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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-semi-definite 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 preprocessing step in covariance estimation. We will present related recovery guarantees and focus on the case of non-negative 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 sub-Gaussian $n$-dimensional complex vectors. We show that, for given $n$ and $N$, one can recover non-negative $s$-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.

Robust recovery of sparse non-negative weights from mixtures of positive-semi-definite matrices