Near Minimax Line Spectral Estimation

TL;DR

This paper introduces a nearly optimal semidefinite programming algorithm for line spectral estimation from noisy samples, achieving near-minimax error rates and superior localization accuracy compared to classical methods.

Contribution

It formulates spectral estimation as a sparse recovery problem with an infinite dictionary and provides theoretical guarantees and practical algorithms with near-minimax optimal performance.

Findings

Algorithm nearly attains minimax lower bound on error

Localization error diminishes as sample size increases

Outperforms classical spectral estimation techniques in experiments

Abstract

This paper establishes a nearly optimal algorithm for estimating the frequencies and amplitudes of a mixture of sinusoids from noisy equispaced samples. We derive our algorithm by viewing line spectral estimation as a sparse recovery problem with a continuous, infinite dictionary. We show how to compute the estimator via semidefinite programming and provide guarantees on its mean-square error rate. We derive a complementary minimax lower bound on this estimation rate, demonstrating that our approach nearly achieves the best possible estimation error. Furthermore, we establish bounds on how well our estimator localizes the frequencies in the signal, showing that the localization error tends to zero as the number of samples grows. We verify our theoretical results in an array of numerical experiments, demonstrating that the semidefinite programming approach outperforms two classical spectral estimation techniques.