## Imaging and Computing Seminar

### Yaniv Plan, Applied and Computational Mathematics, Caltech

**Title:**

A probabilistic and RIPless theory of compressed sensing

**Abstract:**

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.