  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.
