Compressed Sensing off the Grid

TL;DR

This paper introduces an atomic norm minimization method for accurately estimating continuous-valued frequencies of sinusoids from incomplete samples, outperforming grid-based compressed sensing approaches.

Contribution

It proposes a novel atomic norm approach reformulated as a semidefinite program for off-the-grid frequency estimation in compressed sensing.

Findings

Exact recovery with high probability for most sampling sets of size O(s log s log n)

Method works for frequencies in the continuous domain, not on a grid

Numerical experiments confirm effectiveness of the approach

Abstract

We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0,1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples. We reformulate this atomic norm minimization as an exact semidefinite program. Even with this continuous dictionary, we show that most sampling sets of size O(s log s log n) are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Numerical experiments are performed to illustrate the effectiveness of the proposed method.