## Abstract

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 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 ||x||_{1} subject to y =Ax, x ε 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 {[0, 1], R+, R, C}, and the results establish our finding for each of the four associated phase transitions.

Original language | English |
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Pages (from-to) | 1181-1186 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 110 |

Issue number | 4 |

DOIs | |

State | Published - 22 Jan 2013 |

Externally published | Yes |