Subset Selection for Gaussian Markov Random Fields

TL;DR

This paper addresses the challenge of selecting optimal variable subsets in Gaussian Markov random fields to minimize prediction error, introducing approximation algorithms for different graph classes.

Contribution

It proves NP-hardness for exact subset selection and proposes greedy and message passing algorithms for approximate solutions on specific graph types.

Findings

NP-hardness of exact subset selection in Gaussian free fields

Greedy algorithm provides approximation for arbitrary graphs

Message passing algorithm works on bounded tree-width graphs

Abstract

Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, called Gaussian free fields, which arise in semi-supervised learning and computer vision. We then give a simple greedy approximation algorithm for Gaussian free fields on arbitrary graphs. Finally, we give a message passing algorithm for general Gaussian Markov random fields on bounded tree-width graphs.