Coherence and sufficient sampling densities for reconstruction in compressed sensing

TL;DR

This paper introduces a general geometric framework for compressed sensing, providing new bounds on sampling densities that depend on coherence and are applicable to problems like matrix completion.

Contribution

It offers a novel, broad formulation of compressed sensing using analytic varieties and derives universal sampling bounds independent of specific signals.

Findings

Sampling bounds are linear in coherence.

Bounds are logarithmic in ambient dimension.

Applicable to low-rank and distance matrix completion.

Abstract

We give a new, very general, formulation of the compressed sensing problem in terms of coordinate projections of an analytic variety, and derive sufficient sampling rates for signal reconstruction. Our bounds are linear in the coherence of the signal space, a geometric parameter independent of the specific signal and measurement, and logarithmic in the ambient dimension where the signal is presented. We exemplify our approach by deriving sufficient sampling densities for low-rank matrix completion and distance matrix completion which are independent of the true matrix.