Reed Muller Sensing Matrices and the LASSO
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 ≤ 15 and r ≥ 2 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.
KeywordsCompressed Sensing Reed-Muller Codes Delsarte- Goethals Set Random Sub-dictionary LASSO
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