Sparse reconstruction via the Reed-Muller Sieve

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 N² 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 ℓ₂/ℓ₂ error bounds derived in this paper are tighter than the ℓ₂/ℓ₁ bounds arising from random ensembles and the ℓ₁/ℓ₁ bounds arising from expander-based ensembles.