New Insights 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 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.

Original languageEnglish (US)
JournalJournal of Machine Learning Research
StatePublished - 2022
Externally publishedYes

Bibliographical note

Funding Information:
The author thanks three anonymous referees and the action editor for their many helpful comments. The author also thanks Benjamin Stucky and Sara van de Geer for sharing their code and their responses to inquiries; thanks Rohit K. Patra for a helpful conservation; and thanks Daniel J. Eck, Karl Oskar Ekvall, Keshav Motwani, Bradley S. Price, Adam J. Rothman, and Ben Sherwood for their feedback on earlier drafts of this article. This work was supported in part by National Science Foundation grant DMS-2113589.

Publisher Copyright:
© 2022 Aaron J. Molstad.


  • convex optimization
  • covariance matrix estimation
  • multivariate response linear regression
  • pivotal estimation


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