Exact Sparse Representation Recovery in Signal Demixing and Group BLASSO

Abstract: In this short article we present the theory of sparse representations recovery in convex regularized optimization problems introduced in (Carioni and Del Grande, arXiv:2311.08072, 2023). We focus on the scenario where the unknowns belong to Banach spaces and measurements are taken in Hilbert spaces, exploring the properties of minimizers of optimization problems in such settings. Specifically, we analyze a Tikhonov-regularized convex optimization problem, where $y_0$ are the measured data, $w$ denotes the noise, and $\lambda$ is the regularization parameter. By introducing a Metric Non-Degenerate Source Condition (MNDSC) and considering sufficiently small $\lambda$ and $w$, we establish Exact Sparse Representation Recovery (ESRR) for our problems, meaning that the minimizer is unique and precisely recovers the sparse representation of the original data. We then emphasize the practical implications of this theoretical result through two novel applications: signal demixing and super-resolution with Group BLASSO. These applications underscore the broad applicability and significance of our result, showcasing its potential across different domains.