Relaxed Schrödinger Bridges and Robust Network Routing

Chen, Yongxin; Georgiou, Tryphon T.; Pavon, Michele; Tannenbaum, Allen

doi:10.1109/tcns.2019.2935623

Cited by 12 publications

(20 citation statements)

References 61 publications

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“…In Appendix B we established that the exact free energy and internal energy defined in Eqs. ( 23) and (24), respectively, are monotonic functions of the parameter β, and converge to the actual optimal transport distance d (S 1 , S 2 ) when β → ∞. Here we consider the approximation of those quantities obtained with the saddle point approximation, namely the mean-field values F MF and U MF , and we show that they satisfy the same properties.…”

Section: Appendix D: Proof Of Proposition 3: Monotonicity and Limits Of F Mf (β)mentioning

confidence: 74%

“…Finding such a distance and mapping between probability measures is of relevance to most, if not all data science disciplines. As such, applications of OT have exploded in recent years in domains such as signal and image processing [3][4][5][6], machine learning [7][8][9], computer vision and image analysis [10][11][12][13][14][15], linguistics [16,17], differential geometry [18,19], geometric shape matching [20,21], network analyses [22][23][24], gene expression analyses [25], and the analysis of conformational dynamics of biomolecules [26]. In addition, OT has been expanded to matrix-valued and vectorvalued distributions, with applications in three-dimensional (3D) image comparisons [4,6,27,28].…”

Section: Introductionmentioning

confidence: 99%

“…) and(24), respectively. We now list important properties of U MF In addition, both converge to the optimal variable-mass transport energy d u (S 1 , S 2 ).…”

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confidence: 99%

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Physics approach to the variable-mass optimal-transport problem

2021

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Section: Appendix D: Proof Of Proposition 3: Monotonicity and Limits Of F Mf (β)mentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

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Physics approach to the variable-mass optimal-transport problem

2021

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“…The latter can be introduced in order to derive efficient algorithms for approximately solving (5). However, it has also been shown that the introduction of the entropy term in (5) can be interpreted as finding robust transport plans [17]- [19]. This example is further developed in Section V, where we also demonstrate how bimarginal constraints can be used in order to route the flow from certain sources to certain sinks.…”

Section: A Example: Convex Dynamic Network Flow Problemsmentioning

confidence: 94%

Graph-structured tensor optimization for nonlinear density control and mean field games

Ringh¹,

Haasler²,

Chen³

et al. 2021

Preprint

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In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems, which can be seen as a generalization of multi-marginal optimal transport problems with graph-structured costs, appear in many applications. In particular, we show that it can be used to model and solve nonlinear density control problems, including convex dynamic network flow problems and multi-species potential mean field games. The method is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures we derive efficient methods for computing these projections. In particular, these graph structures are the ones that occur in convex dynamic network flow problems and multi-species potential mean field games. We also illustrate the methodology on numerical examples from these problem classes.

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“…3. In some applications such as computer graphics [229,150], interpolation of images [55], spectral morphing [134], machine learning [174], and network routing [64,64,65], the interpolating flow is essential! Let \{ \mu \ast t ; 0 \leq t \leq 1\} and \{ v \ast (t, x); (t, x) \in [0, 1] \times \BbbR n \} be optimal for (3.5).…”

Section: The Dynamic Problemmentioning

confidence: 99%

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

Chen¹,

Georgiou²,

Pavon³

2021

SIAM Rev.

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Schr\" odinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\" odinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\' ar. The problem, known as the Schr\" odinger bridge problem (SBP) with ``uniform"" prior, was more recently recognized as a regularization of the Monge--Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\" odinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938--1940 specifically for Schr\" odinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\" odinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric.In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\" odinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou--Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview \ast

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Relaxed Schrödinger Bridges and Robust Network Routing

Cited by 12 publications

References 61 publications

Physics approach to the variable-mass optimal-transport problem

Physics approach to the variable-mass optimal-transport problem

Graph-structured tensor optimization for nonlinear density control and mean field games

Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge

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