Generalized Iterative Scaling for Regularized Optimal Transport with Affine Constraints: Application Examples

TL;DR

This paper introduces a generalized iterative scaling algorithm for solving regularized optimal transport problems with affine constraints, providing convergence proofs and applications to various OT problems.

Contribution

It extends iterative scaling methods to handle affine constraints in regularized OT, with convergence analysis and multiple practical application examples.

Findings

Proposed a convergent algorithm for affine-constrained regularized OT.

Applied the method to measure minimization under moment constraints.

Extended the approach to martingale and barycentric weak OT problems.

Abstract

We demonstrate the relevance of an algorithm called generalized iterative scaling (GIS) or simultaneous multiplicative algebraic reconstruction technique (SMART) and its rescaled block-iterative version (RBI-SMART) in the field of optimal transport (OT). Many OT problems can be tackled through the use of entropic regularization by solving the Schr\"odinger problem, which is an information projection problem, that is, with respect to the Kullback--Leibler divergence. Here we consider problems that have several affine constraints. It is well-known that cyclic information projections onto the individual affine sets converge to the solution. In practice, however, even these individual projections are not explicitly available in general. In this paper, we exchange them for one GIS iteration. If this is done for every affine set, we obtain RBI-SMART. We provide a convergence proof using an interpretation of these iterations as two-step affine projections in an equivalent problem. This is done in a slightly more general setting than RBI-SMART, since we use a mix of explicitly known information projections and GIS iterations. We proceed to specialize this algorithm to several OT applications. First, we find the measure that minimizes the regularized OT divergence to a given measure under moment constraints. Second and third, the proposed framework yields an algorithm for solving a regularized martingale OT problem, as well as a relaxed version of the barycentric weak OT problem. Finally, we show an approach from the literature for unbalanced OT problems.