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 dimensional tight frames with redundancy 2. 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 tightframes; 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.