Exact Sparse Recovery with L0 Projections

TL;DR

This paper introduces a novel sparse recovery method using L0 projections from an alpha-stable distribution, achieving faster and more accurate exact recovery of sparse signals compared to traditional methods, with robustness to noise.

Contribution

The paper proposes a new sparse recovery algorithm based on L0 projections from an alpha-stable distribution, significantly improving speed and accuracy over existing methods.

Findings

Our algorithm is faster than LP and OMP in decoding.

It achieves exact recovery with fewer measurements, especially when K=2.

The method is robust against measurement noise.

Abstract

Many applications concern sparse signals, for example, detecting anomalies from the differences between consecutive images taken by surveillance cameras. This paper focuses on the problem of recovering a K-sparse signal x in N dimensions. In the mainstream framework of compressed sensing (CS), the vector x is recovered from M non-adaptive linear measurements y = xS, where S (of size N x M) is typically a Gaussian (or Gaussian-like) design matrix, through some optimization procedure such as linear programming (LP). In our proposed method, the design matrix S is generated from an $\alpha$-stable distribution with $\alpha\approx 0$. Our decoding algorithm mainly requires one linear scan of the coordinates, followed by a few iterations on a small number of coordinates which are "undetermined" in the previous iteration. Comparisons with two strong baselines, linear programming (LP) and orthogonal matching pursuit (OMP), demonstrate that our algorithm can be significantly faster in decoding speed and more accurate in recovery quality, for the task of exact spare recovery. Our procedure is robust against measurement noise. Even when there are no sufficient measurements, our algorithm can still reliably recover a significant portion of the nonzero coordinates. To provide the intuition for understanding our method, we also analyze the procedure by assuming an idealistic setting. Interestingly, when K=2, the "idealized" algorithm achieves exact recovery with merely 3 measurements, regardless of N. For general K, the required sample size of the "idealized" algorithm is about 5K.