T-LoHo: A Bayesian Regularization Model for Structured Sparsity and Smoothness on Graphs

Many modern complex data can be represented as a graph. In models dealing with graph-structured data, multivariate parameters are not just sparse but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high dimensional parameters with graphical relations, referred to as a Tree-based Low-rank Horseshoe(T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultaneously. The prior can be embedded in many hierarchical high dimensional models. To illustrate its utility, we apply it to regularize a Bayesian high-dimensional regression problem where the regression coefficients are linked on a graph. The resulting clusters have flexible shapes and satisfy the cluster contiguity constraint with respect to the graph. We design an efficient Markov chain Monte Carlo algorithm that delivers full Bayesian inference with uncertainty measures for model parameters including the number of clusters. We offer theoretical investigations of the clustering effects and posterior concentration results. Finally, we illustrate the performance of the model with simulation studies and real data applications such as anomaly detection in road networks. The results indicate substantial improvements over other competing methods such as sparse fused lasso.