A probabilistic and RIPless theory of compressed sensing

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...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Emmanuel J. Candès, Yaniv Plan
• Keywords: CORR 2010, Education, Measurements, Probability Distribution, Sensing