Exponential convergence of general iterative proportional fitting procedures

TL;DR

This paper proves exponential convergence of iterative proportional fitting procedures (IPFP) for a broad class of optimal transport problems, highlighting the influence of geometric relationships between constraint spaces.

Contribution

It extends convergence results for IPFP to general linear constraints, unifying and broadening prior findings in optimal transport.

Findings

IPFP converges exponentially under linear constraints.

The convergence rate depends on the geometric angle between constraint subspaces.

Stronger convexity in the dual problem enhances convergence speed.

Abstract

Motivated by the success of Sinkhorn's algorithm for entropic optimal transport, we study convergence properties of iterative proportional fitting procedures (IPFP) used to solve more general information projection problems. We establish exponential convergence guarantees for the IPFP whenever the set of probability measures which is projected onto is defined through constraints arising from linear function spaces. This unifies and extends recent results from multi-marginal, adapted and martingale optimal transport. The proofs are based on strong convexity arguments for the dual problem, and the key contribution is to illuminate the role of the geometric interplay between the subspaces defining the constraints. In this regard, we show that the larger the angle (in the sense of Friedrichs) between the linear function spaces, the better the rate of contraction of the IPFP.