Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices
This addresses the problem of slow robust PCA for researchers and practitioners dealing with symmetric positive semidefinite matrices, offering a significant speed-up but being incremental as it builds on existing decomposition methods.
The paper tackles the computational expense of robust PCA for covariance matrices by introducing Denise, a deep learning-based algorithm that learns a function for near-instantaneous decomposition, achieving state-of-the-art quality while being approximately 2000x faster than principal component pursuit and 200x faster than fast PCP.
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.