Low Dimensional Atomic Norm Representations in Line Spectral Estimation

TL;DR

This paper addresses the computational challenges of atomic norm-based line spectral estimation by proposing a low-dimensional approach that enables efficient recovery of spectral signals from sub-sampled data.

Contribution

It introduces conditions under which atomic norm minimization can be performed via low-dimensional semidefinite programs, improving computational efficiency.

Findings

Low-dimensional semidefinite representations are possible under certain sub-sampling conditions.

The proposed method can recover signals in poly-logarithmic time for specific sampling patterns.

The relaxation remains tight, ensuring accurate spectral recovery.

Abstract

The line spectral estimation problem consists in recovering the frequencies of a complex valued time signal that is assumed to be sparse in the spectral domain from its discrete observations. Unlike the gridding required by the classical compressed sensing framework, line spectral estimation reconstructs signals whose spectral supports lie continuously in the Fourier domain. If recent advances have shown that atomic norm relaxation produces highly robust estimates in this context, the computational cost of this approach remains, however, the major flaw for its application to practical systems. In this work, we aim to bridge the complexity issue by studying the atomic norm minimization problem from low dimensional projection of the signal samples. We derive conditions on the sub-sampling matrix under which the partial atomic norm can be expressed by a low-dimensional semidefinite program. Moreover, we illustrate the tightness of this relaxation by showing that it is possible to recover the original signal in poly-logarithmic time for two specific sub-sampling patterns.