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.

翻译：协方差矩阵的稳健主成分分析在隔离关键解释特征方面发挥着重要作用。目前可用的低秩加稀疏分解方法是针对特定矩阵的，这意味着这些算法必须为每个新矩阵重新运行。由于这些算法计算成本高，因此更可取的做法是学习并存储一个函数，该函数在评估时能近乎瞬时地执行此分解。因此，我们引入了Denise——一种基于深度学习的算法，用于协方差矩阵（或更一般地，对称正半定矩阵）的稳健主成分分析，该算法恰好学习了这样一个函数。我们为Denise提供了理论保证，包括一个适用于我们的几何深度学习问题的新型通用逼近定理，以及收敛到学习问题的最优解。实验表明，Denise在分解质量方面达到了最先进的性能，同时比当前最先进的主成分追踪（PCP）方法快约2000倍，比当前的加速优化方法（fast PCP）快200倍。