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Published November 2011 | public
Journal Article

A Probabilistic and RIPless Theory of Compressed Sensing

Abstract

This paper introduces a simple and very general theory of compressive sensing. In this theory, the sensing mechanism simply selects sensing vectors independently at random from a probability distribution F; it includes all standard models—e.g., Gaussian, frequency measurements—discussed in the literature, but also provides a framework for new measurement strategies as well. We prove that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from a minimal number of noisy measurements. The novelty is that our recovery results do not require the restricted isometry property (RIP) to hold near the sparsity level in question, nor a random model for the signal. As an example, the paper shows that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.

Additional Information

© 2011 IEEE. Manuscript received November 26, 2010; revised June 16, 2011; accepted June 27, 2011. Date of current version November 11, 2011. This work was supported in part by the Office of Naval Research (ONR) under Grants N00014-09-1-0469 and N00014-08-1-0749, and by the 2006 Waterman Award from the National Science Foundation (NSF). We would like to thank Deanna Needell for a careful reading of the manuscript.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023