# Riemannian CUR Decompositions for Robust Principal Component Analysis

arXiv: https://arxiv.org/abs/2206.09042 · HTML: https://mlsanity.com/paper/2206.09042/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 17 Jun 2022.

## Builds on (most cited first, 9 of 9)

- [Non-convex Robust PCA](https://mlsanity.com/paper/1410.7660/) (2014)
- [Fast Algorithms for Robust PCA via Gradient Descent](https://mlsanity.com/paper/1605.07784/) (2016)
- [Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection](https://mlsanity.com/paper/2110.05649/) (2021)
- [Accelerated Alternating Projections for Robust Principal Component Analysis](https://mlsanity.com/paper/1711.05519/) (2017)
- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Guarantees of Riemannian Optimization for Low Rank Matrix Completion](https://mlsanity.com/paper/1603.06610/) (2016)
- [Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation](https://mlsanity.com/paper/2010.07422/) (2020)
- [Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion](https://mlsanity.com/paper/2204.03316/) (2022)
- [Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition](https://mlsanity.com/paper/2108.10448/) (2021)

## Built on by (most cited first, 7 of 7)

- [Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling](https://mlsanity.com/paper/2208.09723/) (2022)
- [Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption](https://mlsanity.com/paper/2305.04080/) (2023)
- [On the Robustness of Cross-Concentrated Sampling for Matrix Completion](https://mlsanity.com/paper/2401.15566/) (2024)
- [Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees](https://mlsanity.com/paper/2410.06376/) (2024)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [S3Attention: Improving Long Sequence Attention with Smoothed Skeleton Sketching](https://mlsanity.com/paper/2408.08567/) (2024)
- [Structured Sampling for Robust Euclidean Distance Geometry](https://mlsanity.com/paper/2412.10664/) (2024)

## Abstract (arXiv)

Robust Principal Component Analysis (PCA) has received massive attention in recent years. It aims to recover a low-rank matrix and a sparse matrix from their sum. This paper proposes a novel nonconvex Robust PCA algorithm, coined Riemannian CUR (RieCUR), which utilizes the ideas of Riemannian optimization and robust CUR decompositions. This algorithm has the same computational complexity as Iterated Robust CUR, which is currently state-of-the-art, but is more robust to outliers. RieCUR is also able to tolerate a significant amount of outliers, and is comparable to Accelerated Alternating Projections, which has high outlier tolerance but worse computational complexity than the proposed method. Thus, the proposed algorithm achieves state-of-the-art performance on Robust PCA both in terms of computational complexity and outlier tolerance.