Iterative signal recovery from incomplete and inaccurate measurements
Joel Tropp

Abstract: Compressive Sampling (CoSa) offers a new paradigm for
acquiring signals whose information content is smaller than the
ambient dimension of the signal space. In particular, CoSa applies to
"compressible signals," those that are well approximated by a short
linear combination of vectors from a fixed orthobasis. A key
observation, due to Candès--Romberg--Tao, is that this class of
signals can be acquired efficiently using random linear sampling
methods that satisfy the "Restricted Isometry Property." The major
algorithmic challenge in CoSa is to approximate a compressible signal
from noisy samples. Until recently, all provably correct
reconstruction techniques have relied on large-scale optimization,
which tends to be computationally burdensome. 

New work, which builds on a recent breakthrough of Needell and
Vershynin, has culminated in an iterative, greedy recovery algorithm
that offers the same guarantees as the best optimization-based
approaches. This novel algorithm provides rigorous guarantees on
computational cost and storage, and it is extremely efficient for many
practical problems because it requires only matrix--vector multiplies
with the sampling matrix. 

This talk commences with a short discussion of Compressive Sampling,
the Restricted Isometry Property (RIP), and some random sampling
operators of practical interest. It presents the new algorithm and
offers comparisons with optimization-based approaches. The remainder
of the two presentations will contain a complete proof that the new
algorithm is correct. This proof, surprisingly, is a  simple exercise
that depends on geometric consequences of the RIP. The material is
suitable for anyone familiar with graduate-level functional analysis.

This research is joint with D. Needell and R. Vershynin.