Some Algorithmic Problems and Results in Compressed Sensing

In Compressed Sensing [9], we consider a signal that is compressible with respect to some dictionary of ’s, that is, its information is concentrated in coefficients . The goal is to reconstruct such signals using only a few measurements , for carefully chosen ’s which depend on . Known results [9], [3], [21] prove that there exists a single measurement matrix such that any compressible signal can be reconstructed from these measurements, with error at most times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying mathematics including Linear Algebra [9], Geometry [21] and Uncertainty Principle [3], and because of its potential applications to signal processing, communication theory and compression. In this paper, we focus on algorithmic aspects of compressed sensing and present new problems and results. For example, we 1) Present a simple, deterministic explicit construction of linear measurements that suffice for compressed sensing.1 2) Introduce functional compressed sensing, that is, extend compressed sensing of a signal to that of functions of the signal, for various functions. 3) Extend compressed sensing to a distributed, continuous model of computing. All the results above are obtained by simple combinatorial or number-theoretic ideas.