Exponential convergence rate for Iterative Markovian Fitting

TL;DR

This paper proves that the Iterative Markovian Fitting algorithm converges exponentially fast to the true solution in the discrete-time Schrödinger bridge problem on finite state spaces, providing an explicit rate of convergence.

Contribution

It establishes for the first time that IMF has an exponential convergence rate with an explicit contraction factor.

Findings

IMF converges exponentially in Kullback-Leibler divergence.

Explicit contraction factor for IMF convergence is derived.

Convergence speed of IMF is now quantitatively characterized.

Abstract

We consider the discrete-time Schr\"odinger bridge problem on a finite state space. Although it has been known that the Iterative Markovian Fitting (IMF) algorithm converges in Kullback-Leibler divergence to the ground truth solution, the speed of that convergence remained unquantified. In this work, we establish for the first time that IMF exhibits exponential convergence with an explicit contraction factor.