Weighted $\ell_1$-minimization for generalized non-uniform sparse model

TL;DR

This paper generalizes the non-uniform sparse model in compressed sensing, demonstrating that weighted -minimization with appropriately chosen weights reliably recovers signals from Gaussian measurements.

Contribution

It extends the non-uniform sparse model to a broader setting and provides a method for selecting optimal weights for signal recovery.

Findings

Weighted -minimization recovers signals with high probability.

The method applies to general non-uniform sparse models.

A practical weight selection strategy is proposed.

Abstract

Model-based compressed sensing refers to compressed sensing with extra structure about the underlying sparse signal known a priori. Recent work has demonstrated that both for deterministic and probabilistic models imposed on the signal, this extra information can be successfully exploited to enhance recovery performance. In particular, weighted $\ell_1$-minimization with suitable choice of weights has been shown to improve performance in the so called non-uniform sparse model of signals. In this paper, we consider a full generalization of the non-uniform sparse model with very mild assumptions. We prove that when the measurements are obtained using a matrix with i.i.d Gaussian entries, weighted $\ell_1$-minimization successfully recovers the sparse signal from its measurements with overwhelming probability. We also provide a method to choose these weights for any general signal model from the non-uniform sparse class of signal models.