Diffusion Schr\"odinger Bridge with Applications to Score-Based Generative Modeling

TL;DR

This paper introduces Diffusion Schr"odinger Bridge (DSB), a novel method for generative modeling that efficiently generates data in finite time by solving an entropy-regularized optimal transport problem, improving over existing SDE-based approaches.

Contribution

The paper proposes DSB, an approximation of the IPF procedure for Schr"odinger Bridge problems, enabling finite-time data generation and offering a new computational optimal transport tool.

Findings

DSB recovers previous methods with shorter time intervals.

Subsequent DSB iterations reduce distribution discrepancy.

DSB serves as a continuous analogue of Sinkhorn for optimal transport.

Abstract

Progressively applying Gaussian noise transforms complex data distributions to approximately Gaussian. Reversing this dynamic defines a generative model. When the forward noising process is given by a Stochastic Differential Equation (SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the associated reverse-time SDE may be estimated using score-matching. A limitation of this approach is that the forward-time SDE must be run for a sufficiently long time for the final distribution to be approximately Gaussian. In contrast, solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized optimal transport problem on path spaces, yields diffusions which generate samples from the data distribution in finite time. We present Diffusion SB (DSB), an original approximation of the Iterative Proportional Fitting (IPF) procedure to solve the SB problem, and provide theoretical analysis along with generative modeling experiments. The first DSB iteration recovers the methodology proposed by Song et al. (2021), with the flexibility of using shorter time intervals, as subsequent DSB iterations reduce the discrepancy between the final-time marginal of the forward (resp. backward) SDE with respect to the prior (resp. data) distribution. Beyond generative modeling, DSB offers a widely applicable computational optimal transport tool as the continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi, 2013).