A Unified Framework for Diffusion Bridge Problems: Flow Matching and Schr\"{o}dinger Matching into One

The bridge problem is to find an SDE (or sometimes an ODE) that bridges two given distributions. The application areas of the bridge problem are enormous, among which the recent generative modeling (e.g., conditional or unconditional image generation) is the most popular. Also the famous Schr\"{o}dinger bridge problem, a widely known problem for a century, is a special instance of the bridge problem. Two most popular algorithms to tackle the bridge problems in the deep learning era are: (conditional) flow matching and iterative fitting algorithms, where the former confined to ODE solutions, and the latter specifically for the Schr\"{o}dinger bridge problem. The main contribution of this article is in two folds: i) We provide concise reviews of these algorithms with technical details to some extent; ii) We propose a novel unified perspective and framework that subsumes these seemingly unrelated algorithms (and their variants) into one. In particular, we show that our unified framework can instantiate the Flow Matching (FM) algorithm, the (mini-batch) optimal transport FM algorithm, the (mini-batch) Schr\"{o}dinger bridge FM algorithm, and the deep Schr\"{o}dinger bridge matching (DSBM) algorithm as its special cases. We believe that this unified framework will be useful for viewing the bridge problems in a more general and flexible perspective, and in turn can help researchers and practitioners to develop new bridge algorithms in their fields.