TY - JOUR

T1 - Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices

AU - Monajemi, Hatef

AU - Jafarpour, Sina

AU - Gavish, Matan

AU - Donoho, David L.

PY - 2013/1/22

Y1 - 2013/1/22

N2 - In compressed sensing, one takes n<N samples of an N-dimensional vector x0 using an n×N matrix A, obtaining undersampled measurements y =A×0. For random matrices with independent standard Gaussian 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 ε XN 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 XN for four different sets {[0, 1], R+, R, C}, and the results establish our finding for each of the four associated phase transitions.

AB - In compressed sensing, one takes n<N samples of an N-dimensional vector x0 using an n×N matrix A, obtaining undersampled measurements y =A×0. For random matrices with independent standard Gaussian 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 ε XN 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 XN for four different sets {[0, 1], R+, R, C}, and the results establish our finding for each of the four associated phase transitions.

UR - http://www.scopus.com/inward/record.url?scp=84872876101&partnerID=8YFLogxK

U2 - 10.1073/pnas.1219540110

DO - 10.1073/pnas.1219540110

M3 - Article

AN - SCOPUS:84872876101

SN - 0027-8424

VL - 110

SP - 1181

EP - 1186

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 4

ER -