Localized Schr\"odinger Bridge Sampler

We consider the generative problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. In this paper, we build on previous work combining Schrödinger bridges and Langevin dynamics. A key bottleneck of this approach is the exponential dependence of the required training samples on the dimension, d, of the ambient state space. We propose a localization strategy which exploits conditional independence of conditional expectation values. Localization thus replaces a single high-dimensional Schrödinger bridge problem by d low-dimensional Schrödinger bridge problems over the available training samples. As for the original approach, the localized sampler is stable and geometric ergodic. The sampler also naturally extends to conditional sampling and to Bayesian inference. We demonstrate the performance of our proposed scheme through experiments on a Gaussian problem with increasing dimensions and on a stochastic subgrid-scale parametrization conditional sampling problem. Keywords: generative modeling, Langevin dynamics, Schrödinger bridges, conditional independence, localization, Bayesian inference, conditional sampling, multi-scale closure AMS: 60H10,62F15,62F30,65C05,65C40 1. Introduction In this paper, we consider the problem of sampling from an unknown probability measure ν(dx) on R