Deterministic Matrices Matching The Compressed Sensing Phase Transitions Of Gaussian Random Matrices

In compressed sensing, one takes n < N samples of an N-dimensional vector x(0) using an n x N matrix A, obtaining undersampled measurements y = Ax(0). For random matrices with independent standard Gaussian entries, it is known that, when x(0) 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 parallel to x parallel to(1) subject to y = Ax, x is an element of X-N typically finds the sparsest solution, whereas 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 is an element of {[0, 1], R+, R, C}, and the results establish our finding for each of the four associated phase transitions.