  We consider the problem of approximating the set of eigenvalues of the
covariance matrix of a multivariate distribution (equivalently, the problem of
approximating the "population spectrum"), given access to samples drawn from
the distribution. The eigenvalues of the covariance of a distribution contain
basic information about the distribution, including the presence or lack of
structure in the distribution, the effective dimensionality of the
distribution, and the applicability of higher-level machine learning and
multivariate statistical tools. We consider this fundamental recovery problem
in the regime where the number of samples is comparable, or even sublinear in
the dimensionality of the distribution in question. First, we propose a
theoretically optimal and computationally efficient algorithm for recovering
the moments of the eigenvalues of the population covariance matrix. We then
leverage this accurate moment recovery, via a Wasserstein distance argument, to
show that the vector of eigenvalues can be accurately recovered. We provide
finite--sample bounds on the expected error of the recovered eigenvalues, which
imply that our estimator is asymptotically consistent as the dimensionality of
the distribution and sample size tend towards infinity, even in the sublinear
sample regime where the ratio of the sample size to the dimensionality tends to
zero. In addition to our theoretical results, we show that our approach
performs well in practice for a broad range of distributions and sample sizes.
