='8 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='9 [48] Pose Estimation HRNet [38] AP 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1 [49] Person ReID OSNet [39] Top-1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6 [50] Object detection sees the biggest hit to accuracy, with a 16% drop from SotA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is intuitive, as YOLOv5 [36] is not only IEEE INTERNET OF THINGS JOURNAL 6 the largest model in the algorithmic core, but also the only one that operates on the raw camera stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' So while larger models are available and would be able to produce higher accuracy, even a slight increase in model size or computation would result in a noticeable decrease in throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Human pose estimation sees a decrease in accuracy for a similar reason, though much smaller in scale at only 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' While HRNet [38] is not run on the raw camera stream, it is run individually for each person detected by the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' As such, maintaining a small model size is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Person re-identification sees a slightly larger drop in accuracy than human pose estimation at 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' While this is partly due to using a lightweight model, OSNet [39], the SotA model for person reID is also lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, the SotA uses a centroid based retrieval method not suitable for pen-set reID, of which most surveillance scenarios are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Pedestrian tracking sees almost no drop in accuracy, approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This stems from the comparative ease of tracking pedestrians in a single camera, where a simple, lightweight algorithm like ByteTrack [37] see almost no performance difference from the top of the line SotA approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' High-level Tasks To better understand how the noise generated by the al- gorithmic core effects overall performance, and thus how well Ancilia performs in the realm of real-world intelligent surveillance, we examine the performance of two high-level cognitive surveillance tasks when running on Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For Ancilia to be a benefit to intelligent surveillance tasks, we must ensure that excess false alarms or missed positive events do not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' To assess this, we choose action recognition and anomaly detection, as these tasks can utilize the human pose information generated by the algorithmic core, resulting in faster and less biased inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Since both these methods utilize temporal batches of human poses for each individual, these experiments will directly reflect the quality of the object detection, tracking, re-identification, and pose estimation data generated by Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 1) High-level Task - Action Recognition: We select two state-of-the-art action recognition models, PoseConv3d [40] and CTR-GCN [41], and train them using data generated with Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For each model, we train and test with full (30 FPS) and half (15 FPS) throughput on NTU60-XSub [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Both models use a window size of 30 and are trained for 24 epochs using Stochastic Gradient Descent (SGD) with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='9 and Cosine Annealing scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' PoseConv3d and CTR- GCN have weight decay of 3e−4 and 5e−4 and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The results of these experiments can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We report the Top-1 and Top-5 accuracy and compare the results using data generated by Ancilia to the original data available through the PYSKL toolbox [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We can see that Ancilia is able to provide data of comparable quality to the original;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' action recognition as a high-level task in Ancilia sees around 1% drop in accuracy compared to the original data using PoseConv3D [40] at full throughput, and around 3% at half throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Using CTR-GCN [41], Ancilia sees a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5% TABLE II TOP-1 AND TOP-5 ACCURACIES ON NTU60-XSUB [51] IN FULL AND HALF THROUGHPUT MODES FOR POSECONV3D [40] AND CTR-GCN [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Model Data FPS Top-1 (%) Top-5 (%) PoseConv3D [40] [52] 15 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='47 30 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='76 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='57 Ours 15 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='79 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='82 30 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='99 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='28 CTR-GCN [41] [52] 15 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='36 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='46 30 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='07 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='26 Ours 15 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='58 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='52 30 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='44 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='2 drop at full throughput and a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='8% drop at half throughput, compared to the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' From this we can infer that PoseConv3D is more robust to noise than CTR-GCN, however both performed reasonably well with data generated from Ancilia, demonstrating its efficacy for intelligent surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Another interesting observation is that CTR-GCN [41] actually performed noticeably better at half throughput than at full throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This means that CTR-GCN is more suited to taking advantage of the higher temporal window allowed when using half throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is something to consider when choosing an action recognition model when a real-time throughput of 30 FPS cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 2) High-level Task - Anomaly Detection: Using the Shang- haiTech dataset [53] we train two state-of-the-art anomaly detection models, GEPC [42] and MPED-RNN [43], using both data generated by Ancilia and the data provided by the original authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The same training strategy from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' VI-B1 is used, with both models trained in full (20 FPS) and half (10 FPS) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' GEPC is trained for 25 epochs with a window size of 30 and stride of 20 using Adam optimizer with a learning rate of 1e-4, weight decay of 1e-5, and batch size of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' MPED-RNN is trained with an input window size of 30, a reconstruction window of 12, and a prediction window of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The model is trained for 5 epochs using the Adam optimizer with a learning rate of 1e−3 and a batch size of 265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' TABLE III AUC ROC, AUC PR, AND EER ON SHANGHAITECH DATASET [53] IN FULL AND HALF THROUGHPUT MODES FOR GEPC [42] AND MPED-RNN [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Model Data FPS AUC ROC AUC PR EER GEPC [42] [42] 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='35 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='7372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='31 Ours 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='35 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='7223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='32 MPED-RNN [43] [43] 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='37 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='7023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='36 Ours 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='37 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='37 The results of this experiment can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' In line with current practices, we report Area Under the Receiver Operating Characteristic Cure (AUC ROC), Area IEEE INTERNET OF THINGS JOURNAL 7 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 Normal Heavy Extreme Nodes Throughput (FPS) (a) Server A 1 2 3 4 5 6 7 8 15 20 25 30 Normal Heavy Extreme Nodes Throughput (FPS) (b) Server B 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 Normal Heavy Extreme Nodes Throughput (FPS) (c) Workstation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Throughput of Ancilia across different crowd densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Hardware details can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' TABLE IV SYSTEM CONFIGURATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' STATS ARE PER CPU/GPU OF THE LISTED TYPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Processor GPU Name Model Cores Clock Speed Model CUDA Cores VRAM Server A 2× EPYC 7513 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='6 GHz 4× V100 5120 32 GB Server B 2× Xeon E5-2640 v4 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='4 GHz 2× Titan V 5120 12 GB Workstation Threadripper Pro 3975WX 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='50 GHz 3× A6000 10752 48 GB Under the Precision-recall Curve (AUC PR), and the Equal Error Rate (EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' With GEPC, we can see that Ancilia more than measures up to the task, with only a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5% drop in AUC ROC at full throughput and less than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='2% drop in AUC ROC at half throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' AUC PR shows a more substantial drop of 4% at full throughput, but goes down to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5% at half throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Equal Error Rates are almost identical, seeing almost no change (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='01) when using Ancilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' MPED- RNN, which displayed lower overall accuracy in all regards to begin with, sees a more significant drop in AUC ROC at full throughput, losing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, at half throughput the AUC ROC actually increases when using Ancilia, though only by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The AUC PR results mirror that of GEPC, dropping 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='8% at full throughput and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='7% at half throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The Equal Error Rates are once again nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Being able to perform a high-level task such as anomaly detection while maintaining accuracies so close to current SotA in research, demonstrates Ancilia’s ability to produce quality data, suitable for intelligent surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Real-time System Performance Algorithmic accuracy is vital for ensuring the information provided by high-level cognitive tasks is beneficial for surveil- lance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, Ancilia’s ability to perform in real-time is equally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We conduct a series of ex- periments, evaluating the runtime performance of Ancilia on different hardware, with different scenario intensities, and for increasing number of local nodes per hardware device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We focus on the performance of the local node, as the global node is completely decoupled from the algorithmic pipeline and has no noticeable effect on throughput or latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We choose three different hardware configurations for these experiments: a high-end server, a lower-end server, and a high- end workstation, as seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' For our scenarios, we use the DukeMTMC-video dataset [46] and pick three scenes with different crowd densities: normal density, heavy density, and extreme density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The distribution of detection density in each scenario can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Note that what is considered ”normal density” will change based on application environment, which is why we report on such a wide range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each video lasts for 32k frames, with 7k frames warm-up and cool-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' We test using 1, 2, 4, 6, and 8 local nodes on a single system, showing how throughput and latency scale in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Each experiment is conducted three times, the throughput and latency averaged across runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The results of these experiments can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' V and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The distribution of throughput in these scenes can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Under normal crowd density, Server A and Workstation are both able to achieve over 50 FPS with a single local node, with an end-to-end latency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='39 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='90 seconds respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is well above the 30 and 20 FPS required by action recognition and anomaly detection algorithms at full throughput, and the latency is low enough to be suitable for most surveillance applications where the main concern is notify authorities in time for appropriate response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server A is able to handle 6 local nodes in the normal scenario while maintaining above 30 FPS, while Workstation can do so with 4 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server A is able to maintain above 26 FPS while running all 8 local nodes, while Workstation drops to just below 18 FPS at 8 local nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server B falls just short of 30 FPS even with a single node in normal crowd density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, it is able to maintain above 20 FPS while handling two nodes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Due to having only a single GPU and limited VRAM, Server B was unable to run 4 or more nodes concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Heavy crowd density proves more challenging, with both Server A and Workstation only able to achieve above 30 FPS with up to 4 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The end-to-end latency is also longer than it was under normal crowd density, with Server A seeing almost double the latency and Workstation seeing around a 20% to 80% increase in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server A and Workstation are able to mainatin above 20 FPS at 8 and 6 IEEE INTERNET OF THINGS JOURNAL 8 TABLE V AVERAGE THROUGHPUT AND LATENCY WHEN RUNNING MULTIPLE LOCAL NODES ON A SINGLE SERVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server A Server B Workstation Crowd Density Nodes FPS Latency (s) FPS Latency (s) FPS Latency (s) Normal (70 detections per second) 1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='94 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='39 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='76 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='73 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='90 2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='97 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='07 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='97 4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='91 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='29 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='53 6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='55 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='10 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='61 8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='51 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='94 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='98 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='95 Heavy (216 detections per second) 1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='66 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='48 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='93 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='95 2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='87 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='55 14.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='56 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='43 6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='56 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='71 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='94 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='49 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='49 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='31 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='82 nodes respectively, while Workstation drops to just above 15 FPS at 8 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Interestingly, with heavy crowd density we start to see unusual behavior with Server A having worse performance with a singe node than it does with 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is caused by the abundance of CPU and GPU resources available to the server and a single node being unable to fully utilize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' As such, the behavior of Server A in the heavy and extreme crowd density scenarios does not start to match the expected behavior and mimic the other systems until multiple nodes are being run simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This behavior is not too concerning, considering it does not make sense to purchase such a high-end server class machine for only running a single local node, when a more latency focused workstation would be both cheaper and more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server B behaves similarly to how it did with normal crowd density, except that it falls slightly below 20 FPS when running two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Assuming only half throughput was needed for high- level tasks, Server B would still be suitable for running up to two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' With the extreme crowd density scenario, Ancilia begins to struggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' None of the systems are able to achieve above 30 FPS even with a single camera, putting full throughput action recognition out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server A is able to achieve above 20 FPS with 2 nodes (but notably not with 1) and Workstation is able to do so with 1 or 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Both Server A and Workstation can maintain above 15 FPS at 4 nodes, but both drop to around 9 and 6 FPS at 6 and 8 nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [54] argues that 5 FPS is suitable for tacking pedestrians, and while that is true, high-level tasks that rely on detailed human motion, such as action recognition and anomaly detection, often struggle for accuracy when running below 10 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Another issue is with the increased latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Running only 1 node, Server A and Workstation have latencies of 36 seconds and 25 seconds respectively, which is suitable for many surveillance applications, but might be too much for those that require sharper response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The latency increase to over 2 minutes for both systems with 8 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Combined with the low throughput, it becomes difficult to recommend running more than 4 nodes on a single system with Ancilia when operating under extreme crowd density, expect 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='45 0 5 10 15 20 Extreme Heavy Normal Average Number of Detections/Batch Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='16 20 30 40 50 60 70 80 90 Extreme Heavy Normal Throughput (FPS) Probability Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' The performance of Ancilia in terms of throughput and detection distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' for applications where low throughput and high latency are not as much of a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Server B is unable to achieve 15 FPS, but does stay above 10 FPS for both 1 and 2 nodes, making it suitable for half throughput in anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' However, the latencies of 44 and 48 seconds might be too much for some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' This is the extreme scenario, and it understandably provides quite the challenge for real-time execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Overall, Ancilia is able to meet the needs of high-level cog- nitive tasks while still achieving performance suitable for real- IEEE INTERNET OF THINGS JOURNAL 9 time intelligent surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Exact performance is dependent on both the hardware used and the intensity of the scene, but these results show that even for the most extreme of scenarios, Ancilia can be used to provide intelligent assistance to surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' CONCLUSION In this article we presented Ancilia, an end-to-end scal- able intelligent video surveillance system for the Artificial Intelligence of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Through empirical evaluation, Ancilia has demonstrated its ability to bring state-of-the-art artificial intelligence to real-world surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ancilia performs high-level cognitive tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' action recognition and anomaly detection) in real-time, all while respecting ethical and privacy concerns common to surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research is supported by the National Science Foun- dation (NSF) under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 1831795 and NSF Graduate Research Fellowship Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 1848727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Alexandre, “The public safety implications of the itaewon tragedy,” Dec 2022.