Insights and algorithms for the multivariate square-root lasso
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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 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 adapts to dependent errors and is convex. To justify the use of this estimator, we establish an error bound which illustrates that like the univariate square-root lasso (Belloni et al., 2011), the multivariate square-root lasso is pivotal with respect to the unknown error covariance matrix. Based on our theory, we propose a simple tuning approach which requires fitting the model for only a single value of the tuning parameter, i.e., does not require cross-validation. We propose two algorithms to compute the estimator: a prox-linear alternating direction method of multipliers algorithm, and a fast first order algorithm which can be applied in special cases. In both simulation studies and a real 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.
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