  We consider the problem of estimating the covariance structure of a random
vector $Y\in \mathbb R^d$ from a sample $Y_1,\ldots,Y_n$. We are interested in
the situation when $d$ is large compared to $n$ but the covariance matrix
$\Sigma$ of interest has (exactly or approximately) low rank. We assume that
the given sample is (a) $\epsilon$-adversarially corrupted, meaning that
$\epsilon$ fraction of the observations could have been replaced by arbitrary
vectors, or that (b) the sample is i.i.d. but the underlying distribution is
heavy-tailed, meaning that the norm of $Y$ possesses only finite fourth
moments. We propose an estimator that is adaptive to the potential low-rank
structure of the covariance matrix as well as to the proportion of contaminated
data, and admits tight deviation guarantees despite rather weak assumptions on
the underlying distribution. Finally, we discuss the algorithms that allow to
approximate the proposed estimator in a numerically efficient way.
