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Schrödinger bridge can be viewed as a continuous-time stochastic control problem where the goal is to find an optimally controlled diffusion process whose terminal distribution coincides with a pre-specified target distribution. We propose to generalize this problem by allowing the terminal distribution to differ from the target but penalizing the Kullback-Leibler divergence between the two distributions. We call this new control problem soft-constrained Schrödinger bridge (SSB). The main contribution of this work is a theoretical derivation of the solution to SSB, which shows that the terminal distribution of the optimally controlled process is a geometric mixture of the target and some other distribution. This result is further extended to a time series setting. One application is the development of robust generative diffusion models. We propose a score matching-based algorithm for sampling from geometric mixtures and showcase its use via a numerical example for the MNIST data set. 

We study the Bayesian multi-task variable selection problem, where the goal is to select activated variables for multiple related data sets simultaneously. We propose a new variational Bayes algorithm which generalizes and improves the recently developed “sum of single effects” model of Wang et al. (2020a). Motivated by differential gene network analysis in biology, we further extend our method to joint structure learning of multiple directed acyclic graphical models, a problem known to be computationally highly challenging. We propose a novel order MCMC sampler where our multi-task variable selection algorithm is used to quickly evaluate the posterior probability of each ordering. Both simulation studies and real gene expression data analysis are conducted to show the efficiency of our method. Finally, we also prove a posterior consistency result for multi-task variable selection, which provides a theoretical guarantee for the proposed algorithms. Supplementary materials for this article are available online. 

We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method.