Reed Muller Sensing Matrices and the LASSO

TL;DR

This paper introduces deterministic sensing matrices based on Delsarte-Goethals codes, demonstrating their superior performance over random matrices in sparse signal recovery with LASSO.

Contribution

The authors construct two new families of deterministic sensing matrices with low coherence, spectral norm, and improved recovery performance, including an algorithm for identifying non-orthogonal rows.

Findings

DG frames and sieves outperform Gaussian matrices in signal recovery

DG sieves with m≤15 and r≥2 are tight frames with no duplicate rows

Experimental results confirm improved accuracy and efficiency in sparse recovery

Abstract

We construct two families of deterministic sensing matrices where the columns are obtained by exponentiating codewords in the quaternary Delsarte-Goethals code $DG(m,r)$. This method of construction results in sensing matrices with low coherence and spectral norm. The first family, which we call Delsarte-Goethals frames, are $2^m$ - dimensional tight frames with redundancy $2^{rm}$. The second family, which we call Delsarte-Goethals sieves, are obtained by subsampling the column vectors in a Delsarte-Goethals frame. Different rows of a Delsarte-Goethals sieve may not be orthogonal, and we present an effective algorithm for identifying all pairs of non-orthogonal rows. The pairs turn out to be duplicate measurements and eliminating them leads to a tight frame. Experimental results suggest that all $DG(m,r)$ sieves with $m\leq 15$ and $r\geq2$ are tight-frames; there are no duplicate rows. For both families of sensing matrices, we measure accuracy of reconstruction (statistical 0-1 loss) and complexity (average reconstruction time) as a function of the sparsity level $k$. Our results show that DG frames and sieves outperform random Gaussian matrices in terms of noiseless and noisy signal recovery using the LASSO.