High-dimensional Gaussian model selection on a Gaussian design

TL;DR

This paper introduces a flexible model selection method for high-dimensional Gaussian regression that achieves optimal rates and adapts to various settings, including when the number of variables exceeds observations.

Contribution

It proposes a penalized least-squares procedure for Gaussian model selection that handles high-dimensional data, incorporates prior knowledge, and provides non-asymptotic guarantees.

Findings

Achieves non-asymptotic oracle inequalities.

Handles p > n scenarios effectively.

Demonstrates adaptiveness and minimax optimality.

Abstract

We consider the problem of estimating the conditional mean of a real Gaussian variable $\nolinebreak Y=\sum_{i=1}^p\nolinebreak\theta_iX_i+\nolinebreak \epsilon$ where the vector of the covariates $(X_i)_{1\leq i\leq p}$ follows a joint Gaussian distribution. This issue often occurs when one aims at estimating the graph or the distribution of a Gaussian graphical model. We introduce a general model selection procedure which is based on the minimization of a penalized least-squares type criterion. It handles a variety of problems such as ordered and complete variable selection, allows to incorporate some prior knowledge on the model and applies when the number of covariates $p$ is larger than the number of observations $n$. Moreover, it is shown to achieve a non-asymptotic oracle inequality independently of the correlation structure of the covariates. We also exhibit various minimax rates of estimation in the considered framework and hence derive adaptiveness properties of our procedure.