  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.
