Improved Support Recovery Guarantees for the Group Lasso With Applications to Structural Health Monitoring

This paper considers the problem of estimating an unknown high dimensional signal from noisy linear measurements, {when} the signal is assumed to possess a \emph{group-sparse} structure in a {known,} fixed dictionary. We consider signals generated according to a natural probabilistic model, and establish new conditions under which the set of indices of the non-zero groups of the signal (called the group-level support) may be accurately estimated via the group Lasso. Our results strengthen existing coherence-based analyses that exhibit the well-known "square root" bottleneck, allowing for the number of recoverable nonzero groups to be nearly as large as the total number of groups. We also establish a sufficient recovery condition relating the number of nonzero groups and the signal to noise ratio (quantified in terms of the ratio of the squared Euclidean norms of nonzero groups and the variance of the random additive {measurement} noise), and validate this trend empirically. Finally, we examine the implications of our results in the context of a structural health monitoring application, where the group Lasso approach facilitates demixing of a propagating acoustic wavefield, acquired on the material surface by a scanning laser Doppler vibrometer, into antithetical components, one of which indicates the locations of internal material defects.