Dependent relevance determination for smooth and structured sparse regression

TL;DR

This paper introduces a hierarchical Bayesian model that captures region sparsity in regression weights, improving inference by modeling dependencies and structured priors, with applications demonstrated in brain imaging data.

Contribution

It proposes a novel hierarchical relevance determination framework that models dependencies in sparse regression, combining Gaussian processes and structured priors for enhanced performance.

Findings

Significant improvement over existing methods in simulated data

Effective in brain imaging applications

Provides efficient inference techniques including Laplace and MCMC

Abstract

In many problem settings, parameter vectors are not merely sparse but dependent in such a way that non-zero coefficients tend to cluster together. We refer to this form of dependency as "region sparsity." Classical sparse regression methods, such as the lasso and automatic relevance determination (ARD), which model parameters as independent a priori, and therefore do not exploit such dependencies. Here we introduce a hierarchical model for smooth, region-sparse weight vectors and tensors in a linear regression setting. Our approach represents a hierarchical extension of the relevance determination framework, where we add a transformed Gaussian process to model the dependencies between the prior variances of regression weights. We combine this with a structured model of the prior variances of Fourier coefficients, which eliminates unnecessary high frequencies. The resulting prior encourages weights to be region-sparse in two different bases simultaneously. We develop Laplace approximation and Monte Carlo Markov Chain (MCMC) sampling to provide efficient inference for the posterior. Furthermore, a two-stage convex relaxation of the Laplace approximation approach is also provided to relax the inevitable non-convexity during the optimization. We finally show substantial improvements over comparable methods for both simulated and real datasets from brain imaging.