Exact Sparse Representation Recovery in Signal Demixing and Group BLASSO

TL;DR

This paper establishes conditions under which convex regularized optimization reliably recovers sparse representations exactly, with applications to signal demixing and super-resolution.

Contribution

It introduces the Metric Non-Degenerate Source Condition (MNDSC) and proves exact sparse recovery in Banach and Hilbert space settings, extending prior results.

Findings

Proves exact sparse recovery under MNDSC with small regularization and noise

Demonstrates practical applications in signal demixing and super-resolution

Shows uniqueness and precise recovery of sparse representations

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.