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.
Highlights
Let (X, dX ) and (Y, dY ) be Polish spaces, c : X × Y → R be a cost function, ρ1 ∈ P(X ) and ρ2 ∈ P(Y ) be probability measures
Our method extends 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
We link u∗ and v∗ to the solution of the Schrödinger problem; as a byproduct of our results we are able to provide an alternative proof of the convergence of the Sinkhorn algorithm in the 2-marginal case via a purely optimal transportation approach (Theorem 3.1), seeing it as an alternate maximization procedure
Summary
Journal of Scientific Computing (2020) 85:27 where K is the so-called Gibbs Kernel associated with the cost c: K k(x, y)ρ1. Laborde in [18] show the well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schrödinger system in L∞—see (4.8) in Sect. 4—via a local and global inverse function theorems This is a different approach and orthogonal result compared to the study presented in this paper; their result is restricted to measures ρi which are absolutely continuous with respect to some reference measure, with density bounded from above and below. Despite the different approaches and theoretical guarantees obtained in the 2-marginal problem, in the multi-marginal case for continuous measures the situation changes completely.
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