Discrete Schrödinger Bridges with Applications to Two-Sample Homogeneity Testing

We introduce an entropy-regularized statistic that defines a divergence between probability distributions. The statistic is the transport cost of a coupling which admits an expression as a weighted average of Monge couplings with respect to a Gibbs measure. This coupling is related to the static Schr\"odinger bridge given a finite number of particles. We establish the asymptotic consistency of the statistic as the sample size goes to infinity and show that the population limit is the solution of F\"ollmer's entropy-regularized optimal transport. The proof technique relies on a chaos decomposition for paired samples. We illustrate the interest of the approach on the two-sample homogeneity testing problem.