Sparse Reconstruction via The Reed-Muller Sieve

TL;DR

This paper presents the Reed-Muller Sieve, a deterministic measurement matrix for compressed sensing that improves support detection of sparse signals, enables local detection, and offers noise-resilient reconstruction with tighter error bounds.

Contribution

It introduces the Reed-Muller Sieve, a novel deterministic measurement matrix that enhances support detection and local detection in compressed sensing, with improved noise robustness.

Findings

Supports support detection without independence assumptions

Enables local detection with $N^2 \, \log N$ complexity

Provides tighter noise robustness bounds

Abstract

This paper introduces the Reed Muller Sieve, a deterministic measurement matrix for compressed sensing. The columns of this matrix are obtained by exponentiating codewords in the quaternary second order Reed Muller code of length $N$. For $k=O(N)$, the Reed Muller Sieve improves upon prior methods for identifying the support of a $k$-sparse vector by removing the requirement that the signal entries be independent. The Sieve also enables local detection; an algorithm is presented with complexity $N^2 \log N$ that detects the presence or absence of a signal at any given position in the data domain without explicitly reconstructing the entire signal. Reconstruction is shown to be resilient to noise in both the measurement and data domains; the $\ell_2 / \ell_2$ error bounds derived in this paper are tighter than the $\ell_2 / \ell_1$ bounds arising from random ensembles and the $\ell_1 /\ell_1$ bounds arising from expander-based ensembles.