Discrete Diffusion Schr\"odinger Bridge Matching for Graph Transformation

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 Schrödinger Bridge Matching (DDSBM), a novel framework that utilizes continuous-time Markov chains to solve the SB problem in a highdimensional 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. Transporting an initial distribution to a target distribution is a foundational concept in modern generative modeling. Denoising diffusion models (DDMs) have been highly influential in this area, with a primary focus on generating data distributions from simple prior (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020; Song et al., 2020; Kim et al., 2024b). Despite their promising results, setting the initial distribution as a simple prior makes DDMs hard to work in tasks where the initial distribution becomes a data distribution, such as image-to-image translation. To tackle this, diffusion bridge models (DBMs) extend DDMs to transport data between arbitrary distributions (Liu & Wu, 2023; Liu et al., 2023; Zhou et al., 2023).