Group-level support recovery guarantees for group lasso estimator

Abstract

This paper considers the problem of estimating an unknown high dimensional signal from (typically low-dimensional) noisy linear measurements, where the desired unknown signal is assumed to possess a group-sparse structure, i.e. given a (pre-defined) partition of its entries into groups, only a small number of such groups are non-zero. Assuming the unknown group-sparse signal is generated according to a certain statistical model, we provide guarantees under which it can be efficiently estimated via solving the well-known group Lasso problem. In particular, we demonstrate that the set of indices for non-zero groups of the signal (called the group-level support of the signal) can be exactly recovered by solving the proposed group Lasso problem provided that its constituent non-zero groups are small in number and possess enough energy. Our guarantees rely on the well-conditioning of measurement matrix, which is expressed in terms of the block coherence parameter and can be efficiently computed. Our results are non-asymptotic in nature and therefore applicable to practical scenarios.