Discrete Diffusion Schr\"odinger Bridge Matching for Graph Transformation

TL;DR

This paper introduces Discrete Diffusion Schr"odinger Bridge Matching (DDSBM), a novel method for graph transformation that leverages continuous-time Markov chains to optimize discrete structures like molecules, with proven convergence and effective property optimization.

Contribution

The paper extends Schr"odinger Bridge methods to discrete domains using Markov chains, specifically applying it to graph transformation and molecular optimization.

Findings

Proved convergence of DDSBM to Schr"odinger Bridge in discrete spaces.

Effectively optimized molecular properties with minimal graph modifications.

Retained key features of molecules during optimization.

Abstract

Transporting between arbitrary distributions is a fundamental goal in generative modeling. Recently proposed diffusion bridge models provide a potential solution, but they rely on a joint distribution that is difficult to obtain in practice. Furthermore, formulations based on continuous domains limit their applicability to discrete domains such as graphs. To overcome these limitations, we propose Discrete Diffusion Schr\"odinger Bridge Matching (DDSBM), a novel framework that utilizes continuous-time Markov chains to solve the SB problem in a high-dimensional discrete state space. Our approach extends Iterative Markovian Fitting to discrete domains, and we have proved its convergence to the SB. Furthermore, we adapt our framework for the graph transformation, and show that our design choice of underlying dynamics characterized by independent modifications of nodes and edges can be interpreted as the entropy-regularized version of optimal transport with a cost function described by the graph edit distance. To demonstrate the effectiveness of our framework, we have applied DDSBM to molecular optimization in the field of chemistry. Experimental results demonstrate that DDSBM effectively optimizes molecules' property-of-interest with minimal graph transformation, successfully retaining other features. Source code is available $\href{https://github.com/junhkim1226/DDSBM}{here}$.