Sparse Reconstruction via The Reed-Muller Sieve

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.

To appear in ISIT 2010

Cited by in corpus (1)

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