Approximate Support Recovery of Atomic Line Spectral Estimation: A Tale of Resolution and Precision

TL;DR

This paper analyzes the accuracy of atomic norm minimization in super-resolution line spectral estimation, showing it can reliably recover frequencies with high precision under certain conditions, approaching the Cramér-Rao bound.

Contribution

It provides a theoretical analysis of atomic norm minimization's resolution and accuracy, including error bounds and a primal-dual witness construction, linking resolution to estimation precision.

Findings

Atomic norm estimator localizes frequencies within a neighborhood of size $O(rac{ ootlog n}{n^{3/2}} \sigma)$.

Error bounds match the Cramér-Rao lower bound up to a logarithmic factor.

Analysis reveals atomic norm minimization as a convex approach to a nonlinear, nonconvex least-squares problem.

Abstract

This work investigates the parameter estimation performance of super-resolution line spectral estimation using atomic norm minimization. The focus is on analyzing the algorithm's accuracy of inferring the frequencies and complex magnitudes from noisy observations. When the Signal-to-Noise Ratio is reasonably high and the true frequencies are separated by $O(\frac{1}{n})$, the atomic norm estimator is shown to localize the correct number of frequencies, each within a neighborhood of size $O(\sqrt{{\log n}/{n^3}} \sigma)$ of one of the true frequencies. Here $n$ is half the number of temporal samples and $\sigma^2$ is the Gaussian noise variance. The analysis is based on a primal-dual witness construction procedure. The obtained error bound matches the Cram\'er-Rao lower bound up to a logarithmic factor. The relationship between resolution (separation of frequencies) and precision or accuracy of the estimator is highlighted. Our analysis also reveals that the atomic norm minimization can be viewed as a convex way to solve a $\ell_1$-norm regularized, nonlinear and nonconvex least-squares problem to global optimality.