Joint variable and rank selection for parsimonious estimation of high-dimensional matrices

Abstract

We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor selection and rank reduction are the most popular strategies for obtaining lower-dimensional approximations of the parameter matrix in such models. We show in this article that important gains in prediction accuracy can be obtained by considering them jointly. We motivate a new class of sparse multivariate regression models, in which the coefficient matrix has low rank and zero rows or can be well approximated by such a matrix. Next, we introduce estimators that are based on penalized least squares, with novel penalties that impose simultaneous row and rank restrictions on the coefficient matrix. We prove that these estimators indeed adapt to the unknown matrix sparsity and have fast rates of convergence. We support our theoretical results with an extensive simulation study and two data analyses.

Published in at http://dx.doi.org/10.1214/12-AOS1039 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Estimation of high-dimensional low-rank matrices
• Optimal selection of reduced rank estimators of high-dimensional matrices
• Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements
• Honest variable selection in linear and logistic regression models via ${\ell }_1$ and ${\ell }_1+{\ell }_2$ penalization
• Consistent group selection in high-dimensional linear regression

Cited by in corpus (6)

• Improved Sparse Low-Rank Matrix Estimation
• High-Dimensional Multivariate Posterior Consistency Under Global-Local Shrinkage Priors
• Optimal large-scale quantum state tomography with Pauli measurements
• Parallel integrative learning for large-scale multi-response regression with incomplete outcomes
• Asymptotic properties of adaptive group Lasso for sparse reduced rank regression
• Simultaneous Heterogeneity and Reduced-rank Learning for Multivariate Response Regression