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A survey of the Schr\"odinger problem and some of its connections with optimal transport
TLDR
In this paper, the authors present the Schrodinger problem and some of its connections with optimal transport, and give a user's guide to the problem and a survey of the related literature.Abstract:
This article is aimed at presenting the Schrodinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schrodinger problem. We also give a survey of the related literature. In addition, some new results are proved.read more
Citations
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Iterative Bregman Projections for Regularized Transportation Problems
TL;DR: In this article, a general numerical framework to approximate so-lutions to linear programs related to optimal transport is presented, where the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form.
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On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint
TL;DR: In this article, the relation between the optimal transport problem and the Schrodinger bridge problem from a stochastic control perspective was investigated and connections between the two problems were made.
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Diffusion Models: A Comprehensive Survey of Methods and Applications
Ling Yang,Zhilong Zhang,Shenda Hong,Runsheng Xu,Yue Zhao,Yingxia Shao,Wentao Zhang,Min Yang,Bin Cui +8 more
TL;DR: A comprehensive review of existing variants of the diffusion models and a thorough investigation into the applications of diffusion models, including computer vision, natural language processing, waveform signal processing, multi-modal modeling, molecular graph generation, time series modeling, and adversarial purification.
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Entropic Approximation of Wasserstein Gradient Flows
TL;DR: A novel numerical scheme to approximate gradient flows for optimal transport using an entropic regularization of the transportation coupling, which allows one to trade the initial Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to deal with numerically.
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Entropic and Displacement Interpolation
TL;DR: In this paper, the Schroźdinger bridge problem (SBP) is viewed as a stochastic regularization of OMT and can be cast as a control problem of steering the probability density of the state vector of a dynamical system between two marginals.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Journal ArticleDOI
$I$-Divergence Geometry of Probability Distributions and Minimization Problems
TL;DR: In this article, the minimum discrimination information problem is viewed as projecting a PD onto a convex set of PD's and useful existence theorems for and characterizations of the minimizing PD are arrived at.
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A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
Jean-David Benamou,Yann Brenier +1 more
TL;DR: The Monge-Kantorovich mass transfer problem is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.
Journal ArticleDOI
The variational formulation of the Fokker-Planck equation
TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of the probability density for a stochastic process associated with an Ito Stochastic Differential Equation.
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Dynamical Theories of Brownian Motion
TL;DR: In a course of lectures given by Professor Nelson at Princeton during the spring term of 1966, the authors traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations.