Imaging and Computing Seminar

Yaniv Plan, Applied and Computational Mathematics, Caltech

A probabilistic and RIPless theory of compressed sensing

We introduce a novel, 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 — commonly 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 theory demonstrates that a signal with s nonzero entries can be faithfully recovered from about s log n Fourier coefficients that are contaminated with noise.