New insights for the multivariate square-root lasso

We study the multivariate square-root lasso, a method for fitting the multivariate response (multi-task) linear regression model with dependent errors. This estimator minimizes the nuclear norm of the residual matrix plus a convex penalty. Unlike some existing methods for multivariate response linear regression, which require explicit estimates of the error covariance matrix or its inverse, the multivariate square-root lasso criterion implicitly accounts for error dependence and is convex. To justify the use of this estimator, we establish error bounds which illustrate that like the univariate square-root lasso, the multivariate square-root lasso is pivotal with respect to the unknown error covariance matrix. We propose a new algorithm to compute the estimator: a variation of the alternating direction method of multipliers algorithm; and discuss an accelerated first order algorithm which 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 which estimate both the regression coefficient matrix and error precision matrix.