Title: The Sample Complexity of Toeplitz Covariance Estimation


We study the query complexity of estimating the covariance matrix T of a distribution D over d-dimensional vectors, under the assumption that T is Toeplitz. This assumption is standard in a wide variety of signal processing problems, where the covariance between any two measurements only depends on the time or distance between those measurements. In many of these applications, we are interested in estimation strategies that may choose to view only a subset of entries in each d-dimensional sample from D. We care about minimizing both 1) the number of samples taken and 2) the number of entries accessed in each sample, which often equates to minimizing equipment requirements in applications ranging from wireless transmission to advanced imaging.

We give some of the first nontrivial non-asymptotic bounds on these sample complexity measures. We analyze classical and widely used estimation algorithms, in particular methods based on selecting entries from each sample according to a `sparse ruler'. We explain how sparse ruler based estimation can significantly outperform naive methods when T is close to low-rank, as is often the case in practice. We also develop a novel sampling and estimation strategy that improves on known algorithms in the low-rank case. Our method utilizes tools from random matrix sketching, leverage score based sampling techniques for continuous time signals, and sparse Fourier transform algorithms. 

This work is part of a broader agenda to address fundamental problems in signal processing using tools from theoretical computer science and randomized algorithm design.