Asymptotics of Discrete Schr\"odinger Bridges via Chaos Decomposition

TL;DR

This paper studies the asymptotic behavior of discrete Schr"odinger bridges in optimal transport, providing convergence results, error estimates, and Gaussian chaos limits using a novel chaos decomposition approach.

Contribution

It introduces a new chaos decomposition method for discrete Schr"odinger bridges, extending Hoeffding's U-statistics theory, and derives precise asymptotic error terms and limit theorems.

Findings

Convergence of Gibbs-weighted matchings to Schr"odinger problem solutions.

First two error terms of orders N^{-1/2} and N^{-1} for the approximation.

Gaussian chaos limits in the case of zero Gaussian variance.

Abstract

Consider the problem of matching two independent i.i.d. samples of size $N$ from two distributions $P$ and $Q$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead in this paper the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with $N$ atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as $N\rightarrow \infty$, to the solution of a variational problem, introduced by F\"ollmer, called the Schr\"odinger problem. We also derive the first two error terms of orders $N^{-1/2}$ and $N^{-1}$, respectively. This gives us central limit theorems for integrated test functions, including for the cost of transport, and second order Gaussian chaos limits when the limiting Gaussian variance is zero. The proofs are based on a novel chaos decomposition of the discrete Schr\"odinger bridge by polynomial functions of the pair of empirical distributions as the first and second order Taylor approximations in the space of measures. This is achieved by extending the Hoeffding decomposition from the classical theory of U-statistics.