  We consider a structured estimation problem where an observed matrix is
assumed to be generated as an $s$-sparse linear combination of $N$ given
$n\times n$ positive-semidefinite matrices. Recovering the unknown
$N$-dimensional and $s$-sparse weights from noisy observations is an important
problem in various fields of signal processing and also a relevant
pre-processing step in covariance estimation. We will present related recovery
guarantees and focus on the case of nonnegative weights. The problem is
formulated as a convex program and can be solved without further tuning. Such
robust, non-Bayesian and parameter-free approaches are important for
applications where prior distributions and further model parameters are
unknown. Motivated by explicit applications in wireless communication, we will
consider the particular rank-one case, where the known matrices are outer
products of iid. zero-mean subgaussian $n$-dimensional complex vectors. We show
that, for given $n$ and $N$, one can recover nonnegative $s$--sparse weights
with a parameter-free convex program once $s\leq O(n^2 / \log^2(N/n^2)$. Our
error estimate scales linearly in the instantaneous noise power whereby the
convex algorithm does not need prior bounds on the noise. Such estimates are
important if the magnitude of the additive distortion depends on the unknown
itself.
