Diffusion Bridge Mixture Transports, Schrödinger Bridge Problems and Generative Modeling

The dynamic Schr\"odinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a reference process. We propose a novel sampling-based iterative algorithm, the iterated diffusion bridge mixture transport (IDBM), aimed at solving the dynamic Schr\"odinger bridge problem. The IDBM procedure exhibits the attractive property of realizing a valid coupling between the target measures at each step. We perform an initial theoretical investigation of the IDBM procedure, establishing its convergence properties. The theoretical findings are complemented by numerous numerical experiments illustrating the competitive performance of the IDBM procedure across various applications. Recent advancements in generative modeling employ the time-reversal of a diffusion process to define a generative process that approximately transports a simple distribution to the data distribution. As an alternative, we propose using the first iteration of the IDBM procedure as an approximation-free method for realizing this transport. This approach offers greater flexibility in selecting the generative process dynamics and exhibits faster training and superior sample quality over longer discretization intervals. In terms of implementation, the necessary modifications are minimally intrusive, being limited to the training loss computation, with no changes necessary for generative sampling.