Tight Bounds for Schr\"odinger Potential Estimation in Unpaired Data Translation

Abstract

Modern methods of generative modelling and unpaired data translation based on Schr\"odinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from the initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we choose an Ornstein-Uhlenbeck process as the reference one and estimate the corresponding Schr\"odinger potential. Introducing a risk function as the Kullback-Leibler divergence between couplings, we derive tight bounds on the generalization ability of an empirical risk minimizer over a class of Schr\"odinger potentials, including Gaussian mixtures. Thanks to the mixing properties of the Ornstein-Uhlenbeck process, we almost achieve fast rates of convergence, up to some logarithmic factors, in favourable scenarios. We also illustrate the performance of the suggested approach with numerical experiments.