Support Recovery for Sparse Signals with Unknown Non-stationary Modulation

TL;DR

This paper addresses the challenge of recovering sparse signals with unknown, non-stationary modulations by formulating it as a structured matrix recovery problem and providing theoretical guarantees for support recovery.

Contribution

It introduces a novel lifting-based approach for sparse recovery with non-stationary modulations, deriving conditions for exact support recovery and error bounds.

Findings

The proposed method successfully recovers the support of sparse signals under certain conditions.

Theoretical bounds on recovery errors are established.

Numerical simulations confirm the effectiveness of the approach in practical applications.

Abstract

The problem of estimating a sparse signal from low dimensional noisy observations arises in many applications, including super resolution, signal deconvolution, and radar imaging. In this paper, we consider a sparse signal model with non-stationary modulations, in which each dictionary atom contributing to the observations undergoes an unknown, distinct modulation. By applying the lifting technique, under the assumption that the modulating signals live in a common subspace, we recast this sparse recovery and non-stationary blind demodulation problem as the recovery of a column-wise sparse matrix from structured linear observations, and propose to solve it via block $\ell_{1}$-norm regularized quadratic minimization. Due to observation noise, the sparse signal and modulation process cannot be recovered exactly. Instead, we aim to recover the sparse support of the ground truth signal and bound the recovery errors of the signal's non-zero components and the modulation process. In particular, we derive sufficient conditions on the sample complexity and regularization parameter for exact support recovery and bound the recovery error on the support. Numerical simulations verify and support our theoretical findings, and we demonstrate the effectiveness of our model in the application of single molecule imaging.