Compressed Super-Resolution of Positive Sources

TL;DR

This paper introduces a compressed semidefinite programming approach for positive source super-resolution that reduces computational complexity while maintaining exact recovery guarantees, especially useful in direction finding applications.

Contribution

A novel compressed semidefinite program for positive atomic norm minimization that reduces problem size and computational cost without sacrificing recovery accuracy.

Findings

Achieves exact recovery of positive sources without source separation constraints.

Reduces semidefinite program size from signal dimension to number of sources.

Demonstrates computational savings in direction finding over sparse arrays.

Abstract

Atomic norm minimization is a convex optimization framework to recover point sources from a subset of their low-pass observations, or equivalently the underlying frequencies of a spectrally-sparse signal. When the amplitudes of the sources are positive, a positive atomic norm can be formulated, and exact recovery can be ensured without imposing a separation between the sources, as long as the number of observations is greater than the number of sources. However, the classic formulation of the atomic norm requires to solve a semidefinite program involving a linear matrix inequality of a size on the order of the signal dimension, which can be prohibitive. In this letter, we introduce a novel "compressed" semidefinite program, which involves a linear matrix inequality of a reduced dimension on the order of the number of sources. We guarantee the tightness of this program under certain conditions on the operator involved in the dimensionality reduction. Finally, we apply the proposed method to direction finding over sparse arrays based on second-order statistics and achieve significant computational savings.