Near Minimax Line Spectral Estimation

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.

Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision

“…To solve atomic norm minimizations numerically, the authors of [5,10] (see also [1]) first proposed to reformulate the atomic norm (2.3) as an equivalent semidefinite program. Other numerical schemes are studied in [11,12,13,14].…”

“…Given the noisy observation model (2.1), it is natural to denoise x by solving the atomic norm regularized minimization program [15,5]:…”

“…For the special line spectral estimation problem, where the atomic norm is induced by the set of parameterized complex exponentials, atomic regularizations have been shown to achieve optimal performance for several signal processing tasks. For instance, atomic norm minimization recovers a spectrally sparse signal from a minimal number of random signal samples [4], identifies and removes a maximal number of outliers [6,7], and perform denoising with an error approaching the minimax rate [5]. When multiple measurement vectors are available, a method of exploiting the joint sparsity pattern of different signals to further improve estimation accuracy are proposed in [33] and [34].…”

“…The work [35] developed error bounds on spectral support recovery for bounded noise. In [5], the authors derived suboptimal bounds for the Gaussian noise model. In [36], the authors extended this line of research to general measurement schemes beyond Fourier samples using the Beurling-LASSO (B-LASSO) program.…”

“…This line of work promotes the use of a convex sparse regularizer to solve inverse problems involving spectrally sparse signals, distinguishing them from classical methods based on root finding and singular value decompositions (e.g., Prony's method, MUSIC, ESPIRIT, Matrix Pencil, etc.). The convex regularizer, a special instance of the general atomic norms, has been shown to achieve optimal performance in signal completion [4], denoising [5], and outlier removal [6,7]. For these signal processing tasks, either one can recover the spectral signal exactly (and hence extract the true frequencies precisely), or the error metric is defined using the signal instead of the frequency parameters.…”