Score matching for bridges without learning time-reversals

TL;DR

This paper introduces a novel score-matching algorithm for conditioned diffusion processes that directly learns the score function without requiring time-reversal, outperforming previous methods in accuracy.

Contribution

The paper presents a new score-matching approach for bridged diffusion processes that avoids learning time-reversals, simplifying the process and improving performance.

Findings

Outperforms existing methods using time-reversal in score learning

Avoids the need to learn time-reversal dynamics

Provides a practical algorithm with open-source code

Abstract

We propose a new algorithm for learning bridged diffusion processes using score-matching methods. Our method relies on reversing the dynamics of the forward process and using this to learn a score function, which, via Doob's $h$-transform, yields a bridged diffusion process; that is, a process conditioned on an endpoint. In contrast to prior methods, we learn the score term $\nabla_x \log p(t, x; T, y)$ directly, for given $t, y$, completely avoiding first learning a time-reversal. We compare the performance of our algorithm with existing methods and see that it outperforms using the (learned) time-reversals to learn the score term. The code can be found at https://github.com/libbylbaker/forward_bridge.