Likelihood Training of Schr\"odinger Bridge using Forward-Backward SDEs Theory

Understand

Schr\"odinger Bridge (SB) is an entropy-regularized optimal transport problem that has received increasing attention in deep generative modeling for its mathematical flexibility compared to the Scored-based Generative Model (SGM).

• However, it remains unclear whether the optimization principle of SB relates to the modern training of deep generative models, which often rely on constructing log-likelihood objectives.This raises questions on the suitability of SB models as a principled alternative for generative applications.
• In this work, we present a novel computational framework for likelihood training of SB models grounded on Forward-Backward Stochastic Differential Equations Theory - a mathematical methodology appeared in stochastic optimal control that transforms the optimality condition of SB into a set of SDEs.
• Crucially, these SDEs can be used to construct the likelihood objectives for SB that, surprisingly, generalizes the ones for SGM as special cases.