An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm

Abstract

This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.

Entropy-regularized 2-Wasserstein distance between Gaussian measures

“…Moreover, this problem is numerically more favorable to solve (1) compared, for instance, to the Hungarian and the auction algorithm, due to the Sinkhorn-Knopp algorithm. As shown, for instance in [12,19,25,40,72], the above problem has a unique minimizer given by…”

“…In this subsection we summarize well-known results on the Entropy-Kantorich. For further details and proofs, we refer the reader to [25].…”

“…Given a probability measure μ, the class of Entropy-Kantorovich potentials is defined by the set of measurable functions ϕ on R n satisfying [25,30,36,41,50],…”

Optimal transportation, modelling and numerical simulation

“…Using the discretization of Section 1.3, one iteration of (3.13) is quadratic in N. A nice feature of the algorithm is its formal independence on c, unlike the Monge-Ampère and semidiscrete approaches. Details and references can be found in the survey by Vialard (2019) and the recent papers by Di Marino and Gerolin (2020) and Peyré and Cuturi (2019, §4.2). It gives a linear convergence rate in27…”

“…See Benamou et al (2019a) for more on the convergence of the entropic regularization as → 0. For the convergence rate of multi-marginal Sinkhorn, see Di Marino and Gerolin (2020).…”