Regression Performance of Group Lasso for Arbitrary Design Matrices

In many linear regression problems, explanatory variables are activated in groups or clusters; group lasso has been proposed for regression in such cases. This paper studies the nonasymptotic regression performance of group lasso using `1/`2 regularization for arbitrary (random or deterministic) design matrices. In particular, the paper establishes under a statistical prior on the set of nonzero coefficients that the `1/`2 group lasso has a near-optimal regression error for all but a vanishingly small set of models. The analysis in the paper relies on three easily computable metrics of the design matrix – coherence, block coherence, and spectral norm. Remarkably, under certain conditions on these metrics, the `1/`2 group lasso can perform near-ideal regression even if the model order scales almost linearly with the number of rows of the design matrix. This is in stark contrast with prior work on the regression performance of the `1/`2 group lasso that only provides linear scaling of the model order for the case of random design matrices. Keywords— Group sparsity, linear regression, group lasso, coherence, block coherence