Optimal structured approximation of Fourier subspaces, Toeplitz matrices, and exponential sums

Abstract

This paper studies three structured approximation problems: (1) Recovering the range of a Fourier matrix from a single observation, (2) Recovering a corrupted low-rank Toeplitz/Hankel matrix, and (3) Recovering a finite exponential sum from noisy samples. All three problems are computationally challenging because their structural constraints are difficult to enforce directly. We show that all three tasks can be solved efficiently and optimally by applying the Gradient-MUSIC algorithm for spectral estimation. To provide an example, for a rank-$r$ Toeplitz matrix $T\in {\mathbb C}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary $E\in {\mathbb C}^{n\times n}$ such that $\|E\|_2\leq αn$, our algorithm outputs a Toeplitz matrix $T_\sharp$ of rank exactly $r$ such that $\|T-T_\sharp\|_2 \leq C \|E\|_2$, where $C,α>0$ are absolute constants. This performance guarantee is minimax optimal in $n$, $r$, and $\|E\|_2$. For the other two structured approximation problems, we also provide algorithms that are minimax optimal in the number of samples, rank/sparsity, and noise level. At the heart of this paper is a quantitative transference principle which shows how to convert computational methods and theory for spectral estimation into corresponding methods and theory for the other three problems.