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arXiv · 1004.2926

Sparse Reconstruction via The Reed-Muller Sieve

Robert Calderbank·Stephen Howard·Sina Jafarpour
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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.

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Robert Calderbank, Stephen Howard, Sina Jafarpour. 2010-04-16. Sparse Reconstruction via The Reed-Muller Sieve. https://arxiv.org/abs/1004.2926

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