The Schrödinger bridge problem seeks a process, the Schrödinger bridge, with prescribed initial and terminal distributions, such that the distribution of the Schrödinger bridge minimizes the Kullback-Leibler (KL) divergence to the distribution of a reference process. Schrödinger bridges play a central role in measure transport theory (Marzouk et al., 2016). Notably, it is known that the initial-terminal distribution of a Schrödinger bridge provides a solution to a corresponding entropic optimal transport problem (Peyré & Cuturi, 2020). Schrödinger bridges thus provide an effective framework for finding an alignment between samples from two target distributions. Furthermore, diffusion-based generative models (Ho et al., 2020; Song et al., 2021) can be interpreted as solving trivial instances of the Schrödinger bridge problem (Peluchetti, 2023). Consequently, Schrödinger bridges offer a more general approach to contemporary generative applications. We consider the setting where samples are readily available from both target distributions, and where the reference process is a diffusion process solution to a stochastic differential equation (SDE).

Supervised dimensionality reduction has shown great advantages in finding predictive subspaces. Previous methods rarely consider the popular maximum margin principle and are prone to overfitting to usually small training data, especially for those under the maximum likelihood framework. In this paper, we present a posterior-regularized Bayesian approach to combine Principal Component Analysis (PCA) with the max-margin learning. Based on the data augmentation idea for max-margin learning and the probabilistic interpretation of PCA, our method can automatically infer the weight and penalty parameter of max-margin learning machine, while finding the most appropriate PCA subspace simultaneously under the Bayesian framework. We develop a fast mean-field variational inference algorithm to approximate the posterior. Experimental results on various classification tasks show that our method outperforms a number of competitors.