New insights for the multivariate square-root lasso

TL;DR

This paper introduces the multivariate square-root lasso, a convex optimization method for multivariate linear regression with dependent errors, which is pivotal and computationally efficient, outperforming traditional methods in simulations and genomic data analysis.

Contribution

It proposes a novel multivariate square-root lasso estimator that implicitly handles error dependence without explicit covariance estimation, with new error bounds and efficient algorithms.

Findings

Performs well in simulations and genomic data applications.

Outperforms methods requiring explicit error covariance estimation.

Is computationally efficient and pivotal with respect to error covariance.

Abstract

We study the multivariate square-root lasso, a method for fitting the multivariate response linear regression model with dependent errors. This estimator minimizes the nuclear norm of the residual matrix plus a convex penalty. Unlike existing methods that require explicit estimates of the error precision (inverse covariance) matrix, the multivariate square-root lasso implicitly accounts for error dependence and is the solution to a convex optimization problem. We establish error bounds which reveal that like the univariate square-root lasso, the multivariate square-root lasso is pivotal with respect to the unknown error covariance matrix. In addition, we propose a variation of the alternating direction method of multipliers algorithm to compute the estimator and discuss an accelerated first order algorithm that can be applied in certain cases. In both simulation studies and a genomic data application, we show that the multivariate square-root lasso can outperform more computationally intensive methods that require explicit estimation of the error precision matrix.