Stable generative modeling using Schr\"odinger bridges

TL;DR

This paper introduces a novel generative modeling approach combining Schr"odinger bridges with Langevin dynamics, enabling stable and efficient sampling from complex distributions using a split-step scheme and reference processes.

Contribution

The paper proposes a new generative model that integrates Schr"odinger bridges with Langevin dynamics, addressing stability and convex hull constraints in sampling.

Findings

Effective sampling from high-dimensional distributions.

Stable and efficient due to the split-step scheme.

Versatile extension to conditional sampling and Bayesian inference.

Abstract

We consider the problem of sampling from an unknown distribution for which only a sufficiently large number of training samples are available. Such settings have recently drawn considerable interest in the context of generative modelling and Bayesian inference. In this paper, we propose a generative model combining Schr\"odinger bridges and Langevin dynamics. Schr\"odinger bridges over an appropriate reversible reference process are used to approximate the conditional transition probability from the available training samples, which is then implemented in a discrete-time reversible Langevin sampler to generate new samples. By setting the kernel bandwidth in the reference process to match the time step size used in the unadjusted Langevin algorithm, our method effectively circumvents any stability issues typically associated with the time-stepping of stiff stochastic differential equations. Moreover, we introduce a novel split-step scheme, ensuring that the generated samples remain within the convex hull of the training samples. Our framework can be naturally extended to generate conditional samples and to Bayesian inference problems. We demonstrate the performance of our proposed scheme through experiments on synthetic datasets with increasing dimensions and on a stochastic subgrid-scale parametrization conditional sampling problem as well as generating sample trajectories of a dynamical system using conditional sampling.