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Luo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Gao, “Future frame prediction for anomaly detection – a new baseline,” in 2018 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' [54] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Huynh, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Koniusz, “A comparative review of recent kinect-based action recognition algorithms,” IEEE Transactions on Image Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' 15–28, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' BIOGRAPHY Armin Danesh Pazho (S’22) is currently a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' student at the University of North Carolina at Char- lotte, NC, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' With a focus on Artificial Intelligence, Computer Vision, and Deep Learning, his research delves into the realm of developing AI for practical, real-world applications and addressing the challenges and requirements inherent in these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Specifically, his research covers action recog- nition, anomaly detection, person re-identification, human pose estimation, and path prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Christopher Neff (S’18) is a National Science Foundation Graduate Research Fellow and Doctoral Candidate at the University of North Carolina at Charlotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His dissertation focus is on tackling the challenges of bringing human-centric computer vi- sion to real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His previous work focuses on person re-identification, human pose es- timation, action recognition, real-time system devel- opment, lightweight algorithms, noisy data, domain shift, and real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Ghazal Alinezhad Noghre (S’22) is currently pur- suing her Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' in Electrical and Computer Engi- neering at the University of North Carolina at Char- lotte, NC, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Her research concentrates on Artificial Intelligence, Machine Learning, and Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' She is particularly interested in the applications of anomaly detection, action recogni- tion, and path prediction in real-world environments, and the challenges associated with these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Babak Rahimi Ardabili is a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' student in the Public Policy Analysis program at the University of North Carolina at Charlotte, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His main research area is emerging technologies policy making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' He mainly focuses on the intersection of Artificial Intelligence and policy from a privacy perspective and the challenges of bringing the tech- nology to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Shanle Yao is an Electrical Engineering Graduate student from the University of North Carolina at Charlotte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His dissertation focus is on optimization and application of Computer Vision pipeline perfor- mance and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His areas of interest include object detection, multiple objects tracking, human pose estimation, semantic segmentation and real- world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Mohammedreza Baharani is an ML researcher and edge system deployment engineer at ForesightCares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' He received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' in computer engineering in 2021 from the University of North Carolina at Charlotte, USA, and was a postdoctoral researcher at the TeCSAR Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' His research focuses on the intersection of computer architecture engineering and machine learning, with the goal of enabling AI algorithms on edge devices to have a positive impact in fields such as healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Hamed Tabkhi (S’07–M’14) is an Associate Pro- fessor in the Department of Electrical and Com- puter Engineering, University of North Carolina at Charlotte, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' He was a post-doctoral research associate at Northeastern University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Hamed Tabkhi received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' degree in 2014 from Northeast- ern University under the direction of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Gunar Schirner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Overall, his research focuses on transfor- mative computer systems and architecture for cyber- physical, real-time streaming and emerging machine learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' Award1831795 d PI), Shannon Reid, Dougl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} +page_content=' er,Robert Phocas, Arun Ravi leta,ChristopherNeff,James &Integrative Communit oroach ervice of Public t"policing edge tiple co' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQf9wbT/content/2301.03561v1.pdf'} diff --git a/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf b/CtE2T4oBgHgl3EQf9Ant/content/2301.04225v1.pdf new file mode 100644 index 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The proposed approach exploits Sparse +Gaussian Process to encode pointcloud into a compact form. +Our approach represents both the free space and the occupied +space using only one model (one 2D Sparse Gaussian Process) +instead of the existing two-model framework (two 3D Gaussian +Mixture Models). We achieve this by proposing a variance- +based sampling technique that effectively discriminates between +the free and occupied space. The new representation requires +less memory footprint and can be transmitted across limited- +bandwidth communication channels. The framework is exten- +sively evaluated in simulation and it is also demonstrated using +a real mobile robot equipped with a 3D LiDAR. Our method +results in a 70∼100 times reduction in the communication rate +compared to sending the raw pointcloud. +I. INTRODUCTION +With the rapid advancement of LiDAR technology, we +now can build maps with remarkably high resolution. For +example, each full scan of an only 16-channel 3D LiDAR +can give us 57600 points in the pointcloud that represents +the surrounding obstacles. However, a price for using the +high resolution LiDAR is the computation, storage, and com- +munication costs when mapping the environments. While +one might be able to upgrade the computation and storage +by using a high performance computer system, the com- +munication usually becomes a bottleneck due to the low +communication bandwidth available. In practice, the low +bandwidth communication is considered as a major challenge +for many robotics applications such as occupancy mapping +of underwater and subterranean environments (caves, tunnels, +mines, etc), search-and-rescue missions in disaster scenarios +with a degraded communication infrastructure, and planetary +exploration missions [1]. The low bandwidth can prevent +a robot from real-time sharing its sensor observations, and +this can significantly degrade the system responsiveness if +the robot needs to follow or interact with external control +or supervision platforms. This work tackles the problem +of sharing high-fidelity 3D pointcloud through a limited +bandwidth communication channel. +The system we consider consists of a robot (the scout) +equipped with a LiDAR and a communication apparatus, and +deployed in a low-bandwidth environment. The scout sends +1Mahmoud Ali and Lantao Liu are with the Luddy School of Informatics, +Computing, and Engineering, Indiana University, Bloomington, IN 47408 +USA, {alimaa, lantao}@iu.edu +Occupancy Surface +VSGP +variance-based sampling +SGP +OctoMap +wifi +Scout +Base +Fig. 1: System Overview. +(a) +(b) +(c) +Fig. 2: (a) Gazebo simulated mine tunnel; (b) Original pointcloud +generated by a VLP16 LiDAR in red, and reconstructed pointcloud +from the VSGP model in white; (d) Occupancy Map generated by +OctoMap from the reconstructed pointcloud. +the observations that it acquires to a base for building the +occupancy map of the environment, see Fig. 1. Our approach +exploits the Variational Sparse Gaussian Process (VSGP) [2] +as a generative model to represent the pointcloud in a +compact form. This lightweight representation is transmitted +through low-bandwidth communication to the base where the +original pointcloud is reconstructed. Extensive evaluations +reveal that our approach results in a 70∼100 times reduction +in the memory as well as the communication rate required to +transmit pointcloud data. For example, Fig. 2a shows a scene +of a simulated mine tunnel, where its raw pointcloud (shown +in red, Fig. 2b) requires around 750 KB of memory. Our +approach is able to represent the same observation using only +6 KB of memory and transmit through limited-bandwidth +communication. On the receiver side of the communication +channel, the compact representation is used to reconstruct +the original pointcloud (reconstructed pointcloud shown in +white, Fig. 2b). An occupancy map of the scene can be built +using the reconstructed pointcloud, see Fig. 2c. +II. RELATED WORK +Pointcloud compression algorithms have been investigated +in recent years to cope with the demands to store and +communicate the high-precision 3D points [3]. For example, +the space partitioning trees approaches that exploit the 3D +correlation between pointcloud points are widely used to +arXiv:2301.11251v1 [cs.RO] 26 Jan 2023 + +Velodynecompress the pointcloud data [4]–[9]. Recently, deep learning +based approaches were also proposed to leverage data and +learn or encode the pointcloud compression [10]–[12]. Dif- +ferent from these frameworks, the probabilistic approaches +exploit the compactness of the distributions to compress 3D +sensor observation. For instance, Gaussian Mixture Models +(GMM) [13]–[15] have been proposed as a generative model +to encode 3D occupancy map. The GMM approach encodes +the 3D data as a mixture of Gaussian densities to represent +the occupied and free spaces around the robot. +Gaussian Process (GP) has been proven to be an excellent +framework to model spatial phenomena or features in a +continuous domain [16]–[18]. Unfortunately, the standard +GP has a cubic time complexity and this results in very +limited scalability to large datasets. Methods for reducing the +computing burdens of GPs have been previously investigated. +For example, GP regressions can be done in a real-time +fashion where the problem can be estimated locally with +local data [19]. Sparse GPs (SGPs) [20]–[26] tackle the com- +putational complexity of the normal GP through leveraging +the Bayesian rule with a sequential construction of the most +relevant subset of the data. +We propose a new probabilistic pointcloud compression +approach which is based on the VSGP [2] and inspired by +the GMM approach. While the GMM shares the accumulated +sensory information as a set of accumulated Gaussian den- +sities which are sampled and used as an occupancy map of +the environment, in contrast, the proposed approach relies on +sharing of immediate sensor observation to be reconstructed +on the other side of the communication channel for further +processing based on the required task (e.g. 3D mapping, +object recognition, tracking, etc). +This proposed VSGP-based approach offers a few ad- +vantages over the recent GMM approach: while the GMM +approach uses two 3D GMMs to fit the occupied and free +points [13]–[15], our approach uses only one 2D VSGP to +fit all the occupancy surface, including both the occupied +and free points. The primary reason that our approach uses +one VSGP instead of two is that we are using the variance +calculated by the VSGP at each sampled point during the +reconstruction process to decide if it belongs to the occupied +or the free space. Therefore, the proposed approach results +in a more compact representation of the sensor observation, +which requires less memory than the GMM approach and, +as a consequence, leads to a lower communication rate. +III. BACKGROUND +GP is a non-parametric model described by a mean +function m(x), and a co-variance function (kernel) k(x,x′), +where x is the GP input [27]: +f(x) ∼ GP +� +m(x),k +� +x,x′�� +. +(1) +Considering a data set D = {(xi,yi)}N +i=1 with N training +inputs x and their corresponding scalar outputs (observations) +y. After training the GP using the data set D, the output y∗ for +any new query x∗ can be estimated using the GP prediction: +p(y∗|y) = N(y∗|my(x∗),ky(x∗,x∗)+σ2), +(2) +where my(x) and ky(x,x′) are the posterior mean and co- +variance functions [2]. The GP prediction equation depends +on the values of the hyperparameters (Θ,σ2) where Θ is the +kernel parameters and σ2 is the noise variance. +The computation complexity of a full GP is O(N3). +In order to reduce the computation complexity, different +approximation methods were proposed in the literature by +considering only M input points to represent the entire +training data [27]. These input points are called the inducing +points Xm and their corresponding values of the underlying +function f(x) are called the inducing variables fm. Replacing +the entire data set with only the M-inducing inputs leads to +the SGP which has a computational complexity of O(NM2). +Titsias [2] proposed a variational learning framework to +jointly estimate the kernel hyperparameters and the inducing +points. Titsias’ framework approximates the true exact poste- +rior of a GP p( f|y,Θ) by a variational posterior distribution +q( f, fm), +q(f, fm) = p(f|fm)φ( fm), +(3) +where φ( fm) is the free variational Gaussian distribution. The +Kullback-Leibler (KL) divergence is used to describe the dis- +crepancy between the approximated and the true posteriors. +Minimizing the KL divergence between the approximated +and the true posteriors KL[q( f, fm)||p( f|y,Θ)] is equivalent +to maximizing the variational lower bound of the true log +marginal likelihood: +FV (Xm) = log +� +N +� +y | 0,σ2I +Qnn +�� +− +1 +2σ2 Tr( �K), +Qnn = KnmK−1 +mmKmn, +�K = Cov(f | fm) = Knn −KnmK−1 +mmKmn, +(4) +where FV (Xm) is the variational objective function, Tr( �K) is a +regularization trace term, Knn is the original n×n co-variance +matrix, Kmm is m × m co-variance matrix on the inducing +inputs, Knm is n×m cross-covariance matrix between training +and inducing points, and Knm = KT +mn. More details on VSGP +can be found in Titsias’s work [2]. +IV. METHODOLOGY +The proposed approach exploits the VSGP as a generative +model to encode 3D pointcloud. The VSGP is selected +among different approximation approaches of GP due to +the following reasons: i) The variational approximation dis- +tinguishes between the inducing points M (as a variational +parameter) and the kernel hyperparameters (Θ,σ). ii) The +regularization term Tr( �K) in the variational objective func- +tion (Eq. (4)) regularizes the hyperparameters to avoid over- +fitting of the data. iii) The variational approximation offers +a discrete optimization scheme for selecting the inducing +inputs Xm from the original data1. +A. VSGP as a generative model for the occupancy surface +Inspired by [13], we project the occupied points ob- +served by a ranging sensor, e.g., LiDAR, onto a circular +surface around the sensor origin with a predefined radius +1For more information about the inducing point selection, check [2] + +roc. This surface is called occupancy surface, see Fig. 3. +In our approach, the sensor observation is defined in the +spherical coordinate system, where any observed point xi +is described by the tuple (θi,αi,ri) which represents the +azimuth, elevation, and radius values, respectively. Also, +any pointcloud data can be converted from the cartesian +coordinates (xi,yi,zi) to the spherical coordinates (θi,αi,ri) +using the following equations: +ri = +� +x2 +i +y2 +i +z2 +i , +θi = tan−1(yi,xi), +αi = cos−1(zi/ri). +(5) +All observed points that lie outside the circular occupancy +surface (with a radius ri > roc) or on the surface (with a +radius ri = roc) are neglected and considered as free space. +The rest of the points that are inside the circular surface (with +a radius ri < roc) are projected on the occupancy surface and +called the occupied points. Therefore, the occupancy surface +radius roc acts as the maximum range of the sensor. Each +occupied point xi on the surface is defined by two attributes: +the azimuth and elevation angles xi = (θi,αi), and assigned +an occupancy value f(xi) that is a function of the point radius +ri. The probability of occupancy f(xi) at each point on the +occupancy surface is modeled by a VSGP: +f(x) ∼ VSGP +� +m(x),k +� +x,x′�� +. +(6) +Considering noisy measurements, we add a white noise ε to +the occupancy function f(x), so the observed occupancy is +described as yi = f(xi)+ε where ε follows a Gaussian dis- +tribution N +� +0,σ2 +n +� +. The final model of the occupancy surface +is a 2D VSGP where the input is the azimuth and elevation +angles, x ∈ {(θ,α)}n +i=1, and the corresponding output is the +expected occupancy yi. The three main components of the +final VSGP are: +1) Zero-Mean Function m(x): There are different for- +mulas to describe the relationship between the occupancy +of a point f(xi) on the occupancy surface and its radius +ri [13]. For example, one candidate is f(xi) = 1/ri where ri +is bounded by the minimum and the maximum range of the +sensor rmin < ri < rmax = roc, where rmin > 0. Our approach +relates the occupancy of a point f(xi) to its radius ri by the +following equation f(xi) = roc − ri. This mapping between +the occupancy and the radius of a point is compatible with +the previous assumption that the occupancy surface radius +roc represents the maximum range of the sensor. Moreover, +this mapping is encoded in our VSGP model as a zero-mean +function m(x) = 0 that sets the occupancy value of the non- +observed points to zero. This mapping behavior mimics the +mechanism of the LiDAR itself. +2) Rational Quadratic (RQ) Kernel: The RQ kernel is +selected because a GP prior with an RQ kernel is expected +to have functions that vary across different length scales. +This quality of the RQ kernel copes with the nature of the +occupancy surface, specifically in unstructured environments +where a range of diverse length scales is required, i.e., +kRQ +� +x,x′� += σ2 +� +1+ (x−x′)2 +2αℓ2 +�−α +, +(7) +(a) +(b) +(c) +Fig. 3: (a) Gazebo scene of a robot in a tunnel (black); (b) +The occupancy surface generated from the original pointcloud, +where warmer colors reflect smaller f(xi) values (less occupancy); +(c) The inner surface represents the original occupancy surface +(same as in b), and the middle surface represents the reconstructed +occupancy surface using the VSGP model. The outer grey-coded +surface represents the variance associated with each point on the +reconstructed occupancy surface where brighter colors reflect high +uncertainty. Raw pointcloud is shown in red in (b) and (c). +where σ2 +f is the signal variance, l is the length-scale, and α +sets the relative weighting of large and small scale variations. +The RQ co-variance function is more expressive in terms of +modeling the occupancy surface than the most commonly +used Squared Exponential (SE) co-variance function. This +can be reasoned by the fact that the RQ kernel (when α +and l > 0) is equivalent to a scale mixture of SE kernels +with mixed characteristic length-scales [27]. In practice, we +take into account the resolution of LiDAR along both the +azimuth and elevation axes to initiate different length-scales +along each axis to reflect the LiDAR resolution. +3) Inducing Points Selection: The variational learning +framework proposed in [2] jointly optimizes the variational +parameters (inducing points) and the hyperparameters (Θ,σ) +through a variational Expectation-Maximization (EM) algo- +rithm. In general, the original discrete optimization frame- +work [2] suggests having an incremental set of the inducing +points, so that during the Expectation step (E-step) a point +from the input data is added to the inducing points set to +maximize the variational objective function FV and minimize +the KL divergence between the true and approximated pos- +teriors KL[q( f)||p( f|y,Θ)]. Then the hyperparameters are +updated during the Maximization step (M-step). +Since LiDAR’s field of view is limited within a certain +range, the projection of the observed points on the circular +surface leads to a limited input domain for the VSGP. In +our case, the azimuth and the elevation axes are limited +to (−π to π) and (−15◦ to 15◦), respectively. The limited +input domain is used to initiate a fixed number of inducing +points at evenly distributed locations on the occupied part of +the occupancy surface. In this way, a different combination +of the points is selected at each E-step to maximize the +variational objective function FV and minimize the KL +divergence. Then the hyperparameters are updated during the +M-step. The number of the inducing points M is chosen to +compromise the computational and memory complexity on +one side and the accuracy of the reconstructed pointcloud +on the other side. More inducing points result in higher +computations complexity O(NM2), larger memory to store +the encoded observation, and higher bandwidth to transfer it. +However, more inducing points increase the accuracy of the + +reconstructed pointcloud. We chose M=500 inducing points +to keep the average deviation between the reconstructed +pointcloud and the original pointcloud under 15 cm, see +Section V-A.2 and Fig. 5. After the training phase on the +scout side is completed, the selected inducing points are +combined together with the hyperparameters values of the +VSGP and are transmitted from the scout to the base. +B. Variance-based sampling +On the base side, the inducing points and the values of +the hyperparameters, which are received from the scout, +are used to reconstruct the original occupancy surface. The +reconstruction is done through a GP configured with the +same kernel (RQ) and likelihood (Gaussian) as the VSGP +on the scout side. The base GP is trained on the inducing +points and has a computation complexity of O(M3) where +M is the number of the inducing points, so we refer it as +a sparse GP (SGP) and refer the reconstructed occupancy +surface as the SGP occupancy surface. A grid of query points +x∗ = {(θ,α)}K +i=1 with the same resolution of the LiDAR +along the azimuth and the elevation axes is generated to +reconstruct the original pointcloud from the SGP occupancy +surface – we refer the reconstructed pointcloud as the SGP +pointcloud. If up-sampling of the pointcloud is required for +any reason, a query grid with higher resolution can be used +for the reconstruction process. The SGP occupancy surface +is used to predict the occupancy f(xi) of each point xi of +the query grid x∗. The occupancy is converted back to the +spherical radius ri = roc − f(xi) to restore the 3D spherical +coordinates of each point. +One advantage of the GP and its variants over other +modeling techniques is the uncertainty (variance) associated +with the predicted value at any query point. Considering +the VSGP model of the occupancy surface on the scout +side, the variance associated with the occupied points is low +compared to the variance related to the free points. Selecting +the inducing points as a set from the original occupied +points maintains low-variance values for the occupied part of +the reconstructed SGP occupancy surface on the base side. +Therefore, the variance value associated with any point on +the reconstructed SGP occupancy surface is used to predict +if that point belongs to the occupied or the free part of the +occupancy surface, see Fig. 4. We use a variance threshold +Vth as a judging criterion. In fact, the variance related to +the occupancy surface is different from one observation to +another, and it is affected by both the number of observed +(occupied) points and their distribution over the occupancy +surface. Therefore, we chose the variance threshold Vth as +a variable that changes with the distribution of the variance +over the occupied and free parts of the occupancy surface. +Vth is defined as a linear combination of the variance mean +vm and standard deviation vstd over the surface, i.e., Vth = +Km ∗vm +Kstd ∗vstd where Km and Kstd are constants. These +two constants are tuned by first setting Vth = vm (Km = 1 , +Kstd = 0), then we increase Kstd and decrease Km gradually +till we get the values that give the highest accuracy for the +reconstructed SGP pointcloud (considering a fixed number of +(a) +(b) +(c) +Fig. 4: Variance-based sampling. (a) Gazebo scene shows the +entrance of the tunnel; (b) shows the original (inner), reconstructed +(middle), and variance (outer) surfaces. It also shows the re- +constructed pointcloud (in white) through reconstructing from all +points (free and occupied) of the occupancy surface. (c) shows +reconstructed SGP pointcloud after removing all points that most +likely belong to the free part of the occupancy surface. Raw +pointcloud is shown in red in (b) and (c). +inducing points). Our sampling-based approach is capable +of discriminating between the free points that most likely +belong to the free part of the SGP occupancy surface and +the occupied points that belong to the the occupied part of +the SGP occupancy surface. After removing the free part +of the SGP occupancy surface, the Cartesian coordinates of +the occupied points are calculated using the inverse form of +Eq. (5) to restore the original point cloud, see Fig. 4c. +V. EXPERIMENTAL DESIGN AND RESULTS +The proposed approach is implemented in Python3 on +top of GPflow-v2 [28] and TensorFlow-v2.4 [29] under +ROS framework [30]. Both real-time simulation and real- +time demonstration were considered to evaluate the proposed +approach. In both the simulation and the hardware experi- +ments, a VLP-16 LiDAR was used with a maximum range +of 10m, a frequency of 4Hz, and a resolution of (0.1◦,2◦) +along the azimuth and the elevation axis, respectively. This +configuration results in a maximum pointcloud size of 57600 +points. The query grid, which is used to sample the SGP +occupancy surface on the base side, has the same resolution +as the VLP-16 LiDAR. A 3D occupancy grid map with a +resolution of 5cm is generated from the reconstructed SGP +pointcloud through Octomap [31]. +We investigate the performance of our framework and +compare it with the GMM approach [13]–[15]. While the +GMM approach tackles the occupancy mapping problem as +a whole, our approach focuses on compressing sensor obser- +vations through limited-bandwidth communication channels. +To be able to compare the two approaches, we implemented +the GMM approach in such a way that it is used to encode +one sensor observation at a time instead of generating an +entire occupancy map. We compared our approach with two +versions of the GMM approach: i) A CPU-based implemen- +tation of GMM that follows the same guidelines of [13]. +ii) An upgraded GPU-based implementation of GMM. We +implemented the GPU-GMM to have a fair computation +comparison with our VSGP approach which runs on GPU. +A. Simulation Experiments +1) Simulation Setup:: The simulation setup consists of +two machines that communicate to each other over WiFi: + +The first machine, where the scout and the environment are +simulated, is an Intel® Core™ i7 NUC11 PC equipped with +64 GB RAM and 6 GB Geforce RTX2060 GPU. The second +machine, which acts as the base, is an Intel® Core™ i7 +Alienware Laptop equipped with 32 GB RAM and 8 GB +Geforce RTX2080 GPU. Both are connected using a 2.4 GHz +WiFi router. The network flow is monitored using the ifstat +tool to evaluate the communication performance. The mine +tunnel of the cpr inspection world, which is developed by +ClearPath robotics, is used as our simulation environment. +This environment is selected because it represents one of +the targeted low-bandwidth subterranean environments. The +mine tunnel part of the cpr inspection world fits in a rectan- +gular area with an approximated area of 30×65m2, the tunnel +length is around 135m. The ground elevation and the height +of the tunnel are different from one place to another. The +ClearPath Jackal robot is used as the scout. The proposed +approach was evaluated through 20 real-time simulation +trials. In each trial, the robot starts at the beginning of the +cave and follows a predefined path along the mine using +way-point based navigation. +2) Simulation Results: We evaluate the performance of +our approach based on the reduction in the memory and the +communication rate required to transmit the sensor observa- +tions between the scout and the base. The VSGP representa- +tion requires only 1514 floating points (FP) to represent the +entire pointcloud (3 FP for each inducing point (3x500) + 6 +FP for robot pose + 6 FP for the hyperparameters). This value +is less than the memory needed by the GMM approach which +requires ∼ 2000 FP (10 FP for each component (10x200) +distributed as 6 FP for covariance + 3 FP for mean + 1 +FP for weight) [13]. We send the robot pose to the base +because our approach encodes the observation relative to the +robot body frame, while the GMM approach first transforms +the observation from the robot body frame to a global frame +using the robot current pose and then sends the encoded +Gaussians densties with respect to the global frame. +To quantify the accuracy of the reconstructed SGP point- +cloud, we use the Root Mean Square Deviation (RMSD) +between the radius predicted by our approach and the actual +radius of each point on the occupancy surface. +RMSD = +� +∑N +i=1 (ri − ˆri)2 +N +, +(8) +where N is the size of the pointcloud, ri is the actual radius at +(θi,αi), and ˆri is the estimated radius value at the same point +(θi,αi). Fig. 5a shows the mean and the standard deviation +of the RMSD for each predicted point over 110 observations +(each observation has around 10K to 50K points). Also, +Fig. 5a implicitly reflects the memory required by VSGP and +GMM to store one observation, as described before that the +memory required to store one observation can be calculated +by multiplying the number of inducing points (bottom x-axis) +by 3 and multiplying the number of components (top x-axis) +by 10. We match pairs of the VSGP and GMM models (in +terms of the number of inducing points and components) +based on the memory requirement and the accuracy of the +(a) +(b) +(c) +(d) +Fig. 5: Performance comparisons. (a) shows the RMSD between +the reconstructed and the original pointcloud for VSGP(vs #induc- +ing points) and GMM(vs #components); (b) illustrates the training +time against the pointcloud size (considering 500-inducing points +VSGP, and equivalently, 200-components GMM); (c) represents +the training time versus the #VSGP-inducing points and #GMM- +components; (d) shows the prediction time versus the #VSGP- +inducing points and #GMM-components. +reconstructed pointcloud (reflected by the RMSD) for each +pair, see table I. For example, 500-inducing points VSGP +results in an average RMSD value for each point of 9 cm +with a standard deviation of 10 cm. This corresponds to an +average RMSD of 11 cm with a standard deviation of 25 cm +for a 200-components GMM. +TABLE I: VSGP vs GMM(ind: inducing, cps: components) +VSGP +GMM +# +Memory +RMSD +# +Memory +RMSD +ind +∼FPs +∼cm +cps +∼FPs +∼cm +200 +600 +20±22 +50 +500 +27±50 +300 +900 +14±15 +100 +1000 +16±35 +400 +1200 +12±14 +150 +1500 +13±31 +500 +1500 +9±10 +200 +2000 +11±29 +600 +1800 +9±10 +250 +2500 +11±30 +Now we analyze the results in Fig. 5. Fig. 5a shows the +RMSD values associated with VSGP have a smaller standard +deviation than the GMM’s. It also shows that increasing the +number of the VSGP-inducing points (bottom x-axis) or the +number of the GMM-components (top x-axis) will result in +smaller RMSD (higher accuracy). +An intensive evaluation of the training and the prediction +phases is presented in Figs. 5b-5d. The reduction in the +training time versus the reduction in the size of the raw +pointcloud is presented in Fig. 5b, where 0% removal percent +means a pointcloud size of 57.6K points. Fig. 5c shows +the increase in training time versus the number of inducing +points and the number of components. We compare the +training time of the VSGP, the GMM-CPU (considering the + +#Components +50 +100 +150 +200 +250 +0.75 - +GMM +0.50 +VSGP +RMSI +0.25 +0.00 +0.25 +200 +300 +00f +500 +600 +#Inducing Points103 +Su +VSGP +GNIM +GNIM-GPU +102 +3350606775 +Removal Components +50 +100 +150 +200 +250 +103 +GMM +GMM-GPU +VSGP +102 +200 +400 +600 +Inducing PointsComponents +50 +100 +150 +200 +250 +VSGP +[stu] +GMM +40 +Prediction +20 +200 +400 +600 +Inducing Points(a) +(b) +(c) +(d) +Fig. 6: (a) shows the simulated mine environment in Gazebo; +(b) shows the Octomap of the mine generated from the original +pointcloud; (c) shows the Octomap generated from the recon- +structed SGP pointcloud; (d) shows the communication rate and the +accumulated data sent from the scout to the base in case of sending +raw pointcloud PCL(1750KB/S, 840MB), GMM data(25.8KB/S, +12.4MB), and VSGP data(18.2KB/S, 8.7MB). The y-axis is plotted +in log-scale. +default configuration of the GMM approach used in [13]), +and the GMM-GPU implementation. The results show that +our approach outperforms both the CPU and GPU imple- +mentation of the GMM approach in terms of training time. +Fig. 5d presents the variation of the prediction time of the +VSGP versus the number of the inducing points, where the +values shown in the figure represent the time required to +predict the occupancy value associated with all the points of +the grid query x (57600 points). +Fig. 5d indicates that for a matching pair of GMM and +VSGP (Table I), GMM has a less sampling time than +the paired VSGP. However, the pointcloud reconstruction +process of the VSGP is more convenient than the GMM +approach because a fundamental difference between sam- +pling the VSGP and the GMM is that: when we sample +from a GMM, we get a sample (from a distribution) with +random values (θs,αs,rs), so we do not have control over +the location of the sample on the occupancy surface (θs,αs). +In contrast, for the VSGP approach, we predict the radius +value rs for a certain point on the occupancy surface defined +by (θs,αs). So, we have control over the point location on +the occupancy surface. While constructing the 3D octomap +of the tunnel environment using the scout-base scheme, the +average communication rate was 1750 KB/S, 25.8 KB/S, +and 18.2 KB/S for sending raw point clouds, GMM encoded +data, and VSGP encoded data respectively, see Fig. 6d. The +accumulated data sent through the network is reduced from +840 MB for sending raw pointcloud to 12.4 MB in case +of GMM and 8.7 MB in case of VSGP. This indicates a +(a) +(b) +(c) +Fig. 7: Indoor demonstration. (a) shows octomap of the laboratory +building generated from the original pointcloud. (b) shows octomap +generated from the reconstructed SGP pointcloud. (c) shows the +reduction in the communication rate and the accumulated data sent +from the scout to the base, where log-scale is used for y-axis. PCL +represents the raw pointcloud. +compression ratio of ∼ 96 (840/8.7 ∼ 1750/18.2). +B. Hardware Experiment +A Jackal mobile robot, equipped with a VLP-16 LiDAR +and NUC11 PC, was used as the scout, while the Alien- +ware laptop was used as the base. The demonstration was +conducted in an indoor environment, where the VSGP- +encoded pointcloud data was sent from the scout to the +base to generate a 3D Octomap [31] of the building from +the SGP reconstructed pointcloud in real-time, see Fig. 7. +Fig. 7c shows the reduction in the communication rate for +the hardware experiment. The communication rate dropped +from around 560 KB for transmitting raw pointcloud to +around 8 KB for transmitting the encoded VSGP (this ratio +is equivalent to 70 times smaller rate). The communication +rate of the hardware experiment is low compared to the +simulation experiment because the LiDAR resolution was +halved during the hardware experiment. The total amount +of data transmitted at the end of each trial was around 100 +MB for sending raw pointcloud and only around 1.4 MB for +sending the VSGP encoded observation. +VI. CONCLUSION +In this paper, we introduce a lightweight representation +for the 3D pointcloud using the VSGP. This representation +allows high-fidelity observations to be efficiently stored +and transmitted through limited-bandwidth communication +channels. Based on the results of the simulation and hardware +experiments, our approach results in around 70-100 times +smaller size representation of the sensor observation. This +compact representation can facilitate many of the robotics + +KB +PCI +GMM +Rate +102 +VSGP +MB +PCL +101 +Data +GMM +VSGP +0 +O0T +200 +300 +400 +500 +Time [S]S/ +POL +101 +VSGP +Rate +KB +PCL +Data +102 +VSGP +0 +25 +50 +75 +100 +125 +150 +175 +Time [S]applications which are limited by the communication band- +width such as subterranean and underwater exploration, +search and rescue missions, and planetary exploration. In +addition, our approach can also be beneficial in the context +of multi-robot collaboration where a number of robots are +required to share high-volume information (3D pointcloud) +through low-bandwidth channels. +ACKNOWLEDGEMENT +This work was supported by National Science Foundation +with grant numbers 2006886 and 2047169. +REFERENCES +[1] V. H. Cid et al. 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Octomap: An efficient probabilistic 3d mapping +framework based on octrees. +Autonomous robots, 34(3):189–206, +2013. + diff --git a/JNFIT4oBgHgl3EQfZCuh/content/tmp_files/load_file.txt b/JNFIT4oBgHgl3EQfZCuh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee16b656aff81fc96f6757b91a519324df0ed3d5 --- /dev/null +++ b/JNFIT4oBgHgl3EQfZCuh/content/tmp_files/load_file.txt @@ -0,0 +1,413 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf,len=412 +page_content='This paper has been accepted for publication at 2023 IEEE International Conference on Robotics and Automation (ICRA 2023) Light-Weight Pointcloud Representation with Sparse Gaussian Process Mahmoud Ali and Lantao Liu Abstract— This paper presents a framework to represent high-fidelity pointcloud sensor observations for efficient com- munication and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The proposed approach exploits Sparse Gaussian Process to encode pointcloud into a compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our approach represents both the free space and the occupied space using only one model (one 2D Sparse Gaussian Process) instead of the existing two-model framework (two 3D Gaussian Mixture Models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We achieve this by proposing a variance- based sampling technique that effectively discriminates between the free and occupied space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The new representation requires less memory footprint and can be transmitted across limited- bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The framework is exten- sively evaluated in simulation and it is also demonstrated using a real mobile robot equipped with a 3D LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our method results in a 70∼100 times reduction in the communication rate compared to sending the raw pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' INTRODUCTION With the rapid advancement of LiDAR technology, we now can build maps with remarkably high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, each full scan of an only 16-channel 3D LiDAR can give us 57600 points in the pointcloud that represents the surrounding obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' However, a price for using the high resolution LiDAR is the computation, storage, and com- munication costs when mapping the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' While one might be able to upgrade the computation and storage by using a high performance computer system, the com- munication usually becomes a bottleneck due to the low communication bandwidth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In practice, the low bandwidth communication is considered as a major challenge for many robotics applications such as occupancy mapping of underwater and subterranean environments (caves, tunnels, mines, etc), search-and-rescue missions in disaster scenarios with a degraded communication infrastructure, and planetary exploration missions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The low bandwidth can prevent a robot from real-time sharing its sensor observations, and this can significantly degrade the system responsiveness if the robot needs to follow or interact with external control or supervision platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This work tackles the problem of sharing high-fidelity 3D pointcloud through a limited bandwidth communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The system we consider consists of a robot (the scout) equipped with a LiDAR and a communication apparatus, and deployed in a low-bandwidth environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The scout sends 1Mahmoud Ali and Lantao Liu are with the Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN 47408 USA, {alimaa, lantao}@iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='edu Occupancy Surface VSGP variance-based sampling SGP OctoMap wifi Scout Base Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 1: System Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2: (a) Gazebo simulated mine tunnel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) Original pointcloud generated by a VLP16 LiDAR in red, and reconstructed pointcloud from the VSGP model in white;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (d) Occupancy Map generated by OctoMap from the reconstructed pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' the observations that it acquires to a base for building the occupancy map of the environment, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our approach exploits the Variational Sparse Gaussian Process (VSGP) [2] as a generative model to represent the pointcloud in a compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This lightweight representation is transmitted through low-bandwidth communication to the base where the original pointcloud is reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Extensive evaluations reveal that our approach results in a 70∼100 times reduction in the memory as well as the communication rate required to transmit pointcloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2a shows a scene of a simulated mine tunnel, where its raw pointcloud (shown in red, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2b) requires around 750 KB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our approach is able to represent the same observation using only 6 KB of memory and transmit through limited-bandwidth communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' On the receiver side of the communication channel, the compact representation is used to reconstruct the original pointcloud (reconstructed pointcloud shown in white, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' An occupancy map of the scene can be built using the reconstructed pointcloud, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' RELATED WORK Pointcloud compression algorithms have been investigated in recent years to cope with the demands to store and communicate the high-precision 3D points [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, the space partitioning trees approaches that exploit the 3D correlation between pointcloud points are widely used to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='11251v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='RO] 26 Jan 2023 Velodynecompress the pointcloud data [4]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Recently, deep learning based approaches were also proposed to leverage data and learn or encode the pointcloud compression [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Dif- ferent from these frameworks, the probabilistic approaches exploit the compactness of the distributions to compress 3D sensor observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For instance, Gaussian Mixture Models (GMM) [13]–[15] have been proposed as a generative model to encode 3D occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The GMM approach encodes the 3D data as a mixture of Gaussian densities to represent the occupied and free spaces around the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Gaussian Process (GP) has been proven to be an excellent framework to model spatial phenomena or features in a continuous domain [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Unfortunately, the standard GP has a cubic time complexity and this results in very limited scalability to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Methods for reducing the computing burdens of GPs have been previously investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, GP regressions can be done in a real-time fashion where the problem can be estimated locally with local data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Sparse GPs (SGPs) [20]–[26] tackle the com- putational complexity of the normal GP through leveraging the Bayesian rule with a sequential construction of the most relevant subset of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We propose a new probabilistic pointcloud compression approach which is based on the VSGP [2] and inspired by the GMM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' While the GMM shares the accumulated sensory information as a set of accumulated Gaussian den- sities which are sampled and used as an occupancy map of the environment, in contrast, the proposed approach relies on sharing of immediate sensor observation to be reconstructed on the other side of the communication channel for further processing based on the required task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 3D mapping, object recognition, tracking, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This proposed VSGP-based approach offers a few ad- vantages over the recent GMM approach: while the GMM approach uses two 3D GMMs to fit the occupied and free points [13]–[15], our approach uses only one 2D VSGP to fit all the occupancy surface, including both the occupied and free points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The primary reason that our approach uses one VSGP instead of two is that we are using the variance calculated by the VSGP at each sampled point during the reconstruction process to decide if it belongs to the occupied or the free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Therefore, the proposed approach results in a more compact representation of the sensor observation, which requires less memory than the GMM approach and, as a consequence, leads to a lower communication rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' BACKGROUND GP is a non-parametric model described by a mean function m(x), and a co-variance function (kernel) k(x,x′), where x is the GP input [27]: f(x) ∼ GP � m(x),k � x,x′�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (1) Considering a data set D = {(xi,yi)}N i=1 with N training inputs x and their corresponding scalar outputs (observations) y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' After training the GP using the data set D, the output y∗ for any new query x∗ can be estimated using the GP prediction: p(y∗|y) = N(y∗|my(x∗),ky(x∗,x∗)+σ2), (2) where my(x) and ky(x,x′) are the posterior mean and co- variance functions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The GP prediction equation depends on the values of the hyperparameters (Θ,σ2) where Θ is the kernel parameters and σ2 is the noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The computation complexity of a full GP is O(N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In order to reduce the computation complexity, different approximation methods were proposed in the literature by considering only M input points to represent the entire training data [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' These input points are called the inducing points Xm and their corresponding values of the underlying function f(x) are called the inducing variables fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Replacing the entire data set with only the M-inducing inputs leads to the SGP which has a computational complexity of O(NM2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Titsias [2] proposed a variational learning framework to jointly estimate the kernel hyperparameters and the inducing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Titsias’ framework approximates the true exact poste- rior of a GP p( f|y,Θ) by a variational posterior distribution q( f, fm), q(f, fm) = p(f|fm)φ( fm), (3) where φ( fm) is the free variational Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The Kullback-Leibler (KL) divergence is used to describe the dis- crepancy between the approximated and the true posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Minimizing the KL divergence between the approximated and the true posteriors KL[q( f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' fm)||p( f|y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='Θ)] is equivalent to maximizing the variational lower bound of the true log marginal likelihood: FV (Xm) = log � N � y | 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='σ2I +Qnn �� − 1 2σ2 Tr( �K),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Qnn = KnmK−1 mmKmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' �K = Cov(f | fm) = Knn −KnmK−1 mmKmn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (4) where FV (Xm) is the variational objective function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Tr( �K) is a regularization trace term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Knn is the original n×n co-variance matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Kmm is m × m co-variance matrix on the inducing inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Knm is n×m cross-covariance matrix between training and inducing points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' and Knm = KT mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' More details on VSGP can be found in Titsias’s work [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' METHODOLOGY The proposed approach exploits the VSGP as a generative model to encode 3D pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The VSGP is selected among different approximation approaches of GP due to the following reasons: i) The variational approximation dis- tinguishes between the inducing points M (as a variational parameter) and the kernel hyperparameters (Θ,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' ii) The regularization term Tr( �K) in the variational objective func- tion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (4)) regularizes the hyperparameters to avoid over- fitting of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' iii) The variational approximation offers a discrete optimization scheme for selecting the inducing inputs Xm from the original data1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' VSGP as a generative model for the occupancy surface Inspired by [13], we project the occupied points ob- served by a ranging sensor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=', LiDAR, onto a circular surface around the sensor origin with a predefined radius 1For more information about the inducing point selection, check [2] roc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This surface is called occupancy surface, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In our approach, the sensor observation is defined in the spherical coordinate system, where any observed point xi is described by the tuple (θi,αi,ri) which represents the azimuth, elevation, and radius values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Also, any pointcloud data can be converted