### Ben Recht (University of Wisconsin, Madison, USA)

#### The convex geometry of inverse problems

*Wednesday 27 June 2012 at 15.30, JCMB 6206*

##### Abstract

Deducing the state or structure of a system from partial, noisy measurements
is a fundamental task throughout the sciences and engineering. The resulting
inverse problems are often ill-posed because there are fewer measurements
available than the ambient dimension of the model to be estimated. In practice,
however, many interesting signals or models contain few degrees of freedom
relative to their ambient dimension: a small number of genes may constitute
the signature of a disease, very few parameters may specify the correlation
structure of a time series, or a sparse collection of geometric constraints may
determine a sensor network configuration. Discovering, leveraging, or
recognizing such low-dimensional structure plays an important role in making
inverse problems well-posed.

In this talk, I will propose a unified approach to transform notions of
simplicity and latent low-dimensionality into convex penalty functions. This
approach builds on the success of generalizing compressed sensing to matrix
completion, and greatly extends the catalog of objects and structures that can
be recovered from partial information. I will focus on a suite of data analysis
algorithms designed to decompose general signals into sums of atoms from a
simple - but not necessarily discrete - set. These algorithms are derived in
an optimization framework that encompasses previous methods based on l1-norm
minimization and nuclear norm minimization for recovering sparse vectors and
low-rank matrices. I will provide sharp estimates of the number of generic
measurements required for exact and robust estimation of a variety of
structured models. I will then detail several example applications and
describe how to scale the corresponding algorithms to massive data sets.

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