  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. This leads to a new optimization
principle that inherits the same SB optimality yet without losing applications
of modern generative training techniques, and we show that the resulting
training algorithm achieves comparable results on generating realistic images
on MNIST, CelebA, and CIFAR10. Our code is available at
https://github.com/ghliu/SB-FBSDE.