from the cartesian coordinates (xi,yi,zi) to the spherical coordinates (θi,αi,ri) using the following equations: ri = � x2 i +y2 i +z2 i , θi = tan−1(yi,xi), αi = cos−1(zi/ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (5) All observed points that lie outside the circular occupancy surface (with a radius ri > roc) or on the surface (with a radius ri = roc) are neglected and considered as free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The rest of the points that are inside the circular surface (with a radius ri < roc) are projected on the occupancy surface and called the occupied points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Therefore, the occupancy surface radius roc acts as the maximum range of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Each occupied point xi on the surface is defined by two attributes: the azimuth and elevation angles xi = (θi,αi), and assigned an occupancy value f(xi) that is a function of the point radius ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The probability of occupancy f(xi) at each point on the occupancy surface is modeled by a VSGP: f(x) ∼ VSGP � m(x),k � x,x′�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (6) Considering noisy measurements, we add a white noise ε to the occupancy function f(x), so the observed occupancy is described as yi = f(xi)+ε where ε follows a Gaussian dis- tribution N � 0,σ2 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The final model of the occupancy surface is a 2D VSGP where the input is the azimuth and elevation angles, x ∈ {(θ,α)}n i=1, and the corresponding output is the expected occupancy yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The three main components of the final VSGP are: 1) Zero-Mean Function m(x): There are different for- mulas to describe the relationship between the occupancy of a point f(xi) on the occupancy surface and its radius ri [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, one candidate is f(xi) = 1/ri where ri is bounded by the minimum and the maximum range of the sensor rmin < ri < rmax = roc, where rmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our approach relates the occupancy of a point f(xi) to its radius ri by the following equation f(xi) = roc − ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This mapping between the occupancy and the radius of a point is compatible with the previous assumption that the occupancy surface radius roc represents the maximum range of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Moreover, this mapping is encoded in our VSGP model as a zero-mean function m(x) = 0 that sets the occupancy value of the non- observed points to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This mapping behavior mimics the mechanism of the LiDAR itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2) Rational Quadratic (RQ) Kernel: The RQ kernel is selected because a GP prior with an RQ kernel is expected to have functions that vary across different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This quality of the RQ kernel copes with the nature of the occupancy surface, specifically in unstructured environments where a range of diverse length scales is required, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=', kRQ � x,x′� = σ2 � 1+ (x−x′)2 2αℓ2 �−α , (7) (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 3: (a) Gazebo scene of a robot in a tunnel (black);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) The occupancy surface generated from the original pointcloud, where warmer colors reflect smaller f(xi) values (less occupancy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (c) The inner surface represents the original occupancy surface (same as in b), and the middle surface represents the reconstructed occupancy surface using the VSGP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The outer grey-coded surface represents the variance associated with each point on the reconstructed occupancy surface where brighter colors reflect high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Raw pointcloud is shown in red in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' where σ2 f is the signal variance, l is the length-scale, and α sets the relative weighting of large and small scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The RQ co-variance function is more expressive in terms of modeling the occupancy surface than the most commonly used Squared Exponential (SE) co-variance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This can be reasoned by the fact that the RQ kernel (when α and l > 0) is equivalent to a scale mixture of SE kernels with mixed characteristic length-scales [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In practice, we take into account the resolution of LiDAR along both the azimuth and elevation axes to initiate different length-scales along each axis to reflect the LiDAR resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 3) Inducing Points Selection: The variational learning framework proposed in [2] jointly optimizes the variational parameters (inducing points) and the hyperparameters (Θ,σ) through a variational Expectation-Maximization (EM) algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In general, the original discrete optimization frame- work [2] suggests having an incremental set of the inducing points, so that during the Expectation step (E-step) a point from the input data is added to the inducing points set to maximize the variational objective function FV and minimize the KL divergence between the true and approximated pos- teriors KL[q( f)||p( f|y,Θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Then the hyperparameters are updated during the Maximization step (M-step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Since LiDAR’s field of view is limited within a certain range, the projection of the observed points on the circular surface leads to a limited input domain for the VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In our case, the azimuth and the elevation axes are limited to (−π to π) and (−15◦ to 15◦), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The limited input domain is used to initiate a fixed number of inducing points at evenly distributed locations on the occupied part of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In this way, a different combination of the points is selected at each E-step to maximize the variational objective function FV and minimize the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Then the hyperparameters are updated during the M-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The number of the inducing points M is chosen to compromise the computational and memory complexity on one side and the accuracy of the reconstructed pointcloud on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' More inducing points result in higher computations complexity O(NM2), larger memory to store the encoded observation, and higher bandwidth to transfer it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' However, more inducing points increase the accuracy of the reconstructed pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We chose M=500 inducing points to keep the average deviation between the reconstructed pointcloud and the original pointcloud under 15 cm, see Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' After the training phase on the scout side is completed, the selected inducing points are combined together with the hyperparameters values of the VSGP and are transmitted from the scout to the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Variance-based sampling On the base side, the inducing points and the values of the hyperparameters, which are received from the scout, are used to reconstruct the original occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The reconstruction is done through a GP configured with the same kernel (RQ) and likelihood (Gaussian) as the VSGP on the scout side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The base GP is trained on the inducing points and has a computation complexity of O(M3) where M is the number of the inducing points, so we refer it as a sparse GP (SGP) and refer the reconstructed occupancy surface as the SGP occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' A grid of query points x∗ = {(θ,α)}K i=1 with the same resolution of the LiDAR along the azimuth and the elevation axes is generated to reconstruct the original pointcloud from the SGP occupancy surface – we refer the reconstructed pointcloud as the SGP pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' If up-sampling of the pointcloud is required for any reason, a query grid with higher resolution can be used for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The SGP occupancy surface is used to predict the occupancy f(xi) of each point xi of the query grid x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The occupancy is converted back to the spherical radius ri = roc − f(xi) to restore the 3D spherical coordinates of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' One advantage of the GP and its variants over other modeling techniques is the uncertainty (variance) associated with the predicted value at any query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Considering the VSGP model of the occupancy surface on the scout side, the variance associated with the occupied points is low compared to the variance related to the free points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Selecting the inducing points as a set from the original occupied points maintains low-variance values for the occupied part of the reconstructed SGP occupancy surface on the base side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Therefore, the variance value associated with any point on the reconstructed SGP occupancy surface is used to predict if that point belongs to the occupied or the free part of the occupancy surface, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We use a variance threshold Vth as a judging criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In fact, the variance related to the occupancy surface is different from one observation to another, and it is affected by both the number of observed (occupied) points and their distribution over the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Therefore, we chose the variance threshold Vth as a variable that changes with the distribution of the variance over the occupied and free parts of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Vth is defined as a linear combination of the variance mean vm and standard deviation vstd over the surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=', Vth = Km ∗vm +Kstd ∗vstd where Km and Kstd are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' These two constants are tuned by first setting Vth = vm (Km = 1 , Kstd = 0), then we increase Kstd and decrease Km gradually till we get the values that give the highest accuracy for the reconstructed SGP pointcloud (considering a fixed number of (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 4: Variance-based sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (a) Gazebo scene shows the entrance of the tunnel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) shows the original (inner), reconstructed (middle), and variance (outer) surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' It also shows the re- constructed pointcloud (in white) through reconstructing from all points (free and occupied) of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (c) shows reconstructed SGP pointcloud after removing all points that most likely belong to the free part of the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Raw pointcloud is shown in red in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' inducing points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Our sampling-based approach is capable of discriminating between the free points that most likely belong to the free part of the SGP occupancy surface and the occupied points that belong to the the occupied part of the SGP occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' After removing the free part of the SGP occupancy surface, the Cartesian coordinates of the occupied points are calculated using the inverse form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (5) to restore the original point cloud, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' EXPERIMENTAL DESIGN AND RESULTS The proposed approach is implemented in Python3 on top of GPflow-v2 [28] and TensorFlow-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='4 [29] under ROS framework [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Both real-time simulation and real- time demonstration were considered to evaluate the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In both the simulation and the hardware experi- ments, a VLP-16 LiDAR was used with a maximum range of 10m, a frequency of 4Hz, and a resolution of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='1◦,2◦) along the azimuth and the elevation axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This configuration results in a maximum pointcloud size of 57600 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The query grid, which is used to sample the SGP occupancy surface on the base side, has the same resolution as the VLP-16 LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' A 3D occupancy grid map with a resolution of 5cm is generated from the reconstructed SGP pointcloud through Octomap [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We investigate the performance of our framework and compare it with the GMM approach [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' While the GMM approach tackles the occupancy mapping problem as a whole, our approach focuses on compressing sensor obser- vations through limited-bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' To be able to compare the two approaches, we implemented the GMM approach in such a way that it is used to encode one sensor observation at a time instead of generating an entire occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We compared our approach with two versions of the GMM approach: i) A CPU-based implemen- tation of GMM that follows the same guidelines of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' ii) An upgraded GPU-based implementation of GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We implemented the GPU-GMM to have a fair computation comparison with our VSGP approach which runs on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Simulation Experiments 1) Simulation Setup:: The simulation setup consists of two machines that communicate to each other over WiFi: The first machine, where the scout and the environment are simulated, is an Intel® Core™ i7 NUC11 PC equipped with 64 GB RAM and 6 GB Geforce RTX2060 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The second machine, which acts as the base, is an Intel® Core™ i7 Alienware Laptop equipped with 32 GB RAM and 8 GB Geforce RTX2080 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Both are connected using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='4 GHz WiFi router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The network flow is monitored using the ifstat tool to evaluate the communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The mine tunnel of the cpr inspection world, which is developed by ClearPath robotics, is used as our simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This environment is selected because it represents one of the targeted low-bandwidth subterranean environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The mine tunnel part of the cpr inspection world fits in a rectan- gular area with an approximated area of 30×65m2, the tunnel length is around 135m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The ground elevation and the height of the tunnel are different from one place to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The ClearPath Jackal robot is used as the scout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The proposed approach was evaluated through 20 real-time simulation trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In each trial, the robot starts at the beginning of the cave and follows a predefined path along the mine using way-point based navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 2) Simulation Results: We evaluate the performance of our approach based on the reduction in the memory and the communication rate required to transmit the sensor observa- tions between the scout and the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The VSGP representa- tion requires only 1514 floating points (FP) to represent the entire pointcloud (3 FP for each inducing point (3x500) + 6 FP for robot pose + 6 FP for the hyperparameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This value is less than the memory needed by the GMM approach which requires ∼ 2000 FP (10 FP for each component (10x200) distributed as 6 FP for covariance + 3 FP for mean + 1 FP for weight) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We send the robot pose to the base because our approach encodes the observation relative to the robot body frame, while the GMM approach first transforms the observation from the robot body frame to a global frame using the robot current pose and then sends the encoded Gaussians densties with respect to the global frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' To quantify the accuracy of the reconstructed SGP point- cloud, we use the Root Mean Square Deviation (RMSD) between the radius predicted by our approach and the actual radius of each point on the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' RMSD = � ∑N i=1 (ri − ˆri)2 N , (8) where N is the size of the pointcloud, ri is the actual radius at (θi,αi), and ˆri is the estimated radius value at the same point (θi,αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5a shows the mean and the standard deviation of the RMSD for each predicted point over 110 observations (each observation has around 10K to 50K points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Also, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5a implicitly reflects the memory required by VSGP and GMM to store one observation, as described before that the memory required to store one observation can be calculated by multiplying the number of inducing points (bottom x-axis) by 3 and multiplying the number of components (top x-axis) by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We match pairs of the VSGP and GMM models (in terms of the number of inducing points and components) based on the memory requirement and the accuracy of the (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5: Performance comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (a) shows the RMSD between the reconstructed and the original pointcloud for VSGP(vs #induc- ing points) and GMM(vs #components);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) illustrates the training time against the pointcloud size (considering 500-inducing points VSGP, and equivalently, 200-components GMM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (c) represents the training time versus the #VSGP-inducing points and #GMM- components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (d) shows the prediction time versus the #VSGP- inducing points and #GMM-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' reconstructed pointcloud (reflected by the RMSD) for each pair, see table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' For example, 500-inducing points VSGP results in an average RMSD value for each point of 9 cm with a standard deviation of 10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This corresponds to an average RMSD of 11 cm with a standard deviation of 25 cm for a 200-components GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' TABLE I: VSGP vs GMM(ind: inducing, cps: components) VSGP GMM # Memory RMSD # Memory RMSD ind ∼FPs ∼cm cps ∼FPs ∼cm 200 600 20±22 50 500 27±50 300 900 14±15 100 1000 16±35 400 1200 12±14 150 1500 13±31 500 1500 9±10 200 2000 11±29 600 1800 9±10 250 2500 11±30 Now we analyze the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5a shows the RMSD values associated with VSGP have a smaller standard deviation than the GMM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' It also shows that increasing the number of the VSGP-inducing points (bottom x-axis) or the number of the GMM-components (top x-axis) will result in smaller RMSD (higher accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' An intensive evaluation of the training and the prediction phases is presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5b-5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The reduction in the training time versus the reduction in the size of the raw pointcloud is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5b, where 0% removal percent means a pointcloud size of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='6K points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5c shows the increase in training time versus the number of inducing points and the number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' We compare the training time of the VSGP, the GMM-CPU (considering the #Components 50 100 150 200 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='75 - GMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='50 VSGP RMSI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='25 200 300 00f 500 600 #Inducing Points103 Su VSGP GNIM GNIM-GPU 102 3350606775 Removal Components 50 100 150 200 250 103 GMM GMM-GPU VSGP 102 200 400 600 Inducing PointsComponents 50 100 150 200 250 VSGP [stu] GMM 40 Prediction 20 200 400 600 Inducing Points(a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 6: (a) shows the simulated mine environment in Gazebo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) shows the Octomap of the mine generated from the original pointcloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (c) shows the Octomap generated from the recon- structed SGP pointcloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (d) shows the communication rate and the accumulated data sent from the scout to the base in case of sending raw pointcloud PCL(1750KB/S, 840MB), GMM data(25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='8KB/S, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='4MB), and VSGP data(18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='2KB/S, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='7MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The y-axis is plotted in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' default configuration of the GMM approach used in [13]), and the GMM-GPU implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The results show that our approach outperforms both the CPU and GPU imple- mentation of the GMM approach in terms of training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5d presents the variation of the prediction time of the VSGP versus the number of the inducing points, where the values shown in the figure represent the time required to predict the occupancy value associated with all the points of the grid query x (57600 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 5d indicates that for a matching pair of GMM and VSGP (Table I), GMM has a less sampling time than the paired VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' However, the pointcloud reconstruction process of the VSGP is more convenient than the GMM approach because a fundamental difference between sam- pling the VSGP and the GMM is that: when we sample from a GMM, we get a sample (from a distribution) with random values (θs,αs,rs), so we do not have control over the location of the sample on the occupancy surface (θs,αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In contrast, for the VSGP approach, we predict the radius value rs for a certain point on the occupancy surface defined by (θs,αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' So, we have control over the point location on the occupancy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' While constructing the 3D octomap of the tunnel environment using the scout-base scheme, the average communication rate was 1750 KB/S, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='8 KB/S, and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='2 KB/S for sending raw point clouds, GMM encoded data, and VSGP encoded data respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The accumulated data sent through the network is reduced from 840 MB for sending raw pointcloud to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='4 MB in case of GMM and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='7 MB in case of VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This indicates a (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 7: Indoor demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (a) shows octomap of the laboratory building generated from the original pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (b) shows octomap generated from the reconstructed SGP pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' (c) shows