Weighted $\ell_1$ Minimization for Sparse Recovery with Prior Information

TL;DR

This paper introduces a weighted $ ext{l}_1$ minimization method for sparse signal recovery that leverages prior probability information about entries, optimizing weights to improve recovery success in compressed sensing.

Contribution

It develops a new weighted $ ext{l}_1$ minimization algorithm tailored for signals with prior probability info and derives explicit conditions for near-certain recovery as problem size grows.

Findings

Explicit relationship between system parameters and recovery probability

Optimal weights for different prior probabilities

Demonstrated advantages over standard $ ext{l}_1$ minimization

Abstract

In this paper we study the compressed sensing problem of recovering a sparse signal from a system of underdetermined linear equations when we have prior information about the probability of each entry of the unknown signal being nonzero. In particular, we focus on a model where the entries of the unknown vector fall into two sets, each with a different probability of being nonzero. We propose a weighted $\ell_1$ minimization recovery algorithm and analyze its performance using a Grassman angle approach. We compute explicitly the relationship between the system parameters (the weights, the number of measurements, the size of the two sets, the probabilities of being non-zero) so that an iid random Gaussian measurement matrix along with weighted $\ell_1$ minimization recovers almost all such sparse signals with overwhelming probability as the problem dimension increases. This allows us to compute the optimal weights. We also provide simulations to demonstrate the advantages of the method over conventional $\ell_1$ optimization.