A probabilistic and RIPless theory of compressed sensing

13 years 4 months ago

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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 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) -- they make use of a much weaker notion -- or 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 slog n Fourier coefficients that are contaminated with noise. Keywords. Compressed sensing, 1 minimization, the LASSO, the Dantzig select...

Emmanuel J. Candès, Yaniv Plan

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