A survey of the Schr\"odinger problem and some of its connections with optimal transport

By Christian Léonard. 2013

753 citations recorded by the source corpus.

This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature. In addition, some new results are proved.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6609f2a04229c01f939151e6398c28c86b9ca989

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In Mathematical Physics

Citations in the corpus

Entropic optimal transport: convergence of potentials (2021, background): “1We mention that [19] uses the term Schrödinger potentials for fε/ε, gε/ε in the Schrödinger bridge context, as is natural when no parameter ε is present.”
On the difference between entropic cost and the optimal transport cost (2019, neutral): “See [19] and [4].”
On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint (2014, neutral): “In the case where A(t) ≡ 0, the Schrödinger bridge solution process converges to the solution of optimal mass transport problem [31], [29].”
Computational Optimal Transport (2018, methodology): “This formulation was proposed independently by several authors Liero et al. [2016], Chizat et al.”
Improving and generalizing flow-based generative models with minibatch optimal transport (2023, neutral): “Consider the special case of an ODE whose marginal densities are Gaussian: While the ODE that generates these marginal densities is not unique, one of the simplest is the one that satisfies which is unique by the following theorem.”
Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling (2021, background): “The SB problem is a classical problem appearing in applied mathematics, optimal control and probability; see e.g. Föllmer (1988); Léonard (2014b); Chen et al. (2021b).”
Interpolating between Optimal Transport and MMD using Sinkhorn Divergences (2018, methodology): “The idea of adding an entropic barrier KL( · | α ⊗ β ) to the original linear OT program can be traced back to Schrödinger’s problem (Léonard, 2013) and has been used for instance in social sciences (Galichon and Salanié, 2010).”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “the Schrödinger problem); see [28] and the references therein.”
Stability of entropic optimal transport and Schrödinger bridges (2021, background): “We refer to [33] for a more complete review of the literature which dates back to Schrödinger.”
About the analogy between optimal transport and minimal entropy (2015, background): “We recall some general results from [Léo14b, Thm. 2.12] about the solution of the dynamical Schrödinger problem (3.3).”