  The robust PCA of covariance matrices plays an essential role when isolating
key explanatory features. The currently available methods for performing such a
low-rank plus sparse decomposition are matrix specific, meaning, those
algorithms must re-run for every new matrix. Since these algorithms are
computationally expensive, it is preferable to learn and store a function that
nearly instantaneously performs this decomposition when evaluated. Therefore,
we introduce Denise, a deep learning-based algorithm for robust PCA of
covariance matrices, or more generally, of symmetric positive semidefinite
matrices, which learns precisely such a function. Theoretical guarantees for
Denise are provided. These include a novel universal approximation theorem
adapted to our geometric deep learning problem and convergence to an optimal
solution to the learning problem. Our experiments show that Denise matches
state-of-the-art performance in terms of decomposition quality, while being
approximately $2000\times$ faster than the state-of-the-art, principal
component pursuit (PCP), and $200 \times$ faster than the current
speed-optimized method, fast PCP.
