A Probabilistic and RIPless Theory of Compressed Sensing

Authors:
Emmanuel J. Candes;Yaniv Plan
Affiliations:
Departments of Mathematics and Statistics, Stanford University, Stanford, CA, USA;Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA, USA
Venue:
IEEE Transactions on Information Theory
Year:
2011

Citing 0
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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.