An \ell_1-oracle inequality for the Lasso in finite mixture of multivariate Gaussian regression models

TL;DR

This paper establishes an -oracle inequality for the Lasso estimator in high-dimensional multivariate Gaussian mixture regression models, extending previous results to the multivariate case and focusing on regularization properties.

Contribution

It extends the -oracle inequality for the Lasso to multivariate Gaussian mixture models, emphasizing regularization over variable selection.

Findings

Provides an -oracle inequality for the Lasso in high-dimensional Gaussian mixtures

Extends previous univariate results to multivariate case

Focuses on regularization properties rather than variable selection

Abstract

We consider a multivariate finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates and the size of the response may be much larger than the sample size. We provide an $\ell_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback-Leibler loss. This result is an extension of the $\ell_1$-oracle inequality established by Meynet in \cite{Meynet} in the multivariate case. We focus on the Lasso for its $\ell_1$-regularization properties rather than for the variable selection procedure, as it was done in St\"adler in \cite{Stadler}.