Low-Rank Toeplitz Matrix Estimation via Random Ultra-Sparse Rulers

TL;DR

This paper introduces random ultra-sparse rulers for estimating low-rank Toeplitz matrices, improving sample complexity bounds and robustness over previous methods by avoiding frequency gap assumptions.

Contribution

The paper proposes a novel approach using random ultra-sparse rulers for Toeplitz matrix estimation, removing the need for frequency gap assumptions and enhancing efficiency.

Findings

Improved sample complexity bounds for Toeplitz estimation.

Enhanced robustness and efficiency demonstrated experimentally.

Theoretical analysis for circulant matrices with random hashing.

Abstract

We study how to estimate a nearly low-rank Toeplitz covariance matrix $T$ from compressed measurements. Recent work of Qiao and Pal addresses this problem by combining sparse rulers (sparse linear arrays) with frequency finding (sparse Fourier transform) algorithms applied to the Vandermonde decomposition of $T$. Analytical bounds on the sample complexity are shown, under the assumption of sufficiently large gaps between the frequencies in this decomposition. In this work, we introduce random ultra-sparse rulers and propose an improved approach based on these objects. Our random rulers effectively apply a random permutation to the frequencies in $T$'s Vandermonde decomposition, letting us avoid frequency gap assumptions and leading to improved sample complexity bounds. In the special case when $T$ is circulant, we theoretically analyze the performance of our method when combined with sparse Fourier transform algorithms based on random hashing. We also show experimentally that our ultra-sparse rulers give significantly more robust and sample efficient estimation then baseline methods.