Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices

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 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, convergence to an optimal solution of the learning problem and convergence of the training scheme. Our experiments show that Denise matches state-of-the-art performance in terms of decomposition quality, while being approximately 2000x faster than the state-of-the-art, PCP, and 200x faster than the current speed optimized method, fast PCP.