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 - https://doi.org/10.1073/pnas.1219540110
DO - https://doi.org/10.1073/pnas.1219540110
M3 - مقالة
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 -