So it continues, Florent Krzakala, Jean Barbier and Christophe Schülke have a version of their seeded 'magic matrices' expanded to include Hadamard and Fourier blocks instead of random matrices.
We study compressed sensing of real and complex sparse signals using an optimal and computationally efficient procedure based on the combined use of approximate message-passing and spatially-coupled measurement matrices with fast Fourier and Hadamard operators. We compare the performance of our algorithm, that uses deterministic matrices, with the performance of approximate message-passing using purely random Gaussian matrices, for which the asymptotic behavior is exactly given by the density evolution. We show empirically that after proper randomization, the underlying structure of the operators does not significantly affect the performances of the reconstruction, thus allowing a fast and memory efficient reconstruction up to the information theoretic limit.The results are as impressive as two years ago when they broke the Donoho-Tanner phase transition.
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