Pacific Institute for the Mathematical Sciences - PIMS

One of the central tenets of signal processing is the Shannon/Nyquist
sampling theory: the number of samples needed to reconstruct a signal
without error is dictated by its bandwidth-the length of the shortest
interval which contains the support of the spectrum of the signal under
study. Very recently, an alternative sampling or sensing theory has
emerged which goes against this conventional wisdom. This theory allows
the faithful recovery of signals and images from what appear to be
highly incomplete sets of data, i.e. from far fewer data bits than
traditional methods use. Underlying this methodology is a concrete
protocol for sensing and compressing data simultaneously.

This talk will present the key mathematical ideas underlying this new
sampling or sensing theory, and will survey some of the most important
results. We will argue that this is a robust mathematical theory; not
only is it possible to recover signals accurately from just an
incomplete set of measurements, but it is also possible to do so when
the measurements are unreliable and corrupted by noise. We will see
that the reconstruction algorithms are very concrete, stable (in the
sense that they degrade smoothly as the noise level increases) and
practical; in fact, they only involve solving very simple convex
optimization programs.

An interesting aspect of this theory is that it has bearings on some
fields in the applied sciences and engineering such as statistics,
information theory, coding theory, theoretical computer science, and
others as well. If time allows, we will try to explain these
connections via a few selected examples.

Emmanuel Candes is a Ronald and Maxine Linde Professor of Applied and
Computational Mathematics in the Division of Engineering and Applied
Science at the California Institute of Technology. He is a recent
recipient of the National Science Board's prestigious Alan T. Waterman
Award, the highest honour awarded by the National Science Foundation.
His research interests include compressive sampling, mathematical
signal processing, computational harmonic analysis, approximation
theory, multiscale analysis, and statistical estimation and detection.

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