# Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices

arXiv: https://arxiv.org/abs/2004.13612 · HTML: https://mlsanity.com/paper/2004.13612/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2020.

## Built on by (most cited first, 3 of 3)

- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Transformers Can Solve Non-Linear and Non-Markovian Filtering Problems in Continuous Time For Conditionally Gaussian Signals](https://mlsanity.com/paper/2310.19603/) (2023)
- [Evaluation of data driven low-rank matrix factorization for accelerated solutions of the Vlasov equation](https://mlsanity.com/paper/2501.04024/) (2024)

## Abstract (arXiv)

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× faster than the state-of-the-art, principal component pursuit (PCP), and 200 × faster than the current speed-optimized method, fast PCP.