the reduction in the communication rate and the accumulated data sent from the scout to the base, where log-scale is used for y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' PCL represents the raw pointcloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' compression ratio of ∼ 96 (840/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='7 ∼ 1750/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Hardware Experiment A Jackal mobile robot, equipped with a VLP-16 LiDAR and NUC11 PC, was used as the scout, while the Alien- ware laptop was used as the base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The demonstration was conducted in an indoor environment, where the VSGP- encoded pointcloud data was sent from the scout to the base to generate a 3D Octomap [31] of the building from the SGP reconstructed pointcloud in real-time, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' 7c shows the reduction in the communication rate for the hardware experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The communication rate dropped from around 560 KB for transmitting raw pointcloud to around 8 KB for transmitting the encoded VSGP (this ratio is equivalent to 70 times smaller rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The communication rate of the hardware experiment is low compared to the simulation experiment because the LiDAR resolution was halved during the hardware experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' The total amount of data transmitted at the end of each trial was around 100 MB for sending raw pointcloud and only around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content='4 MB for sending the VSGP encoded observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' CONCLUSION In this paper, we introduce a lightweight representation for the 3D pointcloud using the VSGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This representation allows high-fidelity observations to be efficiently stored and transmitted through limited-bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' Based on the results of the simulation and hardware experiments, our approach results in around 70-100 times smaller size representation of the sensor observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' This compact representation can facilitate many of the robotics KB PCI GMM Rate 102 VSGP MB PCL 101 Data GMM VSGP 0 O0T 200 300 400 500 Time [S]S/ POL 101 VSGP Rate KB PCL Data 102 VSGP 0 25 50 75 100 125 150 175 Time [S]applications which are limited by the communication band- width such as subterranean and underwater exploration, search and rescue missions, and planetary exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' In addition, our approach can also be beneficial in the context of multi-robot collaboration where a number of robots are required to share high-volume information (3D pointcloud) through low-bandwidth channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was supported by National Science Foundation with grant numbers 2006886 and 2047169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' REFERENCES [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} +page_content=' H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFIT4oBgHgl3EQfZCuh/content/2301.11251v1.pdf'} diff --git a/JdE2T4oBgHgl3EQfUgew/content/tmp_files/2301.03814v1.pdf.txt b/JdE2T4oBgHgl3EQfUgew/content/tmp_files/2301.03814v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e3809d040a1d705c385dad75f5481eae1c3168a --- /dev/null +++ b/JdE2T4oBgHgl3EQfUgew/content/tmp_files/2301.03814v1.pdf.txt @@ -0,0 +1,367 @@ +WKB Across Caustics: The Screened-WKB Method +Oscar P. Bruno∗ +Martin D. Maas∗ +Abstract +We present a new methodology, based on the WKB approximation and Fast Fourier Trans- +forms, for the evaluation of wave propagation through inhomogeneous media. This method can +accurately resolve fields containing caustics, while still enjoying the computational advantages +of the WKB approximation, namely, the ability to resolve arbitrarily high-frequency problems in +computing times which are orders-of-magnitude shorter than those required by other algorithms +presently available. For example, the proposed approach can simulate with high accuracy (with +errors such as e.g. 0.1%–0.001%) the propagation of 5 cm radar signals across two-dimensional +configurations resembling atmospheric ducting conditions, spanning hundreds of kilometers and +millions of wavelengths in electrical size, in computing times of a few minutes in a single CPU +core. [Preliminary version] +1 +Introduction +Computations of high-frequency wave propagation through inhomogeneous media play a pivotal +roles in diverse fields such as telecommunications, remote sensing, seismics, quantum mechanics, +and optics. A wide range of methodologies have been developed over the last century for the treat- +ment of high-frequency volumetric-propagation problems. Given that direct numerical simulation +of the configurations of interest, which comprise thousands to millions of wavelengths in acousti- +cal/electrical size, is unfeasible in 2D and even more in 3D, the proposed approaches usually contain +a combination of analytic and numerical approximations. +The celebrated WKB approximation, also known as the Wentzel-Kramers-Brillouin approxima- +tion [2,8], was the first method to obtain accurate solutions to problems involving propagation over +large distances, and is based on the introduction of a system of ray-coordinates, over which the +amplitude and phase of the solution exhibit slow variations. However, the WKB approximation +can break down in certain situations, particularly when the ray mapping becomes singular (i.e. +at caustics). Many approaches have been proposed over time to overcome this limitation, most +notably the KMAH-index theory, according to which a correction can be incorporated after the +caustic, of the form (−i)m, where the constant m depends on the number and type of caustics that +the ray has traversed. This formulation still breaks down at caustics, is inaccurate near caustics, +and, given its complexity, it is seldom used in practice. +Another notable approach to solve these type of problems is provided by the parabolic approxi- +mation introduced in [9], together with its many subsequent versions and improvements, including +the wide-angle approximation [5]. The method of phase-screens [14], in turn, which assumes a con- +stant refractivity profile along each vertical volumetric screen, is applicable for certain restricted +sets of configurations. While, unlike the classical WKB approach, these methods are valid at and +around caustics, their limitations arise from its computational cost. For example, mesh-sizes of +∗Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA +1 +arXiv:2301.03814v1 [physics.comp-ph] 10 Jan 2023 + +the order of ∆z ≈ λ +4 and ∆x ≈ 12.5λ are reported for propagation distances in the order of a +few hundreds of wavelengths (see [7] and references therein). (Here z and x denote the vertical +and range variables, respectively.) The parabolic equation methods are most often based on use +of finite-difference approximations, which gives rise to associated dispersion errors, while Fourier +expansions in the vertical axis are only applicable in the lowest-order parabolic approximations. +The combined effect of large propagation distances, and the presence of dispersion error, and the +requirement of fine spatial discretizations, can lead to extremely large computational cost, under +reasonable error tolerances, for challenging configurations commonly arising in applications. +Other notable approaches include the Gaussian beams formulation, with contributions spanning +from the 60’s, including [1,3,6,13] among many others. This formulation is based on an additional +approximation to WKB, which seeks to obtain the phase in the form of a quadratic polynomial, +whose Hessian matrix is evolved along the ray. This approach eliminates ray-bunching at caustics, +and produces fields which remain bounded. +However, theoretical convergence as k → ∞ has +not been established and is believed to be slow. Moreover, the initial beam representation is a +challenging optimization problem, which leads to errors of a few percent even for propagations +distances of the order of a small number of wavelengths [13]. +An additional approach, known as Dynamic Surface Extension (DSE, see [11,12]), can success- +fully propagate wavefronts in a Cartesian discretization. However, the amplitude computations +present the same limitations as the classical WKB approximation. Finally, the Kinetic Formu- +lation [4] views each ray tracing equation as describing the motion of a ”particle” (e.g. photon, +phonon). This method presents severe computational difficulties, as the initial conditions and so- +lutions are given in terms of Wigner measures, a δ-function that vanishes for incorrect directions p. +The approach put forth in this paper, on the other hand, is based on the WKB approximation, +and overcomes the limitations posed by caustics by resorting to a family of curves (or screens) +on which the total field is decomposed in Fourier modes. Each mode is then propagated for large +electrical distances (i.e. 20,000λ in the example considered in Figure 6) which are also short enough +that the presence of caustics is avoided for each Fourier mode. +2 +The Screened WKB Method +We consider, as a model problem, the scalar Helmholtz equation +∆u(r) + k2ε(r)u(r) = 0 +(1) +u = us + uinc +(2) +lim +r→∞ r +�∂us +∂r − ikus +� += 0, +(3) +whose character at high frequencies presents challenges often found in diverse fields, such high- +frequency electromagnetism, acoustics, seismics and quantum mechanics. The proposed screened- +WKB approach first introduces a family of curves (or screens) Γq, for q = 0, 1, . . . , Ns, as depicted +in Figure 1. The method proceeds by propagating the solution from one screen to the next on the +basis of Fourier expansions on the screens Γq and applications of the classical WKB approach for +each separate Fourier mode. +For conciseness, we consider planar screens of the form Γq = {(xq, z) : z ∈ (za, zb)}. The initial +conditions on Γ0 are user-prescribed, and given by: +u|Γ0 = uinc|Γ0 +(4) +2 + +Figure 1: Schematics underlying the proposed S-WKB method. Here and throughout this paper +flat screens Γq are used, but curved screens could alternatively be utilized, if convenient. +We then represent the incident field on each screen Γq, arising from propagation of the field from +Γq−1 (or given by (4) for q = 0) by exploiting certain expressions of the form +u(x, z) ≈ +N/2−1 +� +n=−N/2 +Aq +n(x, z) exp(ikψq +n(x, z)), +(5) +valid between Γq and Γq+1, together with the WKB approximation. +For simplicity, in our description we consider configurations which may accurately be expressed +in terms of z-periodic functions, of period [za, zb], which, in particular, can be used to treat cases +wherein the solution decays rapidly outside a bounded interval in the z variable. (Other arrange- +ments, including rough top and bottom surfaces and other irregularities, can also be incorporated +in this framework, but are not considered in this paper at any length.) Under such assumptions, +for a given screen Γq, a “vertical” DFT can be used by introducing an equi-spaced grid +{zm : m = −N/2, . . . , N/2 − 1} +(6) +in the interval [za, zb] and performing an FFT—which yields +wq +j = +N/2−1 +� +m=−N/2 +u(xq, zm)e−ijzm. +(7) +Then, re-expressing the field u(xq, z) in terms of an inverse DFT, we obtain +u|Γq ≈ 1 +N +N/2−1 +� +m=−N/2 +wq +jeijzm, +(8) +3 + +2 +I1'T2 +'13Figure 2: Example 1: Ray-tracing leading to a single cusp caustic. +which may be expressed in terms of (5) by requiring that +ψq +n(xq, z) = +� +z + zb − za +2 +� +2nπ +k(zb − za). +(9) +As a second step, each term in the expansion (5) is obtained up to the next screen Γq+1 by +means of the classical WKB expansion (see e.g. [7, chapter 3]), which, in particular, requires the +solution of the Eikonal and Transport equations +(∇ψ)2 = ε(r) +(10) +and +2∇ψ · ∇A + A∆ψ = 0. +(11) +In the present case, the initial conditions for each Fourier mode on Γq are obtained from (9) and +(10): +∂zψn(xq, z) = +2nπ +k(zb − za) +(12) +∂xψn(xq, z) = +� +ε(xq, z) − +� +2nπ +k(zb − za) +�2 +(13) +This procedure yields a finite number of adequately spaced geometrical-optics rays, and corre- +sponding values of ψn and An along the rays for the n-th mode. By adequately selecting the spacing +of the screens Γq it can ensured that all the modes − N +2 ≤ n ≤ N +2 − 1 propagate to the next screen +4 + +Figure 3: S-WKB solution (top), and physically-exact separation-of-variables solution with super- +imposed geometrical-optics rays (bottom), with k = 125, along a propagation domain 40 km +(800, 000 wavelengths) in range. +5 + +Figure 4: Error for the “single-caustic” solution displayed in Figure (3): a relative error of the +order of 10−5 was obtained throughout the propagation domain. +Γq+1 without incurring caustics. Interpolation can then be used on Γq+1 to obtain approximate +values of u on the 1D Cartesian grid (xq+1, zm) (− N +2 ≤ n ≤ N +2 − 1) on Γq+1: +u(xq+1, zm) ≈ +N/2−1 +� +n=−N/2 +Aq +n(xq+1, zm) exp(ikψq +n(zm)). +(14) +Expanding u(xq+1, z) in a Fourier series along Γq+1 the next iteration of the algorithm can then +be initiated. Repeating this procedure for all screens the field u over the domain of interest can be +obtained. +3 +Numerical Results +This section presents results of applications of the proposed algorithm to problems of wave propa- +gation through inhomogeneous media, in two-dimensional configurations, and through wide ranges +of problem parameters. In order to evaluate the accuracy of the proposed S-WKB method by +comparisons with solutions obtainable by means of separation of variables, we first consider x- +invariant permittivities (i.e. permittivities of the form ε(x, z) = ε(z)), as described in what follows, +for which a simple high-order spectral solver can be used to obtained reference solutions that are +physically-exact—i.e., which contain no approximations to (1) other than those inherent in the well +established numerical solver utilized. +6 + +Figure 5: Geometrical optics rays for a “ducting” configuration. +3.1 +High-order reference solutions for x-invariant permittivities +In order to obtain a valid solution to (1) via separation of variables we seek a solution of the form +u(x, z) = +∞ +� +i=0 +aieiαixφi(z). +(15) +Substituting (15) in (1) leads to +∞ +� +i=0 +(−α2 + φ′′ +i (z) + k2ε(z))aieiαix = 0. +(16) +Using the orthogonality of the complex exponentials we then obtain +(−α2 +i + φ′′ +i (z) + k2ε(z))ai = 0. +(17) +It follows that the non-zero coefficients αi in (15) satisfy the Sturm-Liouville problem: +φ′′ +i (z) + k2ε(z)φi(z) = α2 +i φi(z) +(18) +with radiation boundary conditions: +lim +z→±∞ +� +zφ′ +i − ikφi +� += 0, +(19) +Numerically, the radiation boundary conditions can be imposed by considering a sufficiently +large interval (za, zb) and imposing either Dirichlet or periodic boundary conditions at such points. +The resulting Sturm-Liouville problem can be discretized with high-order spectral methods. For +the purposes of the present paper we utilized the spectral eigensolver [10], which is available in the +ApproxFun.jl Julia package. +3.2 +Evaluation of the S-WKB accuracy for a Gaussian permittivity model +In this section we consider the exponential permittivity model +ε(z) = 1 + ae−bz2 +(20) +7 + +Figure 6: “Multiple caustics” test case depicting an idealized “ducting” configuration. S-WKB field +values (top), and field values with super-imposed geometrical-optics rays (bottom). The geometrical +optics rays are depicted in Figure 5. +8 + +whose physically-exact solution can be obtained by relying on the method described in Section 3.1. +For our example, an incident field given by a Gaussian beam +uinc(x, z) = +� ∞ +−∞ +ei√ +k2−β2x+iβze− β2 +σ2 dβ +(21) +is utilized, wherein the integral in the variable β is evaluated via standard numerical integration +techniques. +In our first example we consider the permittivity model (20) with parameters a = 10−4 and +b = 10−4—which, at C-band, results in a configuration that gives rise to a single caustic of cusp +type for the first 40km (800, 000 wavelengths) in horizontal propagation range. The geometrical +optics rays are displayed in Fig. 2. The S-WKB solution alongside the Sturm-Liouville solution +with superimposed ray-tracing are depicted in Fig. 3. As shown in Fig. 4, the relative errors for +this configuration are of the order of 10−5. Employing N = 400 Fourier modes and a total of 40 +screens, the S-WKB solution in this case was obtained in a computing times of 2 minutes in a +single-core, whereas the separation-of-variables solution required single-core runs of approximately +1.5 hours. +Figure 7: Smooth convex lens simulations produced by the S-WKB method. Note the fine-scale +fields behind caustics, whose simulation is otherwise quite challenging; cf. e.g. [7, Fig. 6.10]. +For our next example we consider a “ducting” configuration, in which the incident Gaussian +beam is tilted by an angle of 0.2◦, and where the Gaussian permittivity (20) was used with param- +eters a = 10−4 and b = 10−3—in such a way that the energy is contained within a bounded interval +along the z axis. The geometrical-optics rays form a complex system with multiple caustics, as +9 + +Figure 8: Permittivity and geometrical-optics rays (left), and rays superimposed on the field de- +picted in Figure 7. +depicted in Fig. 5. We consider the propagation of this signal over a range of 200Km (4 million +wavelengths) in range. A total of n = 800 Fourier modes and 200 of the order of 0.1%. +4 +Smooth Lens +We now consider the test case of a smooth convex lens proposed in [4] on the basis of the permittivity +function given by +ε(r) = +� +1 +d2 > L +� +a +b−cos(πd2) +�2 +d2 ≤ L +d2 = +�x − xc +xd +�2 ++ +�z − zc +zd +�2 +(22) +For the numerical example depicted in Fig. 7, we have set L = 1, a = 4, b = 3, xc = 0.5, xd = +0.2, zc = 0, zd = 0.8. The displayed result compares favorably to that presented in [4] on the basis +of a kinetic formulation, as well as the similar problem demonstrated in [7, Fig. 6.10]. A separation +of variables solution is not available in this case, and the error could be evaluated by means of +S-WKB implementation of higher order. In presence of the previous examples, however, we may +estimate the error in the range of 0.1% to 0.001%. +References +[1] Vasilii M Babich and Vladimir Sergeevich Buldyrev. +Short-wavelength diffraction theory: +asymptotic methods. Springer, 1991. +[2] Max Born and Emil Wolf. Principles of optics: electromagnetic theory of propagation, inter- +ference and diffraction of light. Elsevier, 2013. +[3] Vlastislav ˇCerven`y, Mikhail M Popov, and Ivan Pˇsenˇc´ık. Computation of wave fields in inhomo- +geneous media—gaussian beam approach. Geophysical Journal International, 70(1):109–128, +1982. +[4] Bj¨orn Engquist and Olof Runborg. Computational high frequency wave propagation. Acta +numerica, 12:181–266, 2003. +10 + +[5] RH Hardin. Applications of the split-step fourier method to the numerical solution of nonlinear +and variable coefficient wave equations. SIAM Review (Chronicles), 15, 1973. +[6] Lars H¨ormander. Linear partial differential operators. Springer, 1963. +[7] Finn B Jensen, William A Kuperman, Michael B Porter, Henrik Schmidt, and Alexandra +Tolstoy. Computational ocean acoustics, volume 794. Springer, 2011. +[8] Joseph B Keller. Geometrical theory of diffraction. Josa, 52(2):116–130, 1962. +[9] Mikhail Aleksandrovich Leontovich and Vladimir Aleksandrovich Fock. Solution of the problem +of propagation of electromagnetic waves along the earth’s surface by the method of parabolic +equation. J. Phys. Ussr, 10(1):13–23, 1946. +[10] Sheehan Olver and Alex Townsend. A fast and well-conditioned spectral method. siam RE- +VIEW, 55(3):462–489, 2013. +[11] Steven J Ruuth, Barry Merriman, and Stanley Osher. A fixed grid method for capturing the +motion of self-intersecting wavefronts and related pdes. Journal of Computational Physics, +163(1):1–21, 2000. +[12] John Steinhoff, Meng Fan, and Lesong Wang. A new eulerian method for the computation +of propagating short acoustic and electromagnetic pulses. Journal of Computational Physics, +157(2):683–706, 2000. +[13] Nicolay M Tanushev, Bj¨orn Engquist, and Richard Tsai. Gaussian beam decomposition of +high frequency wave fields. Journal of Computational Physics, 228(23):8856–8871, 2009. +[14] Ru-Shan Wu. +Wide-angle elastic wave one-way propagation in heterogeneous media and +an elastic wave complex-screen method. +Journal of Geophysical Research: +Solid Earth, +99(B1):751–766, 1994. +11 + diff --git a/JdE2T4oBgHgl3EQfUgew/content/tmp_files/load_file.txt b/JdE2T4oBgHgl3EQfUgew/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5d9486b1b8b09d31a3b27eb86a4f8d6f6e8b405 --- /dev/null +++ b/JdE2T4oBgHgl3EQfUgew/content/tmp_files/load_file.txt @@ -0,0 +1,176 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf,len=175 +page_content='WKB Across Caustics: The Screened-WKB Method Oscar P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Bruno∗ Martin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Maas∗ Abstract We present a new methodology, based on the WKB approximation and Fast Fourier Trans- forms, for the evaluation of wave propagation through inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' This method can accurately resolve fields containing caustics, while still enjoying the computational advantages of the WKB approximation, namely, the ability to resolve arbitrarily high-frequency problems in computing times which are orders-of-magnitude shorter than those required by other algorithms presently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For example, the proposed approach can simulate with high accuracy (with errors such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='1%–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='001%) the propagation of 5 cm radar signals across two-dimensional configurations resembling atmospheric ducting conditions, spanning hundreds of kilometers and millions of wavelengths in electrical size, in computing times of a few minutes in a single CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' [Preliminary version] 1 Introduction Computations of high-frequency wave propagation through inhomogeneous media play a pivotal roles in diverse fields such as telecommunications, remote sensing, seismics, quantum mechanics, and optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' A wide range of methodologies have been developed over the last century for the treat- ment of high-frequency volumetric-propagation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Given that direct numerical simulation of the configurations of interest, which comprise thousands to millions of wavelengths in acousti- cal/electrical size, is unfeasible in 2D and even more in 3D, the proposed approaches usually contain a combination of analytic and numerical approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The celebrated WKB approximation, also known as the Wentzel-Kramers-Brillouin approxima- tion [2,8], was the first method to obtain accurate solutions to problems involving propagation over large distances, and is based on the introduction of a system of ray-coordinates, over which the amplitude and phase of the solution exhibit slow variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' However, the WKB approximation can break down in certain situations, particularly when the ray mapping becomes singular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' at caustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Many approaches have been proposed over time to overcome this limitation, most notably the KMAH-index theory, according to which a correction can be incorporated after the caustic, of the form (−i)m, where the constant m depends on the number and type of caustics that the ray has traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' This formulation still breaks down at caustics, is inaccurate near caustics, and, given its complexity, it is seldom used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Another notable approach to solve these type of problems is provided by the parabolic approxi- mation introduced in [9], together with its many subsequent versions and improvements, including the wide-angle approximation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The method of phase-screens [14], in turn, which assumes a con- stant refractivity profile along each vertical volumetric screen, is applicable for certain restricted sets of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' While, unlike the classical WKB approach, these methods are valid at and around caustics, their limitations arise from its computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For example, mesh-sizes of ∗Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='03814v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='comp-ph] 10 Jan 2023 the order of ∆z ≈ λ 4 and ∆x ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='5λ are reported for propagation distances in the order of a few hundreds of wavelengths (see [7] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (Here z and x denote the vertical and range variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=') The parabolic equation methods are most often based on use of finite-difference approximations, which gives rise to associated dispersion errors, while Fourier expansions in the vertical axis are only applicable in the lowest-order parabolic approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The combined effect of large propagation distances, and the presence of dispersion error, and the requirement of fine spatial discretizations, can lead to extremely large computational cost, under reasonable error tolerances, for challenging configurations commonly arising in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Other notable approaches include the Gaussian beams formulation, with contributions spanning from the 60’s, including [1,3,6,13] among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' This formulation is based on an additional approximation to WKB, which seeks to obtain the phase in the form of a quadratic polynomial, whose Hessian matrix is evolved along the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' This approach eliminates ray-bunching at caustics, and produces fields which remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' However, theoretical convergence as k → ∞ has not been established and is believed to be slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Moreover, the initial beam representation is a challenging optimization problem, which leads to errors of a few percent even for propagations distances of the order of a small number of wavelengths [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' An additional approach, known as Dynamic Surface Extension (DSE, see [11,12]), can success- fully propagate wavefronts in a Cartesian discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' However, the amplitude computations present the same limitations as the classical WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Finally, the Kinetic Formu- lation [4] views each ray tracing equation as describing the motion of a ”particle” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' photon, phonon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' This method presents severe computational difficulties, as the initial conditions and so- lutions are given in terms of Wigner measures, a δ-function that vanishes for incorrect directions p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The approach put forth in this paper, on the other hand, is based on the WKB approximation, and overcomes the limitations posed by caustics by resorting to a family of curves (or screens) on which the total field is decomposed in Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Each mode is then propagated for large electrical distances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 20,000λ in the example considered in Figure 6) which are also short enough that the presence of caustics is avoided for each Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 2 The Screened WKB Method We consider, as a model problem, the scalar Helmholtz equation ∆u(r) + k2ε(r)u(r) = 0 (1) u = us + uinc (2) lim r→∞ r �∂us ∂r − ikus � = 0, (3) whose character at high frequencies presents challenges often found in diverse fields, such high- frequency electromagnetism, acoustics, seismics and quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The proposed screened- WKB approach first introduces a family of curves (or screens) Γq, for q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' , Ns, as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The method proceeds by propagating the solution from one screen to the next on the basis of Fourier expansions on the screens Γq and applications of the classical WKB approach for each separate Fourier mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For conciseness, we consider planar screens of the form Γq = {(xq, z) : z ∈ (za, zb)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The initial conditions on Γ0 are user-prescribed, and given by: u|Γ0 = uinc|Γ0 (4) 2 Figure 1: Schematics underlying the proposed S-WKB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Here and throughout this paper flat screens Γq are used, but curved screens could alternatively be utilized, if convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' We then represent the incident field on each screen Γq, arising from propagation of the field from Γq−1 (or given by (4) for q = 0) by exploiting certain expressions of the form u(x, z) ≈ N/2−1 � n=−N/2 Aq n(x, z) exp(ikψq n(x, z)), (5) valid between Γq and Γq+1, together with the WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For simplicity, in our description we consider configurations which may accurately be expressed in terms of z-periodic functions, of period [za, zb], which, in particular, can be used to treat cases wherein the solution decays rapidly outside a bounded interval in the z variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (Other arrange- ments, including rough top and bottom surfaces and other irregularities, can also be incorporated in this framework, but are not considered in this paper at any length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=') Under such assumptions, for a given screen Γq, a “vertical” DFT can be used by introducing an equi-spaced grid {zm : m = −N/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' , N/2 − 1} (6) in the interval [za, zb] and performing an FFT—which yields wq j = N/2−1 � m=−N/2 u(xq, zm)e−ijzm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=" (7) Then, re-expressing the field u(xq, z) in terms of an inverse DFT, we obtain u|Γq ≈ 1 N N/2−1 � m=−N/2 wq jeijzm, (8) 3 2 I1'T2 '13Figure 2: Example 1: Ray-tracing leading to a single cusp caustic." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' which may be expressed in terms of (5) by requiring that ψq n(xq, z) = � z + zb − za 2 � 2nπ k(zb − za).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (9) As a second step, each term in the expansion (5) is obtained up to the next screen Γq+1 by means of the classical WKB expansion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' [7, chapter 3]), which, in particular, requires the solution of the Eikonal and Transport equations (∇ψ)2 = ε(r) (10) and 2∇ψ · ∇A + A∆ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (11) In the present case, the initial conditions for each Fourier mode on Γq are obtained from (9) and (10): ∂zψn(xq, z) = 2nπ k(zb − za) (12) ∂xψn(xq, z) = � ε(xq, z) − � 2nπ k(zb − za) �2 (13) This procedure yields a finite number of adequately spaced geometrical-optics rays, and corre- sponding values of ψn and An along the rays for the n-th mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' By adequately selecting the spacing of the screens Γq it can ensured that all the modes − N 2 ≤ n ≤ N 2 − 1 propagate to the next screen 4 Figure 3: S-WKB solution (top), and physically-exact separation-of-variables solution with super- imposed geometrical-optics rays (bottom), with k = 125, along a propagation domain 40 km (800, 000 wavelengths) in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 5 Figure 4: Error for the “single-caustic” solution displayed in Figure (3): a relative error of the order of 10−5 was obtained throughout the propagation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Γq+1 without incurring caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Interpolation can then be used on Γq+1 to obtain approximate values of u on the 1D Cartesian grid (xq+1, zm) (− N 2 ≤ n ≤ N 2 − 1) on Γq+1: u(xq+1, zm) ≈ N/2−1 � n=−N/2 Aq n(xq+1, zm) exp(ikψq n(zm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (14) Expanding u(xq+1, z) in a Fourier series along Γq+1 the next iteration of the algorithm can then be initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Repeating this procedure for all screens the field u over the domain of interest can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 3 Numerical Results This section presents results of applications of the proposed algorithm to problems of wave propa- gation through inhomogeneous media, in two-dimensional configurations, and through wide ranges of problem parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' In order to evaluate the accuracy of the proposed S-WKB method by comparisons with solutions obtainable by means of separation of variables, we first consider x- invariant permittivities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' permittivities of the form ε(x, z) = ε(z)), as described in what follows, for which a simple high-order spectral solver can be used to obtained reference solutions that are physically-exact—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=', which contain no approximations to (1) other than those inherent in the well established numerical solver utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 6 Figure 5: Geometrical optics rays for a “ducting” configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='1 High-order reference solutions for x-invariant permittivities In order to obtain a valid solution to (1) via separation of variables we seek a solution of the form u(x, z) = ∞ � i=0 aieiαixφi(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (15) Substituting (15) in (1) leads to ∞ � i=0 (−α2 + φ′′ i (z) + k2ε(z))aieiαix = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (16) Using the orthogonality of the complex exponentials we then obtain (−α2 i + φ′′ i (z) + k2ε(z))ai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' (17) It follows that the non-zero coefficients αi in (15) satisfy the Sturm-Liouville problem: φ′′ i (z) + k2ε(z)φi(z) = α2 i φi(z) (18) with radiation boundary conditions: lim z→±∞ � zφ′ i − ikφi � = 0, (19) Numerically, the radiation boundary conditions can be imposed by considering a sufficiently large interval (za, zb) and imposing either Dirichlet or periodic boundary conditions at such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The resulting Sturm-Liouville problem can be discretized with high-order spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For the purposes of the present paper we utilized the spectral eigensolver [10], which is available in the ApproxFun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='jl Julia package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='2 Evaluation of the S-WKB accuracy for a Gaussian permittivity model In this section we consider the exponential permittivity model ε(z) = 1 + ae−bz2 (20) 7 Figure 6: “Multiple caustics” test case depicting an idealized “ducting” configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' S-WKB field values (top), and field values with super-imposed geometrical-optics rays (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The geometrical optics rays are depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 8 whose physically-exact solution can be obtained by relying on the method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For our example, an incident field given by a Gaussian beam uinc(x, z) = � ∞ −∞ ei√ k2−β2x+iβze− β2 σ2 dβ (21) is utilized, wherein the integral in the variable β is evaluated via standard numerical integration techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' In our first example we consider the permittivity model (20) with parameters a = 10−4 and b = 10−4—which, at C-band, results in a configuration that gives rise to a single caustic of cusp type for the first 40km (800, 000 wavelengths) in horizontal propagation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The geometrical optics rays are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The S-WKB solution alongside the Sturm-Liouville solution with superimposed ray-tracing are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 4, the relative errors for this configuration are of the order of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Employing N = 400 Fourier modes and a total of 40 screens, the S-WKB solution in this case was obtained in a computing times of 2 minutes in a single-core, whereas the separation-of-variables solution required single-core runs of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Figure 7: Smooth convex lens simulations produced by the S-WKB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' Note the fine-scale fields behind caustics, whose simulation is otherwise quite challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' [7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' For our next example we consider a “ducting” configuration, in which the incident Gaussian beam is tilted by an angle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='2◦, and where the Gaussian permittivity (20) was used with param- eters a = 10−4 and b = 10−3—in such a way that the energy is contained within a bounded interval along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The geometrical-optics rays form a complex system with multiple caustics, as 9 Figure 8: Permittivity and geometrical-optics rays (left), and rays superimposed on the field de- picted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' We consider the propagation of this signal over a range of 200Km (4 million wavelengths) in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' A total of n = 800 Fourier modes and 200 of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 4 Smooth Lens We now consider the test case of a smooth convex lens proposed in [4] on the basis of the permittivity function given by ε(r) = � 1 d2 > L � a b−cos(πd2) �2 d2 ≤ L d2 = �x − xc xd �2 + �z − zc zd �2 (22) For the numerical example depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 7, we have set L = 1, a = 4, b = 3, xc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='5, xd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='2, zc = 0, zd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' The displayed result compares favorably to that presented in [4] on the basis of a kinetic formulation, as well as the similar problem demonstrated in [7, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE2T4oBgHgl3EQfUgew/content/2301.03814v1.pdf'} +page_content=' A separation of variables solution is not available in this case, and the error could be evaluated by means of S-WKB implementation of higher order.' 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state-of-the-art accuracy, when their +assumptions are satisfied concerning the noise dis- +tributions. However, under misspecification their +accuracy deteriorates sharply, especially when the +conditional variance in the anti-causal direction is +smaller than that in the causal direction. In this pa- +per, we demonstrate the misspecification problem +and analyze why and when it occurs. We show +that residual independence testing is much more +robust to misspecification than likelihood-based +cause-effect inference. Our empirical evaluation +includes 580 synthetic and 99 real-world datasets. +1. Introduction +Distinguishing cause and effect is a fundamental problem +in many disciplines, such as biology, healthcare and fi- +nance (Zhang & Chan, 2006; Huang, 2021; Mansouri et al., +2022). Randomized controlled trials (RCTs) are the gold +standard for finding causal relationships. However, it may be +unethical, expensive or even impossible to perform RCTs in +real-world domains (Peters et al., 2017; Pearl & Mackenzie, +2018). Causal discovery algorithms aim to find causal rela- +tionships given observational data alone. Traditional causal +discovery algorithms can only identify causal relationships +up to Markov equivalence classes (MECs) (Spirtes & Gly- +mour, 1991; Kalisch & B¨uhlman, 2007; Colombo et al., +2012). To break the symmetry in a MEC, additional as- +sumptions are needed (Peters et al., 2017; Sch¨olkopf, 2022), +such as the type of functional dependence of effect on cause. +Structural causal models (SCMs) specify a functional class +for the causal relations in the data (Pearl, 2009; Peters et al., +2017). In this work, we focus on one particular type of SCM +1Simon Fraser University, Burnaby, Canada. Correspondence +to: Xiangyu Sun . +Preprint. +called location-scale noise models (LSNMs) (Tagasovska +et al., 2020; Khemakhem et al., 2021; Xu et al., 2022; Strobl +& Lasko, 2022; Immer et al., 2022): +Y := f(X) + g(X) · ZY +(1) +where X is the cause, Y is the effect, written X → Y , and +ZY is a latent noise variable independent of X (i.e., X ⊥ +⊥ ZY ). The functions f and g are twice-differentiable on +the domain of X, and g is strictly positive on the domain of +X. LSNMs are closely related to affine flow models, where +g(X) = exp s(X) (Khemakhem et al., 2021). LSNMs +generalize the widely studied additive noise models (ANMs, +where g(x) = 1 for all x) and allow heteroscedasticity +where the variance of Y conditional on X, (i.e. V[Y |X]) +depends on the value of X. +Two major approaches for cause-effect inference in SCMs +are maximum likelihood (ML) and independence testing +(IT) of residuals vs. the (putative) cause (Mooij et al., 2016). +Both have been recently evaluated for LSNMs (Khemakhem +et al., 2021; Immer et al., 2022), with good accuracy, espe- +cially when the f and g functions are estimated by neural +networks. Immer et al. note, however, that ML can be less +robust than IT, in the sense that accuracy deteriorates when +the noise distribution is not Gaussian. +In this paper we investigate the robustness of ML vs. IT. +Our analysis shows that ML cause-effect inference performs +poorly when two factors coincide: (1) Noise Misspecifica- +tion: the ML method assumes a different noise distribution +from the true one. (2) Misleading Conditional Variances +(CVs): V[Y |X] > V[X|Y ] in the data generated by causal +direction X → Y . For example, in an experiment on syn- +thetic datasets shown in Table 1 below, (i) changing the true +noise distribution from Gaussian to uniform and (ii) manip- +ulating the CV of effect Y given cause X (i.e. V[Y |X]), +while keeping other settings equal, can decrease the rate of +identifying the true causal direction from 100% to 10%. In +contrast, IT approaches maintain a perfect 100% accuracy. +Both conditions (1) and (2) often hold in practice. For real- +world domains, assumptions about the noise distribution +can be hard to determine or verify. It is also common to +have misleading CVs in real-world datasets. For example, +in the T¨ubingen Cause-Effect Pairs Benchmark (Mooij et al., +arXiv:2301.12930v1 [cs.LG] 26 Jan 2023 + +Cause-Effect Inference in Location-Scale Noise Models +2016), about 40% of the real-world datasets exhibit a mis- +leading CV (see Appendix Table 4). +We make the following contributions: +• Describe experiments and theoretical analysis to show +that ML methods succeed when the noise distribution +is known. +• Demonstrate empirically that ML methods often fail +when the noise distribution is misspecified and CV is +misleading. +• Introduce a new IT method based on an affine flow +model. +• Demonstrate empirically that our IT method is robust +to noise misspecification and misleading CVs. +The paper is structured as follows. We discuss related works +and preliminaries in Sections 2 and 3, respectively. Sec- +tion 4 examines when ML methods succeed and when they +fail. Section 5 demonstrates the robustness of the IT ap- +proach. Evaluations on synthetic and real-world datasets are +given in Section 6. The code and scripts to reproduce all the +results can be found online 1. +2. Related Works +Causal Discovery. Causal discovery methods have been +well studied in machine learning (Spirtes & Glymour, 1991; +Kalisch & B¨uhlman, 2007; Colombo et al., 2012). Assum- +ing causal sufficiency (no latent confounders) and faithful- +ness, they find causal structure up to a MEC. To identify +causal relations within a MEC, additional assumptions are +needed (Peters et al., 2017; Sch¨olkopf, 2022). SCMs exploit +constraints that result from assumptions about the functional +dependency of effects on causes. Functional dependencies +are often studied in the fundamental case of two variables +X and Y , the simplest MEC (Mooij et al., 2016). We follow +this line of work and study two-variable LSNMs assuming +causal sufficiency. +Structural Causal Models. Assuming a linear non-Gaussian +acyclic model (LINGAM) (Shimizu et al., 2006), the causal +direction was proved to be identifiable. The key idea in +LINGAM is that in the true causal direction X → Y , the +model residuals are independent of X. We refer to methods +derived from the LINGAM approach as independence test- +ing (IT) methods. The more general additive noise model +(ANM) (Hoyer et al., 2008) allows for nonlinear cause- +effect relationships and is generally identifiable, except for +some special cases. There are other identifiable SCMs, such +as post-nonlinear models (Zhang & Hyvarinen, 2012) and +Poisson generalized linear models (Park & Park, 2019). +1https://github.com/xiangyu-sun-789/CAREFL-H +LSNM identifiability. There are several identifiability re- +sults for the causal direction in LSNMs. Xu et al. (2022) +prove LSNMs are identifiable for linear causal dependen- +cies. Khemakhem et al. (2021) show that nonlinear LSNMs +are identifiable with Gaussian noise. Strobl & Lasko (2022); +Immer et al. (2022) prove LSNMs are identifiable except in +some pathological cases (cf. Section 4.1). +Cause-Effect Inference in LSNMs. The blueprint is to fit two +LSNM models for each direction X → Y and X ← Y and +select the direction with a higher model score. HECI (Xu +et al., 2022) bins putative cause values and selects a direc- +tion based on a Bayesian information criterion based score. +BQCD (Tagasovska et al., 2020) uses nonparamatric quan- +tile regression to approximate minimum description length +to select the direction. GRCI (Strobl & Lasko, 2022) finds +the direction based on mutual information between the pu- +tative cause and the model residuals. LOCI (Immer et al., +2022) models the conditional distribution of effect given +cause with Gaussian natural parameters. Then, it chooses +the direction based on either likelihood (LOCI-M) or in- +dependence (LOCI-H). CAREFL-M (Khemakhem et al., +2021) fits affine flow models and scores them by likelihood. +CAREFL-M is more general than LOCI since LOCI uses a +fixed Gaussian noise distribution prior, whereas CAREFL- +M can utilize different prior distributions. We introduce +CAREFL-H as a new IT method that scores the fitted affine +flow models based on independence. DECI (Geffner et al., +2022) generalizes CAREFL-M to multivariate cases but is +designed for ANMs. To our knowledge, the problem of mis- +leading CVs is neither identified nor analyzed in previous +work. +3. Preliminaries +In this section, we define the cause-effect inference problem +and review the LSNM data likelihood. +3.1. Problem Definition: Cause-Effect Inference +Cause-effect inference takes as input a dataset D over two +random variables X, Y with N observational data pairs +(X, Y ) = {(x1, y1), (x2, y2), . . . , (xN, yN)} generated by +a ground-truth LSNM in Equation (1). The binary output +decision indicates whether X causes Y (X → Y ) or Y +causes X (X ← Y ). We assume no latent confounders, +selection bias or feedback cycles. +3.2. Definition of Maximum Likelihood Approach for +LSNMs +An LSNM model for two variables (X, Y ) is a pair (→, PZ). +The → represents the direction X → Y with parameter +space f ≡ fθ, g ≡ gψ, PX ≡ PX,ζ. The ← represents +the direction X ← Y with parameter space h ≡ hθ′, k ≡ + +Cause-Effect Inference in Location-Scale Noise Models +kψ′, PY ≡ PY,ζ′. For notational simplicity, we treat the +model functions directly as model parameters and omit ref- +erence to their parameterizations. For example, we write +f ≡ fθ for a function f that is implemented by a neural +network with weights θ. We refer to PZ as the model prior +noise distribution. +A parameterized LSNM model defines a data distribution as +follows (Immer et al., 2022) (derivation in Appendix A): +P→,PZ(X, Y ; f, g, PX) +=PX(X) · PZY (Y − f(X) +g(X) +) · +1 +g(X) +P←,PZ(X, Y ; h, k, PY ) +=PY (Y ) · PZX(X − h(Y ) +k(Y ) +) · +1 +k(Y ) +(2) +For conciseness, we use shorter notation P→,PZ(X, Y ) and +P←,PZ(X, Y ) in later sections. The likelihood of a dataset +D for a parametrized → model is given by +P→,PZ(D; f, g, PX) = +N +� +i=1 +P→,PZ(xi, yi; f, g, PX) +The ML approach estimates the ML parameters and utilizes +them to score the → model (similarly for the ← model ): +�f, �g, �PX := arg max +f,g,PX P→,PZ(D; f, g, PX) +(3) +L→,PZ(D) := P→,PZ(D; �f, �g, � +PX) +(4) +4. Analysis of Maximum Likelihood +Approach +In this section, we state existing and provide new identi- +fiability results for LSNMs. We use theoretical analysis +to understand why misspecified ML methods fail in the +presence of misleading CVs (cf. Table 1). +4.1. Identifiability of LSNMs with Correct Noise +Distribution +Strobl & Lasko (2022); Immer et al. (2022) prove the iden- +tifiability of LSNMs, assuming a correctly specified noise +distribution. That is, given that the data generating distribu- +tion (X, Y ) follows a LSNM in the direction X → Y , the +same distribution with equal likelihood cannot be induced +by a LSNM in the backward direction X ← Y , except in +some pathological cases. In terms of our notation, direction +identifiability means that if (→, PZ) is the data generating +model, then +P([L→,PZ(D) − L←,PZ(D)] > 