In compressed sensing, one takes n < N samples of an N -dimensional vector x0 using an n × N matrix A, obtaining undersampled measurements y = Ax0. For random matrices with Gaussian i.i.d entries, it is known that, when x0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )-phase diagram, convex optimization min ||x||1 subject to y = Ax, x ∈ X N typically finds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property -with the same phase transition location -holds for a wide range of non-Gaussian random matrix ensembles. We report extensive experiments showing that the Gaussian phase transition also describes numerous deterministic matrices, including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Namely, for each of these deterministic matrices in turn, for a typical k-sparse object, we observe that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices. Our experiments considered coefficients constrained to X N for four different sets X ∈ {[0, 1], R+, R, C}, and the results establish our finding for each of the four associated phase transitions.
This class of deterministic matrices is shown to have the same phase transition phenomenon as observed in the Gaussian random matrix case, see REF for more details.
Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America
Medicine Engineering