Group Sparse Priors for Covariance Estimation

TL;DR

This paper introduces a hierarchical Bayesian approach for learning block-structured sparse Gaussian graphical models, enabling adaptive group regularization and improved inference on real-world datasets.

Contribution

It proposes a novel prior called group l1 and l1,2 positive definite matrix distributions, allowing flexible group-specific regularization in GGMs.

Findings

Outperforms fixed block structure methods on real data

Enables learning of unknown group assignments in GGMs

Uses variational inference with bounds on partition functions

Abstract

Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation under a novel prior which we call the group l1 and l1,2 positivedefinite matrix distributions. This enables us to build a hierarchical model in which the l1 regularization terms vary depending on which group the entries are assigned to, which in turn allows us to learn block structured sparse GGMs with unknown group assignments. Exact inference in this hierarchical model is intractable, due to the need to compute the normalization constant of these matrix distributions. However, we derive upper bounds on the partition functions, which lets us use fast variational inference (optimizing a lower bound on the joint posterior). We show that on two real world data sets (motion capture and financial data), our method which infers the block structure outperforms a method that uses a fixed block structure, which in turn outperforms baseline methods that ignore block structure.