Sparse Methods for Automatic Relevance Determination

TL;DR

This paper analyzes and compares methods for inducing sparsity in Bayesian regression, focusing on automatic relevance determination (ARD) and its extensions, with theoretical insights and empirical evaluations for nonlinear system identification.

Contribution

It provides a detailed analysis of ARD, introduces regularization and thresholding methods to improve sparsity, and compares their performance in linear problems with many basis functions.

Findings

Regularization and thresholding enhance ARD's sparsity.

Favorable performance of methods with orthogonal covariates.

Empirical results demonstrate advantages and limitations of proposed methods.

Abstract

This work considers methods for imposing sparsity in Bayesian regression with applications in nonlinear system identification. We first review automatic relevance determination (ARD) and analytically demonstrate the need to additional regularization or thresholding to achieve sparse models. We then discuss two classes of methods, regularization based and thresholding based, which build on ARD to learn parsimonious solutions to linear problems. In the case of orthogonal covariates, we analytically demonstrate favorable performance with regards to learning a small set of active terms in a linear system with a sparse solution. Several example problems are presented to compare the set of proposed methods in terms of advantages and limitations to ARD in bases with hundreds of elements. The aim of this paper is to analyze and understand the assumptions that lead to several algorithms and to provide theoretical and empirical results so that the reader may gain insight and make more informed choices regarding sparse Bayesian regression.