One-Step Estimator for Permuted Sparse Recovery

This paper considers the unlabeled sparse recovery under multiple measurements, i.e., ${\mathbf{Y}} = {\mathbf{\Pi}}^{\natural} {\mathbf{X}} {\mathbf{B}}^{\natural} + {\mathbf{W}}$, where ${\mathbf{Y}} \in \mathbb{R}^{n\times m}, {\mathbf{\Pi}}^{\natural}\in \mathbb{R}^{n\times n}, {\mathbf{X}} \in \mathbb{R}^{n\times p}, {\mathbf{B}} ^{\natural}\in \mathbb{R}^{p\times m}, {\mathbf{W}}\in \mathbb{R}^{n\times m}$ represents the observations, missing (or incomplete) correspondence information, sensing matrix, sparse signals, and additive sensing noise, respectively. Different from the previous works on multiple measurements ($m > 1$) which all focus on the sufficient samples regime, namely, $n > p$, we consider a sparse matrix $\mathbf{B}^{\natural}$ and investigate the insufficient samples regime (i.e., $n \ll p$) for the first time. To begin with, we establish the lower bound on the sample number and *signal-to-noise ratio* ($ {\mathsf{SNR}}$) for the correct permutation recovery. Moreover, we present a simple yet effective estimator. Under mild conditions, we show that our estimator can restore the correct correspondence information with high probability. Numerical experiments are presented to corroborate our theoretical claims.