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

Hatef Monajemi, Sina Jafarpour, Matan Gavish, David L. Donoho*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

68 Scopus citations


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.

Original languageAmerican English
Pages (from-to)1181-1186
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number4
StatePublished - 22 Jan 2013
Externally publishedYes


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