Estimation of Gaussian graphs by model selection

TL;DR

This paper presents a non-asymptotic method for estimating Gaussian graphs from limited data by model selection, focusing on controlling the maximal degree of the estimated graphs.

Contribution

It introduces a penalized empirical risk approach for Gaussian graph estimation in high-dimensional, small-sample settings, with theoretical performance guarantees.

Findings

Performance bounds depend on sample size and graph degree

Method effectively estimates graphs with degree up to roughly n/(2 log p)

Provides non-asymptotic guarantees for model selection in Gaussian graph estimation

Abstract

We investigate in this paper the estimation of Gaussian graphs by model selection from a non-asymptotic point of view. We start from a n-sample of a Gaussian law P_C in R^p and focus on the disadvantageous case where n is smaller than p. To estimate the graph of conditional dependences of P_C, we introduce a collection of candidate graphs and then select one of them by minimizing a penalized empirical risk. Our main result assess the performance of the procedure in a non-asymptotic setting. We pay a special attention to the maximal degree D of the graphs that we can handle, which turns to be roughly n/(2 log p).