Support Recovery for Sparse Recovery and Non-stationary Blind Demodulation

The problem of estimating a sparse signal from its modulated low-dimensional observations appears naturally in many applications like signal deconvolution and self-calibration. In the presence of noise, however, the sparse signal and modulation parameters cannot be exactly recovered. Therefore, in this paper, we study the support recovery problem for the sparse recovery and non-stationary blind demodulation model, in which each dictionary atom composing the low-dimension observations undergoes a distinct modulation process. Using the lifting technique and with the assumption that the modulating signals live in a common subspace, we reformulate this problem as one of recovering a column-wise sparse matrix from structured linear observations. We propose to solve the reformulated problem via block ℓ
₁
norm regularized quadratic minimization. We derive sufficient conditions on the sample complexity and regularization parameter such that the support of the ground truth sparse signal can be exactly recovered with overwhelming probability, and we bound the recovery error of the sparse signal and modulation process on the support. Simulations illustrate and support our theoretical results.