Estimation of Shortest Path Covariance Matrices

We study the sample complexity of estimating the covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{d\times d}$ of a distribution $\mathcal D$ over $\mathbb{R}^d$ given independent samples, under the assumption that $\mathbf{\Sigma}$ is graph-structured. In particular, we focus on shortest path covariance matrices, where the covariance between any two measurements is determined by the shortest path distance in an underlying graph with $d$ nodes. Such matrices generalize Toeplitz and circulant covariance matrices and are widely applied in signal processing applications, where the covariance between two measurements depends on the (shortest path) distance between them in time or space. We focus on minimizing both the vector sample complexity: the number of samples drawn from $\mathcal{D}$ and the entry sample complexity: the number of entries read in each sample. The entry sample complexity corresponds to measurement equipment costs in signal processing applications. We give a very simple algorithm for estimating $\mathbf{\Sigma}$ up to spectral norm error $\epsilon \left\|\mathbf{\Sigma}\right\|_2$ using just $O(\sqrt{D})$ entry sample complexity and $\tilde O(r^2/\epsilon^2)$ vector sample complexity, where $D$ is the diameter of the underlying graph and $r \le d$ is the rank of $\mathbf{\Sigma}$. Our method is based on extending the widely applied idea of sparse rulers for Toeplitz covariance estimation to the graph setting. In the special case when $\mathbf{\Sigma}$ is a low-rank Toeplitz matrix, our result matches the state-of-the-art, with a far simpler proof. We also give an information theoretic lower bound matching our upper bound up to a factor $D$ and discuss some directions towards closing this gap.