# Robust PCA by Manifold Optimization

arXiv: https://arxiv.org/abs/1708.00257 · HTML: https://mlsanity.com/paper/1708.00257/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2017.

## Built on by (most cited first, 13 of 13)

- [LQ-LoRA: Low-rank Plus Quantized Matrix Decomposition for Efficient Language Model Finetuning](https://mlsanity.com/paper/2311.12023/) (2023)
- [Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach](https://mlsanity.com/paper/2109.12701/) (2021)
- [GaLore+: Boosting Low-Rank Adaptation for LLMs with Cross-Head Projection](https://mlsanity.com/paper/2412.19820/) (2024)
- [Unlabeled Principal Component Analysis and Matrix Completion](https://mlsanity.com/paper/2101.09446/) (2021)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [On the Robustness of Cross-Concentrated Sampling for Matrix Completion](https://mlsanity.com/paper/2401.15566/) (2024)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [Robust Regularized Low-Rank Matrix Models for Regression and Classification](https://mlsanity.com/paper/2205.07106/) (2022)
- [Structured Sampling for Robust Euclidean Distance Geometry](https://mlsanity.com/paper/2412.10664/) (2024)
- [Accurate and fast matrix factorization for low-rank learning](https://mlsanity.com/paper/2104.10785/) (2021)
- [SKI to go Faster: Accelerating Toeplitz Neural Networks via Asymmetric Kernels](https://mlsanity.com/paper/2305.09028/) (2023)

## Abstract (arXiv)

Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on manifold optimization. It is shown that, with a proper designed initialization, the proposed algorithms are guaranteed to converge to the underlying low-rank matrix linearly. Compared with a previous work based on the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms reduce the dependence on the conditional number of the underlying low-rank matrix theoretically. Simulations and real data examples confirm the competitive performance of our method.