0) → 1 as N → ∞ +Immer et al. prove the following identifiability result: +Theorem 4.1 (Theorem 1 from (Immer et al., 2022)). For +data (X, Y ) that follows a LSNM in both direction X → Y +and X ← Y , i.e., +Y = f(X) + g(X) · ZY , where X ⊥⊥ ZY +X = h(Y ) + k(Y ) · ZX, where Y ⊥⊥ ZX +The following condition must be true: +(log p(y))′′ + +g′(x) +G(x, y) · (log p(y))′ ++ ∂2 +∂y2 · νX|Y (x|y) + +g(x) +G(x, y) · +∂2 +∂y∂x · νX|Y (x|y) ++ +g′(x) +G(x, y) · ∂ +∂y · νX|Y (x|y) = 0 +(5) +where G(x, y) = g(x) · f ′(x) + g′(x) · [y − f(x)] ̸= 0 and +νX|Y (x|y) = log pZX( x−h(y) +k(y) ) − log k(y). +They state that Equation (5) will be false except for “patho- +logical cases”. In addition, Khemakhem et al. (2021) pro- +vide sufficient conditions for LSNMs with Gaussian noise +to be identifiable. Our next theorem provides identifiability +results for some non-Gaussian noise distributions: +Theorem 4.2. Suppose that the true data-generating dis- +tribution follows an LSNM model in both X → Y and +X ← Y directions: +1. If the noise distribution is Uniform(a, b), then both +g(X) and k(Y ) are constant functions. +2. If the noise distribution is ContinuousBernoulli(λ ̸= +0.5)2 or Exponential(λ), then one of the following +conditions holds: +• g(X)−1 and k(Y )−1 are constant functions. +• g(X)−1 and k(Y )−1 are linear functions with the +same coefficients on X and Y , respectively. +The proof is in Appendix B. Essentially, the theorem shows +that for Uniform, Exponential, and ContinuousBernoulli +noise distributions, the true LSNM model can be identified +unless it degenerates to (i) a homoscedastic additive noise +model or (ii) a heteroscedastic model with the same linear +scale in both directions. +4.2. Non-Identifiability of LSNMs with Misspecified +Noise Distribution +Existing identifiability results for ML methods require know- +ing the ground-truth noise distribution. We find that when +2The case of ContinuousBernoulli(λ = 0.5) is equivalent +to Uniform(0, 1)). + +Cause-Effect Inference in Location-Scale Noise Models +Table 1: Accuracy over 10 datasets generated by SCM LSNM-sine-tanh (definition in Appendix G) with N = 10, 000 +samples. The task is a binary decision whether X causes Y or Y causes X. Rewrite Equation (1) as Y = f(X)+α·g(X)·ZY , +where α is a scale factor to alter the CV. CVs are computed by binning the putative cause. X denotes the ground-truth cause +and Y denotes the ground-truth effect. We used Gaussian(0, 1) as model noise prior for both CAREFL and LOCI. The +suffix -M denotes a ML method. The suffix -H denotes the corresponding IT method (more in Section 5). +True Noise +Gaussian(0, 1) +Uniform(−1, 1) +α +0.1 +0.5 +1 +5 +10 +0.1 +0.5 +1 +5 +10 +V[Y |X] +vs. +V[X|Y ] +0.166 +vs. +0.455 +0.615 +vs. +0.709 +0.834 +vs. +0.793 +0.990 +vs. +0.821 +0.997 +vs. +0.817 +0.044 +vs. +0.047 +0.404 +vs. +0.375 +0.673 +vs. +0.566 +0.975 +vs. +0.681 +0.994 +vs. +0.677 +CAREFL-M +1.0 +1.0 +1.0 +1.0 +1.0 +0.7 +0.6 +0.7 +0.1 +0.1 +LOCI-M +1.0 +1.0 +1.0 +1.0 +1.0 +0.7 +0.6 +0.7 +0.1 +0.1 +CAREFL-H +1.0 +1.0 +1.0 +1.0 +1.0 +0.9 +1.0 +1.0 +1.0 +1.0 +LOCI-H +1.0 +1.0 +1.0 +1.0 +1.0 +0.7 +1.0 +1.0 +1.0 +1.0 +the noise distribution is misspecified for both causal and +anti-causal models, the anti-causal model may achieve a +higher data likelihood, even in the sample size limit. In +other words, ML model selection is not consistent when +the noise distribution is misspecified. In terms of our nota- +tion, if (→, PZ) is the data generating model, but the noise +distribution is misspecified as P ′ +Z, then +P([L→,P ′ +Z(D) − L←,P ′ +Z(D)] > 0) ̸→ 1 as N → ∞ +We conducted a simple experiment to show that with a +misspecified noise distribution, ML model selection can +fail badly. Table 1 shows results for CAREFL-M (Khe- +makhem et al., 2021) and LOCI-M (Immer et al., 2022) with +model prior distribution Gaussian(0, 1). For the left half +of the table, the data was generated with Gaussian(0, 1) +noise, matching the model specification. In this case, both +CAREFL-M and LOCI-M work quite well, and increasing +CV in the causal direction does not affect their accuracy. +For the right half of the table, the data was generated with +Uniform(−1, 1) noise, contradicting the model specifica- +tion. With the misspecified noise, the accuracy of both +CAREFL-M and LOCI-M decreases to 70%. With both +misspecified noise and misleading CVs, their accuracy be- +comes even lower. For example, in the last column when +V[Y |X] = 0.994 and V[X|Y ] = 0.677, both CAREFL-M +and LOCI-M give an accuracy of just 10%. Thus they select +the incorrect anti-causal direction 9 out of 10 times. +To understand why higher CV corresponds to lower data +likelihood, note that the model Equation (2) entails the +following relationships for the X → Y model: +V [Y |X] = g2(X) · V [ZY ] +P→,PZ(X, Y ) = PX(X) · PZY (ZY ) · +1 +g(X) +where ZY = Y −f(X) +g(X) . Therefore we can expect V [Y |X] +and P→,PZ(X, Y ) to be negatively related: +1. Increasing V [Y |X] by enlarging g(X) reduces +1 +g(X), +which in turn reduces P→,PZ(X, Y ). +2. High variance typically means small densities. There- +fore, increasing V [Y |X] by enlarging V [ZY ] often re- +duces PZY (ZY ), which in turn reduces P→,PZ(X, Y ). +Figure 1 illustrates these relationships in actual datasets. +1. CV and likelihood are negatively related (Figures 1a +and 1b), with correctly or incorrectly specified noise. +2. When the noise distribution is correctly specified, the +causal model always has a higher likelihood, even with +misleading CVs (Figure 1a). +3. When the noise distribution is misspecified, the anti- +causal model often has a higher likelihood under mis- +leading CVs (Figure 1b). +5. Robustness of Independence Testing +This section describes IT approaches for cause-effect learn- +ing in LSNMs, including a new IT method based on affine +flows. A theoretical analysis explains why IT approaches +are robust to noise misspecification and misleading CVs. +5.1. The Independence Testing Approach +Inspired by the breakthrough LINGAM approach, indepen- +dence testing has been used in existing methods for various +SCMs (Hoyer et al., 2008; Shimizu et al., 2011; Peters et al., +2014; Strobl & Lasko, 2022; Immer et al., 2022). Like the + +Cause-Effect Inference in Location-Scale Noise Models +(a) Log-Likelihood Difference Under Correct Noise +Specification: Gaussian(0, 1) noise +(b) Log-Likelihood Difference Under Noise Misspecifi- +cation: Uniform(−1, 1) noise +(c) HSIC Difference Under Correct Noise Specification: +Gaussian(0, 1) noise +(d) HSIC Difference Under Noise Misspecification: +Uniform(−1, 1) noise +Figure 1: Visualization of Table 1. First row (1a,1b): ML methods. Second row (1c,1d): IT methods. Y-axis < 0.0: A ML +method returns the incorrect anti-causal direction. Y-axis > 0.0: An IT method returns the incorrect anti-causal direction. +ML methods may fail under misspecification and misleading CVs (1b). IT methods are more robust (1d). +ML approach (Equation (4)), IT methods fit the model pa- +rameters in both directions, typically maximizing the data +likelihood (Equation (3)). The difference is in the model +selection step: While ML approaches select the direction +with the highest likelihood, IT approaches select the direc- +tion with the highest degree of independence between the +fitted model residuals and the putative cause. Algorithm 1 +provides pseudo-code. +As in previous work (e.g., Mooij et al. (2016)), we use +the Hilbert-Schmidt independence criterion (HSIC) (Gret- +ton et al., 2005) to measure (in)dependence throughout the +paper. HSIC measures the squared distance between the +joint probability of the two variables and the product of +their marginals embedded in the reproducible kernel Hilbert +space. We have HSIC(U, V ) = 0 if and only if U ⊥⊥ V . +To fit the functions in a LSNM, we use the affine flow esti- +mator T from CAREFL-M (Khemakhem et al., 2021). We +refer to the resulting IT method as CAREFL-H. This com- +bination of affine flow with IT appears to be new. Details +on CAREFL-M and learning the flow transformation T are +given in Appendix C. +Another approach to IT methods is to test the independence +of residuals for both X and Y variables (He et al., 2021). +We found that this performs similarly to CAREFL-H and +therefore report experimental results only for the more com- +mon LINGAM-style approach. More details on IT with +residuals are in Appendix D. +5.2. Suitability Theory +With a consistent HSIC estimator, Mooij et al. (2016) show +that an IT approach consistently selects the causal direction +for ANMs if the regression method is suitable. A regres- +sion method is suitable if the expected mean squared error +between the predicted residuals �E and the true residuals E +approaches 0 in the limit of N → ∞: +lim +N→∞ ED,D′ +� 1 +N || �E1...N − E1...N||2 +� += 0 +(6) +where D and D′ denote training set and testing set, respec- +tively. Hence, with enough data a suitable regression method +reconstructs the ground-truth noise. +If an HSIC estimator is consistent, the estimated HSIC value +converges in probability to the population HSIC value. +� +HSIC(X, Y ) +P−→ HSIC(X, Y ). +Mooij et al. (2016) show that even a biased HSIC estimator +with a fixed bounded kernel is consistent. +With a consistent HSIC estimator and a suitable regression +method, the consistency result for ANMs in Mooij et al. +(2016) extends naturally to LSNMs. +Proposition 5.1. For identifiable LSNMs with an indepen- +dent noise term in one causal direction only, if an IT ap- + +: Causal - Anti-Causal +2.00 +1.75 +1.50 +1.25 +1.00 +Log-Likelihood: +0.75 +0.50 +0.25 +0.00 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +Mean CV: Causal - Anti-Causal-Causal +0.6 +Anti. +0.5 +0.4 +Causal +0.3 +0.2 +Log-Likelihood: +0.1 +0.0 +-0.1 +-0.2 +0.0 +0.1 +0.2 +0.3 +0.4 +Mean CV: Causal - Anti-CausalAnti-Causal +0.000 +-0.005 +一 +-0.010 +: Causal +-0.015 +HSIC-Value: +0.020 +-0.025 +0.6-0.4-0.2 +0.0 +0.2 +Mean CV: Causal - Anti-CausalAnti-Causal +0.0000 +-0.0005 +0.0010 +: Causal +-0.0015 +HSIC-Value: +-0.0020 +-0.0025 +-0.0030 +0.0 +0.1 +0.2 +0.3 +0.4 +Mean CV: Causal - Anti-CausalCause-Effect Inference in Location-Scale Noise Models +Algorithm 1 CAREFL-H +1: Input: data pairs D := (X, Y ), the flow estimator T +of CAREFL-M with prior PZ, and an HSIC estimator +2: Output: estimated causal direction dir +3: Split D into training set Dtrain := (Xtrain, Ytrain) and +testing set Dtest := (Xtest, Ytest) +4: Optimize T�θ, � +ψ,�ζ(Dtrain; PZ) in X → Y direction via +ML to estimate �f and �g +5: Compute the residual �ZY := Ytest− � +f(Xtest) +�g(Xtest) +6: Optimize T � +θ′,� +ψ′, � +ζ′(Dtrain; PZ) in X ← Y direction +via ML to estimate �h and �k +7: Compute the residual �ZX := Xtest−�h(Ytest) +�k(Ytest) +8: if HSIC(Xtest, �ZY ) < HSIC(Ytest, �ZX) then +9: +dir := X → Y +10: else if HSIC(Xtest, �ZY ) > HSIC(Ytest, �ZX) then +11: +dir := X ← Y +12: else +13: +dir := no conclusion +14: end if +proach is used with a suitable regression method for LSNMs +and a consistent HSIC estimator, then the IT approach is +consistent for inferring causal direction for LSNMs. +5.3. Suitability: Empirical Results +The suitability value S is the left-hand side of Equation (6). +Table 2 shows an empirical evaluation of S for the flow +estimator T in the causal direction. We generate data from +3 synthetic LSNMs (definition in Appendix G), and eval- +uate T under noise misspecification and misleading CVs. +We find that as the sample size grows, S approaches 0. +In other words, T is empirically suitable under noise mis- +specification and misleading CVs. Therefore, according +to Proposition 5.1, CAREFL-H based on the flow estimator +T and a consistent HSIC estimator is empirically consis- +tent for inferring causal direction in LSNMs under noise +misspecification and misleading CVs. +Because the T in CAREFL-M (Khemakhem et al., 2021) +uses neural networks to approximate the observed data dis- +tribution, it is difficult to provide a theoretical guarantee of +suitability. Therefore, although our experiments indicate +that T is often suitable in practice, we do not claim that it +is suitable for all LSNMs. For example, we have found it +to be not suitable in the low-noise regime when the LSNMs +are close to deterministic. +6. Experiments +The code and scripts to reproduce all the results are given +online 3. We show that across hyperparameter choices the IT +approach (i.e., CAREFL-H) produces much higher accuracy +than the ML approach (i.e,. CAREFL-M) in the difficult +settings with noise misspecification and misleading CVs, +and produces comparable accuracy in the easier settings +without noise misspecification or misleading CVs. Also the +IT approach is more robust with real-world data, where the +ground-truth noise distribution is unknown. +For all experiments, we start with the same default hyperpa- +rameter values for both CAREFL-M and CAREFL-H and +alter one hyperparameter value at a time. The default hyper- +parameter values are those specified in CAREFL-M (Khe- +makhem et al., 2021) for the T¨ubingen Cause-Effect Pairs +Benchmark (Mooij et al., 2016). Please see Appendix F for +more details on default and alternative hyperparameter val- +ues. Previous work (Mooij et al., 2016; Immer et al., 2022) +reported that ML methods perform better with data splitting +(split data into training set for model fitting and testing set +for model selection) and IT methods perform better with +data recycling (the same data is used for both model fitting +and selection). Therefore, we use both splitting methods: +(i) CAREFL(0.8): 80% as training and 20% as testing. (ii) +CAREFL(1.0): training = testing = 100%. +We use a consistent HSIC estimator with Gaussian ker- +nels (Pfister et al., 2018). A summary of experimental +datasets is provided in Appendix Table 4. All the datasets +are normalized to have mean 0 and variance 1. +6.1. Synthetic Datasets +Please see Appendix G for the definition of the 3 ground- +truth LSNM SCMs and details on how synthetic datasets are +generated from them. The sample sizes in each synthetic +dataset are 500 or 5,000. As shown in Appendix Table 4, +most synthetic datasets generated by such SCMs have mis- +leading CVs. Based on the analysis of Section 4.2 we formu- +late the following hypotheses. (i) We expect CAREFL-M +to be accurate given a correct noise specification with or +without misleading CVs. (ii) With noise misspecification +but without misleading CVs, we expect the accuracy of +CAREFL-M to be reduced. (iii) With both noise misspeci- +fication and misleading CVs, we expect the accuracy to be +very low, often below 50%. Overall, the results from the +experiments confirm our hypotheses. +6.1.1. NOISE MISSPECIFICATION +We +evaluate +CAREFL-M +and +CAREFL-H +against +data generated with Uniform(−1, 1), Exponential(1), +3https://github.com/xiangyu-sun-789/CAREFL-H + +Cause-Effect Inference in Location-Scale Noise Models +Table 2: Suitability of the flow estimator T of CAREFL-M in the causal direction under noise misspecification and +misleading CVs. T is trained with a Laplace prior. The original dataset with size 2N is split into two: 50% as training set +and 50% as testing set. V[Y |X] > V[X|Y ] indicates misleading CVs in the dataset. +(a) LSNM-tanh-exp-cosine and ContinuousBernoulli(0.9) noise. V[Y |X] vs. V[X|Y ]: 0.324 vs. 0.291. +N=50 +N=500 +N=1000 +N=5000 +[SZ1, SZ2] +[0.02406, 0.01283] +[0.00194, 0.00204] +[0.00094, 0.00092] +[0.0002, 0.00018] +(b) LSNM-sine-tanh and Uniform(−1, 1) noise. V[Y |X] vs. V[X|Y ]: 0.422 vs. 0.367. +N=50 +N=500 +N=1000 +N=5000 +[SZ1, SZ2] +[0.0081, 0.00285] +[0.00039, 0.00031] +[0.0002, 0.00017] +[0.00003, 0.00003] +(c) LSNM-sigmoid-sigmoid and Exponential(1) noise. V[Y |X] vs. V[X|Y ]: 0.927 vs. 0.657. +N=50 +N=500 +N=1000 +N=5000 +[SZ1, SZ2] +[0.00979, 0.00443] +[0.00074, 0.00058] +[0.00045, 0.00026] +[0.00005, 0.00005] +Figure 2: Weighted accuracy over 99 datasets from T¨ubingen Cause-Effect Pairs Benchmark. +ContinuousBernoulli(0.9) +or +Beta(0.5, 0.5) +noise, +covered by our identifiability Theorem 4.2 (except +Beta(0.5, 0.5)). +Khemakhem et al. (2021) claim that +CAREFL-M is robust to noise misspecification. We show +that it may fail remarkably. +We summarize findings from the 336 settings here; the de- +tailed results are given in Appendix Figures 3 to 14. In 289 +settings (86.01%), both CAREFL-M(0.8) and CAREFL- +M(1.0) select the correct causal direction with less than 50% +random accuracy. Furthermore, in 110 settings (32.74%) +both CAREFL-M(0.8) and CAREFL-M(1.0) fail catastroph- +ically with an accuracy of 0%. These experiments also +show that the accuracy of CAREFL-M often decreases as +N increases. In contrast, CAREFL-H(1.0) achieves better +accuracy than CAREFL-M in 333 settings (99.11%). The +accuracy of CAREFL-H(1.0) goes below 50% in only 6 set- +tings (1.79%). The results demonstrate the robustness of the +IT approach under noise misspecification and misleading +CVs, across different hyperparameter choices. +Appendix Table 4 and Appendix Figure 14 show that with +misleading CVs, the accuracy of CAREFL-M is close to 0%. +This is much lower than the corresponding cases without +misleading CVs in Appendix Figures 6 and 10. +6.1.2. CORRECT NOISE SPECIFICATION +These experiments show that CAREFL-H is comparable +with CAREFL-M under correct noise specification, with +or without misleading CVs, especially on larger datasets, +as long as the affine model capacity is sufficient. We eval- +uate CAREFL-M and CAREFL-H against data generated +with Gaussian(0, 1) and Laplace(0, 1) noise. The detailed +results are in Appendix Figures 15 to 20. We find that +CAREFL-M is more sample efficient than CAREFL-H +when the model prior matches the data. Consistent with the +suitability results in Section 5.3, the accuracy of CAREFL- +H improves with more data. For example, with N = 500, +there are 49 out of 84 settings (58.33%) where CAREFL-M +outperforms CAREFL-H(1.0). However, with N = 5, 000, +CAREFL-H(1.0) achieves similar accuracy as CAREFL- +M on all datasets (except LSNM-sigmoid-sigmoid with +Laplace(0, 1) noise.) In addition, CAREFL-H(1.0) may +underperform CAREFL-M when the number of hidden neu- + +0.8 +0.8 × +Accuracy +0.8 +0.6 +0.6 +0.6 +F9'0 +0.6 +0.4 +0.4 +0.4 +0.4 +0.4 +2 +5 +10 +20 +1 +4 +10 +500 +750 +1000 +2000 +0.0 +0.0001 0.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +Table 3: The best accuracy of each method on the SIM and T¨ubingen Cause-Effect Pairs benchmarks. For methods other +than CAREFL-M and CAREFL-H, we use the results reported in Immer et al. (2022). +(a) SIM Benchmarks +LOCI-M +LOCI-H +GRCI +BQCD +HECI +CAM +RESIT +CAREFL-M +CAREFL-H +SIM +0.53 +0.79 +0.77 +0.62 +0.49 +0.57 +0.77 +0.55 +0.80 +SIM-c +0.50 +0.83 +0.77 +0.72 +0.55 +0.60 +0.82 +0.58 +0.85 +SIM-ln +0.79 +0.72 +0.77 +0.80 +0.65 +0.87 +0.87 +0.84 +0.83 +SIM-G +0.78 +0.82 +0.70 +0.64 +0.56 +0.81 +0.78 +0.82 +0.79 +(b) Weighted Accuracy for T¨ubingen Cause-Effect Pairs Benchmark +LOCI-M +LOCI-H +GRCI +BQCD +HECI +CAM +RESIT +CAREFL-M +CAREFL-H +T¨ubingen +Cause-Effect +Pairs +0.57 +0.64 +0.82 +0.77 +0.71 +0.58 +0.57 +0.73 +0.82 +rons, sub-flows or training epochs is low. The reason is +that an IT approach requires more power to fit the LSNM +functions and produce a good reconstruction of the noise. +6.2. Synthetic Benchmark +Similar to Tagasovska et al. (2020); Immer et al. (2022), +we compare CAREFL-M and CAREFL-H against the SIM +benchmark suite (Mooij et al., 2016). SIM comprises 4 +sub-benchmarks: default (SIM), with one confounder (SIM- +c), low noise levels (SIM-ln) and Gaussian noise (SIM-G). +In this benchmark, most datasets do not have misleading +CVs (Appendix Table 4), which favors ML methods. Each +sub-benchmark contains 100 datasets and each dataset has +N = 1000 data pairs. As shown in Appendix Figure 21, +CAREFL-M and CAREFL-H(1.0) achieve similar accu- +racy on SIM-ln and SIM-G across different hyperparame- +ter choices. For SIM and SIM-c, CAREFL-H, especially +CAREFL-H(1.0), outperforms CAREFL-M by 20%-30% +in all settings. The accuracy of CAREFL-M is only about +random guess (40%-60%) on SIM and SIM-c. +Table 3a compares CAREFL-H with SOTA methods. For +each CAREFL method, we report the best accuracy ob- +tained with the hyperparameter settings considered in Ap- +pendix Figure 21, without further tuning. +CAREFL-H +achieves the best accuracy on SIM and SIM-c, and achieves +competitive accuracy on SIM-ln and SIM-G. +6.3. Real-World Benchmark: T¨ubingen Cause-Effect +We compare CAREFL-M and CAREFL-H against real- +world datasets from the T¨ubingen Cause-Effect Pairs Bench- +mark (Mooij et al., 2016). The benchmark is commonly +used to evaluate cause-effect inference algorithms (Khe- +makhem et al., 2021; Xu et al., 2022; Immer et al., 2022). +To be consistent with previous work (Tagasovska et al., +2020; Strobl & Lasko, 2022; Immer et al., 2022), we ex- +clude 6 multivariate and 3 discrete datasets (#47, #52-#55, +#70, #71, #105, #107) and utilize the remaining 99 bivariate +datasets. As recommended by Mooij et al. (2016), we re- +port weighted accuracy. 40% of datasets in the benchmark +feature misleading CVs. As shown in Figure 2, CAREFL- +H(1.0) outperforms CAREFL-M in all configurations by +large margins (7%-30%). +We also compare CAREFL-H with SOTA methods. Fol- +lowing Khemakhem et al. (2021), we use a single set of +hyperparameters for all 99 datasets, found by grid search +(Appendix H). Table 3b shows that CAREFL-H achieves +the SOTA accuracy (82%). With the respectively best hyper- +parameter settings, CAREFL-H is 9% more accurate than +CAREFL-M (Khemakhem et al., 2021). +7. Conclusion and Future Work +We identified a failure of maximum-likelihood (ML) meth- +ods for cause-effect inference in location-scale noise models. +Our analysis shows that the failure mode occurs when the +noise distribution is misspecified and conditional effect vari- +ances are misleading (i.e., higher in the causal direction). +Selecting causal models by independence tests (IT) is robust +even in this difficult setting. Extensive empirical evalu- +ation compared the ML approach and a new IT method +based on affine flows, using both synthetic and real-world +datasets. The IT flow method achieves better accuracy under +noise misspecification and misleading CVs, with robust per- +formance across different hyperparameter choices. 