Asymptotics of Discrete Schr\"odinger Bridges via Chaos Decomposition

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.