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For a LSNM model X → Y defined in Equation (1), we have PY |X(Y |X) = PZY ( Y −f(X) +g(X) ) · +1 +g(X), where +PZY is the noise distribution. +For the data distribution in the X → Y direction: +P→,PZ(X, Y ) = PX(X) · PY |X(Y |X) = PX(X) · PZY (Y − f(X) +g(X) +) · +1 +g(X) +Similarly, in the X ← Y direction: +P←,PZ(X, Y ) = PY (Y ) · PX|Y (X|Y ) = PY (Y ) · PZX(X − h(Y ) +k(Y ) +) · +1 +k(Y ) +Proof for Lemma A.1. For a LSNM model defined in Equation (1), we have ZY = Y −f(X) +g(X) . Hence, ∂ZY +∂Y += +1 +g(X). Since +g(X) > 0, so | ∂ZY +∂Y | = | +1 +g(X)| = +1 +g(X). +PY |X(Y |X) +=PZY (ZY |X) · | det ∂ZY +∂Y | +(via change of variables) +=PZY (ZY ) · | det ∂ZY +∂Y | +=PZY (Y − f(X) +g(X) +) · |∂ZY +∂Y | +=PZY (Y − f(X) +g(X) +) · +1 +g(X) +B. Identifiability Proofs +In this section, we prove Theorem 4.2. +If the data (x, y) follows a LSNM in the forward (i.e. causal) model: +y := f(x) + g(x) · zY +where zX and zY are the noise terms, e ⊥⊥ zX, x ⊥⊥ zY , g(x) > 0 for all x on its domain. We assume f(·) and g(·) are +twice-differentiable on the domain of x. +If the data (x, y) follows a LSNM in the backward (i.e. anti-causal) model: +x := h(y) + k(y) · mX +where mY and mX are the noise terms, e′ ⊥⊥ mY , y ⊥⊥ mX, k(y) > 0 for all y on its domain. We assume h(·) and k(·) +are twice-differentiable on the domain of y. +zX, zY , mY and mX follow one of Uniform(a, b), Exponential(λ) or ContinuousBernoulli(λ) distribution accordingly. +Proof for Uniform(a, b) Noise. For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x) +g(x) ) · +1 +g(x) = PZY (zY ) · +1 +g(x) = +1 +b−a · +1 +g(x) = +1 +(b−a)·g(x). Similarly, for the backward model, we have PX|Y (x|y) = +1 +(b−a)·k(y). + +Cause-Effect Inference in Location-Scale Noise Models +The joint likelihood of the observation (x, y) in the causal model is: +P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) · +1 +(b − a) · g(x) +The joint likelihood of the observation (x, y) in the backward model is: +P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) · +1 +(b − a) · k(y) +If the data follows both models: +P→,PZ(x, y) = P←,PZ(x, y) +PX(x) · +1 +(b − a) · g(x) = PY (y) · +1 +(b − a) · k(y) +PX(x) · +1 +g(x) = PY (y) · +1 +k(y) +Take the derivative of both sides with respect to x: +PX(x) · (g(x)−1)′ + (PX(x))′ · g(x)−1 = 0 +PX(x) · −1 · g(x)−2 · g(x)′ + 0 · g(x)−1 = 0 +PX(x) · −1 · g(x)−2 · g(x)′ = 0 +PX(x) > 0 +g(x)−2 = +1 +g(x)2 > 0 +Therefore, g(x)′ = 0 +Similarly, if we take the derivative of both sides with respect to y instead, we have: +k(y)′ = 0 +These imply that both g(x) and k(y) are constant functions. +Proof for Exponential(λ) Noise. For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x) +g(x) ) · +1 +g(x) = λ · e−λ· y−f(x) +g(x) +· +1 +g(x) = +λ +g(x) · e− +λ +g(x) ·(y−f(x)). Similarly, for the backward model, we have PX|Y (x|y) = +λ +k(y) · e− +λ +k(y) ·(x−h(y)). +The joint likelihood of the observation (x, y) in the causal model is: +P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) · +λ +g(x) · e− +λ +g(x) ·(y−f(x)) +log P→,PZ(x, y) = log PX(x) + log λ − log g(x) − +λ +g(x) · (y − f(x)) +The joint likelihood of the observation (x, y) in the backward model is: +P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) · +λ +k(y) · e− +λ +k(y) ·(x−h(y)) +log P←,PZ(x, y) = log PY (y) + log λ − log k(y) − +λ +k(y) · (x − h(y)) + +Cause-Effect Inference in Location-Scale Noise Models +If the data follows both models: +log P→,PZ(x, y) = log P←,PZ(x, y) +log PX(x) + log λ − log g(x) − +λ +g(x) · (y − f(x)) += log PY (y) + log λ − log k(y) − +λ +k(y) · (x − h(y)) +log PX(x) − log g(x) − λ · g(x)−1 · y + λ · g(x)−1 · f(x) += log PY (y) − log k(y) − λ · k(y)−1 · x + λ · k(y)−1 · h(y) +Take the derivative of both sides with respect to x: +(log PX(x))′ − (log g(x))′ − λ · y · (g(x)−1)′ + λ · (g(x)−1 · f(x))′ += −λ · k(y)−1 +Take the derivative of both sides with respect to y: +−λ · (g(x)−1)′ = −λ · (k(y)−1)′ +∂g(x)−1 +∂x += ∂k(y)−1 +∂y +They can be equal only if both sides are constants. Therefore, g(x)−1 and k(y)−1 are both constants or both linear functions +with the same coefficient on x and y, respectively. +Proof for ContinuousBernoulli(λ ̸= 0.5) Noise. Please refer to Uniform for ContinuousBernoulli(λ = 0.5), which +equals to Uniform(0, 1). For the causal model, according to Lemma A.1, we have PY |X(y|x) = PZY ( y−f(x) +g(x) ) · +1 +g(x) = +Cλ·λ +y−f(x) +g(x) ·(1−λ)1− y−f(x) +g(x) · +1 +g(x), where Cλ is the normalizing constant of the continuous Bernoulli distribution. Similarly, +for the backward model, we have PX|Y (x|y) = Cλ · λ +x−h(y) +k(y) +· (1 − λ)1− x−h(y) +k(y) +· +1 +k(y). +The joint likelihood of the observation (x, y) in the causal model is: +P→,PZ(x, y) = PX(x) · PY |X(y|x) = PX(x) · +1 +g(x) · Cλ · λ +y−f(x) +g(x) +·(1 − λ)1− y−f(x) +g(x) +log P→,PZ(x, y) = log PX(x) − log g(x) + log Cλ + log λ +y−f(x) +g(x) ++ log(1 − λ)1− y−f(x) +g(x) +The joint likelihood of the observation (x, y) in the backward model is: +P←,PZ(x, y) = PY (y) · PX|Y (x|y) = PY (y) · +1 +k(y) · Cλ · λ +x−h(y) +k(y) +·(1 − λ)1− x−h(y) +k(y) +log P←,PZ(x, y) = log PY (y) − log k(y) + log Cλ + log λ +x−h(y) +k(y) ++ log(1 − λ)1− x−h(y) +k(y) + +Cause-Effect Inference in Location-Scale Noise Models +If the data follows both models: +log P→,PZ(x, y) = log P←,PZ(x, y) +log PX(x) − log g(x) + log Cλ + log λ +y−f(x) +g(x) ++ log(1 − λ)1− y−f(x) +g(x) += log PY (y) − log k(y) + log Cλ + log λ +x−h(y) +k(y) ++ log(1 − λ)1− x−h(y) +k(y) +log PX(x) − log g(x) + log Cλ + y − f(x) +g(x) +· log λ + (1 − y − f(x) +g(x) +) +· log(1 − λ) += log PY (y) − log k(y) + log Cλ + x − h(y) +k(y) +· log λ + (1 − x − h(y) +k(y) +) +· log(1 − λ) +log PX(x) − log g(x) + log Cλ + g(x)−1 · log λ · y − g(x)−1· +log λ · f(x) + log(1 − λ) − log(1 − λ) · g(x)−1 · y + log(1 − λ) · g(x)−1 · f(x) += log PY (y) − log k(y) + log Cλ + k(y)−1 · log λ · x − k(y)−1· +log λ · h(y) + log(1 − λ) − log(1 − λ) · k(y)−1 · x + log(1 − λ) · k(y)−1 · h(y) +Take the derivative of both sides with respect to x: +(log PX(x))′ − g(x)−1 · g(x)′ − 1 · g(x)−2 · g(x)′ · log λ · y +− (g(x)−1 · log λ · f(x))′ − log(1 − λ) · −1 · g(x)−2 · g(x)′ · y ++ (log(1 − λ) · g(x)−1 · f(x))′ = k(y)−1 · log λ − log(1 − λ) · k(y)−1 +Take the derivative of both sides with respect to y: +− g(x)−2 · g(x)′ · log λ − log(1 − λ) · −1 · g(x)−2 · g(x)′ += −1 · k(y)−2 · k(y)′ · log λ − log(1 − λ) · −1 · k(y)−2 · k(y)′ +g(x)−2 · g(x)′ · log λ − log(1 − λ) · g(x)−2 · g(x)′ += k(y)−2 · k(y)′ · log λ − log(1 − λ) · k(y)−2 · k(y)′ +g(x)−2 · g(x)′ · (log λ − log(1 − λ)) = k(y)−2 · k(y)′ · (log λ − log(1 − λ)) +Since λ ̸= 0.5, therefore log λ − log(1 − λ) ̸= 0 +g(x)−2 · g(x)′ = k(y)−2 · k(y)′ +∂g(x)−1 +∂x += ∂k(y)−1 +∂y +They can be equal only if both sides are constants. Therefore, g(x)−1 and k(y)−1 are both constants or both linear functions +with the same coefficient on x and y, respectively. +C. CAREFL-M +CAREFL-M (Khemakhem et al., 2021) models a LSNM in Equation (1) via affine flows T. Each sub-flow Tk ∈ T is +defined as the following: +X = t1 + es1 · ZX +Y = t2(X) + es2(X) · ZY +(7) +where X is the putative cause and Y is the putative effect in X → Y direction. t1 and s1 are constants. t2 and s2 are +functions parameterized using neural networks. Without loss of generality, X is assumed to be a function of latent noise + +Cause-Effect Inference in Location-Scale Noise Models +Algorithm 2 CAREFL-M +1: Input: data pairs D := (X, Y ), and the flow estimator T with prior PZ +2: Output: estimated causal direction dir +3: Split D into training set Dtrain := (Xtrain, Ytrain) and testing set Dtest := (Xtest, Ytest) +4: Optimize T�t1,�s1,�t2,�s2(Dtrain; PZ) in X → Y direction via ML +5: Compute the likelihood �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) in X → Y direction +6: Optimize T�t′1,�s′1,�t′2,�s′2(Dtrain; PZ) in X ← Y direction via ML +7: Compute the likelihood �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) in X ← Y direction +8: if �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) > �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) then +9: +dir := X → Y +10: else if �L→,PZ(Dtest; T�t1,�s1,�t2,�s2) < �L←,PZ(Dtest; T�t′1,�s′1,�t′2,�s′2) then +11: +dir := X ← Y +12: else +13: +dir := no conclusion +14: end if +variable ZX. If t1 = 0 and s1 = 0, then X = ZX. The exponential function e ensures the multipliers to Z are positive +without expression loss. Similarly, for the backward direction X ← Y : +Y = t′ +1 + es′ +1 · ZY +X = t′ +2(Y ) + es′ +2(Y ) · ZX +(8) +where Y is the putative cause and X is the putative effect in X ← Y direction. t′ +1 and s′ +1 are constants. t′ +2 and s′ +2 are +functions parameterized using neural networks. +Given Equation (7), the joint log-likelihood of (x, y) in X → Y direction is: +log P→,PZ(x, y) = log PZX +� +e−s1 · (x − t1) +� ++ log PZY +� +e−s2(x) · (y − t2(x)) +� +− s1 − s2(x) +(9) +Similarly for the X ← Y direction. Note that the priors PZ = {PZX, PZY } in Equation (9) may mismatch the unknown +ground-truth noise distribution P ∗ +Z = {P ∗ +ZX, P ∗ +ZY }. Both CAREFL-M and CAREFL-H optimizes Equation (9) for each +direction over the training set. For CAREFL-M, it chooses the direction with ML Score L over the testing set as the +estimated causal direction. Detailed procedure for CAREFL-M is given in Algorithm 2. To map the parameters θ, ψ, ζ, θ′, +ψ′ and ζ′ in LSNM (Equation (2)) to the flow estimator T of CAREFL (Equations (7) and (8)), we have fθ ≡ t2, gψ ≡ es2, +PX,ζ ≡ {t1, es1}, fθ′ ≡ t′ +2, gψ′ ≡ es′ +2 and PY,ζ′ ≡ {t′ +1, es′ +1}. +D. CAREFL-H Alternative Independence Testing +In Algorithm 1, CAREFL-H tests independence between the putative cause and the residual of the putative effect in each +direction, i.e., between X and �ZY in X → Y direction, and between Y and �ZX in X ← Y direction. An alternative way of +testing independence is to test between the residual of the putative cause and the residual of the putative effect (He et al., +2021), i.e., between �ZX and �ZY in both directions. Please see Algorithm 3 for complete steps. +Theorem D.1. By optimizing the log-likelihood in the causal direction, may or may not under noise misspecification, +CAREFL-H gives the reconstructed residual of the ground-truth cause (i.e. �ZX) identical to the ground-truth cause (i.e. X) +and to the latent noise variable of the cause (i.e. ZX) up to shifting and scaling. +Proof is in Appendix E. Although we prove only for the cause, empirically we find that in the causal direction the +reconstructed residual of the ground-truth effect (i.e., �ZY ) is also close to the latent noise variable of the effect (i.e., +ZY ). According to Theorem D.1, testing HSIC(X, �ZY ) in Algorithm 1 and HSIC( �ZX, �ZY ) in Algorithm 3 for the +causal direction are equivalent. The difference comes from testing HSIC(Y, �ZX) in Algorithm 1 and HSIC( �ZX, �ZY ) +in Algorithm 3 for the anti-causal direction. Although in our experiments the two algorithms often produce the same +estimation of causal direction, we prefer Algorithm 1, since it relies on few estimations of the residuals. + +Cause-Effect Inference in Location-Scale Noise Models +Algorithm 3 CAREFL-H (Between Residuals) +1: Input: data pairs D := (X, Y ), the flow estimator T of CAREFL-M with prior PZ, and an HSIC estimator +2: Output: estimated causal direction dir +3: Split D into training set Dtrain := (Xtrain, Ytrain) and testing set Dtest := (Xtest, Ytest) +4: Optimize T�t1,�s1,�t2,�s2(Dtrain; PZ) in X → Y direction via ML to estimate �t1, �s1, �t2 and �s2 +5: Compute the residuals �ZX,→ := Xtest−�t1 +e�s1 +and �ZY,→ := Ytest−�t2(X) +e�s2(X) +6: Optimize T�t′1,�s′1,�t′2,�s′2(Dtrain; PZ) in X ← Y direction via ML to estimate �t′1, �s′1, �t′2 and �s′2 +7: Compute the residuals �ZY,← := Ytest−�t′1 +e +� +s′1 +and �ZX,← := Xtest−�t′2(Y ) +e +� +s′2(Y ) +8: if HSIC( �ZX,→, �ZY,→) < HSIC( �ZX,←, �ZY,←) then +9: +dir := X → Y +10: else if HSIC( �ZX,→, �ZY,→) > HSIC( �ZX,←, �ZY,←) then +11: +dir := X ← Y +12: else +13: +dir := no conclusion +14: end if +E. Proof for Theorem D.1 +In this section, we prove Theorem D.1. +The flow estimator T in CAREFL-M (Khemakhem et al., 2021) models a LSNM in Equation (1) as the following: +X = t1 + es1 · ZX +Y = t2(X) + es2(X) · ZY +(10) +where X and Y are putative cause and effect, respectively. ZX and ZY are assumed to follow a prior distribution, e.g. +Gaussian(0, 1) or Laplace(0, 1). t1 and s1 are constants. It also means ZX is identical to X up to shifting and scaling. +If t1 = 0 and s1 = 0, then X = ZX. t2 and s2 are functions parameterized using neural networks. Let (xn, yn), where +n ∈ {1, . . . , N}, be the n-th data pair. +Fact 1, invert Equation (10): +zn +X = e−s1 · (xn − t1) +zn +Y = e−s2(xn) · (yn − t2(xn)) +Fact 2, how �zn +X and �zn +Y are computed in the flow estimator T: +�zn +X = e−�s1 · (xn − �t1) +�zn +Y = e−�s2(xn) · (yn − �t2(xn)) + +Cause-Effect Inference in Location-Scale Noise Models +Fact 3, how the flow estimator T is trained via ML (may under noise misspecification): +pX(xn, yn) = pZX(�zn +X) · pZY (�zn +Y ) · | det +� ∂�zn +X +∂xn +∂�zn +X +∂yn +∂�zn +Y +∂xn +∂�zn +Y +∂yn +� +| += pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn))) +· | det +�e−�s1 +0 +∂�zn +Y +∂xn +e−�s2(xn) +� +| += pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn))) +· |e−�s1 · e−�s2(xn) − 0| += pZX(e−�s1 · (xn − �t1)) · pZY (e−�s2(xn) · (yn − �t2(xn))) · e−�s1 · e−�s2(xn) += pZX(e−�s1 · (es1 · zn +X + t1 − �t1)) +· pZY (e−�s2(xn) · (es2(xn) · zn +Y + t2(xn) − �t2(xn))) +· e−�s1 · e−�s2(xn) += pZX(e−�s1 · (es1 · zn +X + t1 − �t1)) · e−�s1 +· pZY (e−�s2(xn) · (es2(xn) · zn +Y + t2(xn) − �t2(xn))) · e−�s2(xn) += pZX(es1−�s1 · zn +X + e−�s1 · (t1 − �t1)) · e−�s1 +· pZY (es2(xn)−�s2(xn) · zn +Y + e−�s2(xn) · (t2(xn) − �t2(xn))) · e−�s2(xn) +Assume the priors pZX and pZY are standard Gaussian N(0, 1) for math convenience. The proof is analogous with other +prior distributions, e.g., standard Laplace. We do not make assumptions on the ground-truth noise distribution, which allows +noise misspecification. +pX(xn, yn) = N(es1−�s1 · zn +X + e−�s1 · (t1 − �t1); µ = 0, σ2 = 1) · e−�s1 +· N(es2(xn)−�s2(xn) · zn +Y + e−�s2(xn) · (t2(xn) − �t2(xn)); µ = 0, σ2 = 1) +· e−�s2(xn) +(the PDF of N(x; 0, 1) is +1 +√ +2π · e− 1 +2 ·(x)2) += +1 +√ +2π · e− 1 +2 ·(es1−�s1·zn +X+e−�s1·(t1−�t1))2 · e−�s1 +· +1 +√ +2π · e− 1 +2 ·(es2(xn)−�s2(xn)·zn +Y +e−�s2(xn)·(t2(xn)−�t2(xn)))2 · e−�s2(xn) +ln pX(xn, yn) = ln +1 +√ +2π − 1 +2 · (es1−�s1 · zn +X + e−�s1 · (t1 − �t1))2 − �s1 ++ ln +1 +√ +2π − 1 +2 · (es2(xn)−�s2(xn) · zn +Y + e−�s2(xn) · (t2(xn) − �t2(xn)))2 +− �s2(xn) +Lemma E.1. To maximize the log-likelihood, may or may not under noise misspecification, �t1 = E[X], a constant. +Lemma E.2. To maximize the log-likelihood, may or may not under noise misspecification, �s1 = E[ln |�t1 − X|], a constant. +�zn +X = e−�s1 · (xn − �t1) +�zn +X = e−E[ln |�t1−X|] · (xn − �t1) +�zn +X = e−E[ln |E[X]−X|] · (xn − E[X]) +�zn +X = C1 · (xn − C2) + +Cause-Effect Inference in Location-Scale Noise Models +Therefore, �ZX and X are identically distributed up to shifting and scaling. +Since xn := t1 + es1 · zn +X, we have: +�zn +X = C1 · (xn − C2) += C1 · (es1 · zn +X + t1 − C2) += C1 · es1 · zn +X + C1 · t1 − C1 · C2 += C3 · zn +X + C4 − C5 += C3 · zn +X + C6 +Therefore, �ZX and ZX are also identically distributed up to shifting and scaling. +Proof for Lemma E.1. +∂ ln pX(xn, yn) +∂�t1 += e−�s1 · (zn +X · es1−�s1 + e−�s1 · (t1 − �t1)) +Let e−�s1 · (zn +X · es1−�s1 + e−�s1 · (t1 − �t1)) = 0 +�t1 = t1 + zn +X · es1 = xn +Similarly, for ∂ ln pX(xn=1,yn=1) +∂�t1 +, �t1 = xn=1; for ∂ ln pX(xn=2,yn=2) +∂�t1 +, �t1 = xn=2. Therefore, +�t1 = 1 +n · +N +� +n=1 +xn = E[X] +Since ∂2 ln pX(xn,yn) +∂2�t1 +is negative, �t1 maximizes the log-likelihood. +Proof for Lemma E.2. +∂ ln pX(xn, yn) +∂�s1 += (−zn +X · es1−�s1 − (t1 − �t1) · e−�s1)2 − 1 +Let (−zn +X · es1−�s1 − (t1 − �t1) · e−�s1)2 − 1 = 0 +± e�s1 = �t1 − xn +e�s1 = ±(�t1 − xn) +e· must be positive. +e�s1 = |�t1 − xn| +�s1 = ln |�t1 − xn| +Similarly, for ∂ ln pX(xn=1,yn=1) +∂�s1 +, �s1 = ln |�t1 − xn=1|; for ∂ ln pX(xn=2,yn=2) +∂�s1 +, �s1 = ln |�t1 − xn=2|. Therefore, +�s1 = 1 +n · +N +� +n=1 +ln |�t1 − xn| = E[ln |�t1 − X|] +Since ∂2 ln pX(xn,yn) +∂2�s1 +is negative, �s1 maximizes the log-likelihood. + +Cause-Effect Inference in Location-Scale Noise Models +F. Default and Alternative Hyperparameter Values Used in Section 6 +We use the reported hyperparameter values in CAREFL-M (Khemakhem et al., 2021) for the T¨ubingen Cause-Effect Pairs +Benchmark (Mooij et al., 2016) as the default hyperparameter values in all our experiments: +• The flow estimator T is parameterized with 4 sub-flows (alternatively: 1, 7 and 10). +• For each sub-flow, f, g, h and k are modelled as four-layer MLPs with 5 hidden neurons in each layer (alternatively: 2, +10 and 20). +• Prior distribution is Laplace (alternatively: Gaussian prior). +• Adam optimizer (Kingma & Ba, 2014) is used to train each model for 750 epochs (alternatively: 500, 1000 and 2000). +• L2-penalty strength is 0 by default (alternatively: 0.0001, 0.001, 0.1). +Although we also observe that LOCI-H is more robust than LOCI-M under noise misspecification and misleading CVs, we +omit their results because: (1) CAREFL often outperforms LOCI (see Tables 1 and 3). (2) LOCI is fixed as a Gaussian +distribution, whereas CAREFL can specify different prior distributions. (3) For conciseness. LOCI uses different set of +hyperparameters than CAREFL. So their results cannot be merged into the same figures. +G. Synthetic SCMs Used in Section 6.1 +We use the following SCMs to generate synthetic datasets: +• LSNM-tanh-exp-cosine: Y := tanh(X · θ1) · θ2 + ecos(X·ψ1)·ψ2 · ZY +• LSNM-sine-tanh: Y := sin(X · θ1) · θ2 + (tanh(X · ψ) + φ) · ZY +• LSNM-sigmoid-sigmoid: Y := σ(X · θ1) · θ2 + σ(X · ψ1) · ψ2 · ZY +where ZY is the ground-truth noise sampled from one of the following distributions: ContinuousBernoulli(0.9), +Uniform(−1, 1), Exponential(1), Beta(0.5, 0.5), Gaussian(0, 1) and Laplace(0, 1), and σ is the sigmoid function. Al- +though we did not prove identifiability with Beta(0.5, 0.5) noise in Theorem 4.2, empirically we find it is identifiable. +Following (Zheng et al., 2018; 2020), each θ and ψ are sampled uniformly from range [−2, −0.5] ∪ [0.5, 2]. φ is sampled +uniformly from range [1, 2] to make the tanh function positive. The number of data pairs in each synthetic dataset is +N ∈ {500, 5000}. +H. Hyperparameter Values of CAREFL-H For Table 3b +To acquire the result of CAREFL-H in Table 3b, the hyperparameter values used are as follows: +• Number of hidden neurons in each layer of the MLPs: 2 +• Number of sub-flows: 10 +• Training dataset = testing dataset = 100%. +The rest of the hyperparameter values are identical to the default ones. + +Cause-Effect Inference in Location-Scale Noise Models +Table 4: (Best viewed in color) Summary of datasets. Settings in Section 6.1 are color-coded. (1) Blue: with noise +misspecification; (2) Red: with more than 50% of datasets having misleading CVs; (3) Brown: both (1) and (2). +Type +Name +Noise +α +Number of +Datasets +Percentage of Datasets +With Misleading CVs +(%) +Synthetic +(Table 1) +LSNM-sine-tanh +Gaussian(0, 1) +0.1 +10 +30 +0.5 +10 +50 +1 +10 +70 +5 +10 +100 +10 +10 +100 +LSNM-sine-tanh +Uniform(−1, 1) +0.1 +10 +30 +0.5 +10 +90 +1 +10 +100 +5 +10 +100 +10 +10 +100 +Synthetic +(Section 6.1) +LSNM-tanh-exp-cosine +Uniform(−1, 1) +N/A +10 +60 +Beta(0.5, 0.5) +10 +80 +ContinuousBernoulli(0.9) +10 +70 +Exponential(1) +10 +20 +Gaussian(0, 1) +10 +40 +Laplace(0, 1) +10 +40 +LSNM-sine-tanh +Uniform(−1, 1) +N/A +10 +100 +Beta(0.5, 0.5) +10 +100 +ContinuousBernoulli(0.9) +10 +60 +Exponential(1) +10 +20 +Gaussian(0, 1) +10 +60 +Laplace(0, 1) +10 +80 +LSNM-sigmoid-sigmoid +Uniform(−1, 1) +N/A +10 +90 +Beta(0.5, 0.5) +10 +100 +ContinuousBernoulli(0.9) +10 +80 +Exponential(1) +10 +70 +Gaussian(0, 1) +10 +90 +Laplace(0, 1) +10 +100 +Synthetic +(Section 6.2) +SIM +N/A +N/A +100 +40 +SIM-c +100 +46 +SIM-ln +100 +23 +SIM-G +100 +29 +Real-World +(Section 6.3) +T¨ubingen Cause-Effect Pairs +N/A +N/A +99 +40 + +Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 3: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Uniform(−1, 1) noise. +(a) N = 500 +(b) N = 5000 +Figure 4: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Beta(0.5, 0.5) noise. + +1.0- +1.0 +1.0 +1.0 +0.5 - +0.5 - +0.5 - +0.5 - +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)1.0 +1.0 +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 - +0.5 +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 - +1.0 +0.5 - +0.5 - +0.5 - +0.5 +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 5: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and ContinuousBernoulli(0.9) noise. +(a) N = 500 +(b) N = 5000 +Figure 6: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Exponential(1) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +X +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 - +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +T +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 - +0.5 +0.5 +X +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +NumberofSub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0 +0.5 +0.5 - +0.5 - +0.5 - +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +4 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 7: Accuracy over 10 datasets: LSNM-sine-tanh and Uniform(−1, 1) noise. +(a) N = 500 +(b) N = 5000 +Figure 8: Accuracy over 10 datasets: LSNM-sine-tanh and Beta(0.5, 0.5) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +NumberofSub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 - +0.5 +0.5 +0.0 +0.0- +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +X +0.5 +0.5 +0.5 +0.5 +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 - +0.5 - +0.5 - +0.0 +0.0- +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 9: Accuracy over 10 datasets: LSNM-sine-tanh and ContinuousBernoulli(0.9) noise. +(a) N = 500 +(b) N = 5000 +Figure 10: Accuracy over 10 datasets: LSNM-sine-tanh and Exponential(1) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +X +0.5 +0.5 +X +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 ++ +x +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 - +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +NumberofSub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 11: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Uniform(−1, 1) noise. +(a) N = 500 +(b) N = 5000 +Figure 12: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Beta(0.5, 0.5) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +4 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 - +1.0 +1.0 +1.0 +Accuracy +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 ++ +4 +7 +10 +500 +75010002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 - +0.5 - +0.5 - +0.5 - +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0- +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 - +0.5 - +0.5 +0.0 +0.0- +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 13: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and ContinuousBernoulli(0.9) noise. +(a) N = 500 +(b) N = 5000 +Figure 14: Accuracy over 10 datasets: LSNM-sigmoid-sigmoid and Exponential(1) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 - +0.0- +0.0- +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 - +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 - +0.5 - +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Numberof Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 - +0.5 +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 15: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Gaussian(0, 1) noise. +(a) N = 500 +(b) N = 5000 +Figure 16: Accuracy over 10 datasets: LSNM-tanh-exp-cosine and Laplace(0, 1) noise. + +1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 - +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 - +1.0 +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +X +0.5 +0.5 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 - +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 +0.5 - +0.5 - +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +PriorsCAREFL-M (0.8) +CAREFL-M (1.0) +X +CAREFL-M (0.8) +CAREFL-M (1.0) +CAREFL-H (0.8) +CAREFL-H (1.0) +6. +CAREFL-H (0.8) +CAREFL-H (1.0)Cause-Effect Inference in Location-Scale Noise Models +(a) N = 500 +(b) N = 5000 +Figure 17: Accuracy over 10 datasets: LSNM-sine-tanh and Gaussian(0, 1) noise. +(a) N = 500 +(b) N = 5000 +Figure 18: Accuracy over 10 datasets: LSNM-sine-tanh and Laplace(0, 1) noise. + +1.0 +1.0 +1.0 +1.0- +1.0 +0.5 +S +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 - +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 +1 +7 +10 +500 +750 +1000 +2000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +NumberofSub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 - +0.5 - +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 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+(a) Accuracy over 100 datasets from SIM sub-benchmark. +(b) Accuracy over 100 datasets from SIM-c sub-benchmark. +(c) Accuracy over 100 datasets from SIM-ln sub-benchmark. +(d) Accuracy over 100 datasets from SIM-G sub-benchmark. +Figure 21: Results with SIM benchmarks. + +1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 - +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +5 +10 +20 ++ +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs +L2-Penalty +Priors1.0 +1.0 +1.0 +1.0 +1.0 +0.5 +0.5 - +0.5 +0.5 +0.5 +0.0 +0.0 +0.0 +0.0 +0.0 +2 +10 +20 +1 +4 +7 +10 +500 +750 +10002000 +0.0 +0.00010.001 +0.1 +laplace +gaussian +Number of Hidden Neurons +Number of Sub-Flows +Number of Epochs 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Independence Testing Xiangyu Sun 1 Oliver Schulte 1 Abstract Location-scale noise models (LSNMs) are a class of heteroscedastic structural causal models with wide applicability, closely related to affine flow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Recent likelihood-based methods de- signed for LSNMs that infer cause-effect relation- ships achieve state-of-the-art accuracy, when their assumptions are satisfied concerning the noise dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' However, under misspecification their accuracy deteriorates sharply, especially when the conditional variance in the anti-causal direction is smaller than that in the causal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' In this pa- per, we demonstrate the misspecification problem and analyze why and when it occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' We show that residual independence testing is much more robust to misspecification than likelihood-based cause-effect inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Our empirical evaluation includes 580 synthetic and 99 real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Introduction Distinguishing cause and effect is a fundamental problem in many disciplines, such as biology, healthcare and fi- nance (Zhang & Chan, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Huang, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Mansouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Randomized controlled trials (RCTs) are the gold standard for finding causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' However, it may be unethical, expensive or even impossible to perform RCTs in real-world domains (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Pearl & Mackenzie, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Causal discovery algorithms aim to find causal rela- tionships given observational data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Traditional causal discovery algorithms can only identify causal relationships up to Markov equivalence classes (MECs) (Spirtes & Gly- mour, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Kalisch & B¨uhlman, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' To break the symmetry in a MEC, additional as- sumptions are needed (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Sch¨olkopf, 2022), such as the type of functional dependence of effect on cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Structural causal models (SCMs) specify a functional class for the causal relations in the data (Pearl, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' In this work, we focus on one particular type of SCM 1Simon Fraser University, Burnaby, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFOT4oBgHgl3EQfzDRI/content/2301.12930v1.pdf'} +page_content=' Correspondence to: Xiangyu Sun