# Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

arXiv: https://arxiv.org/abs/1711.09492 · HTML: https://mlsanity.com/paper/1711.09492/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2017.

## Built on by (most cited first, 25 of 25)

- [The Ideal Continual Learner: An Agent That Never Forgets](https://mlsanity.com/paper/2305.00316/) (2023)
- [Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices](https://mlsanity.com/paper/2201.12592/) (2022)
- [SubTrack++ : Gradient Subspace Tracking for Scalable LLM Training](https://mlsanity.com/paper/2502.01586/) (2025)
- [Graph-Embedded Subspace Support Vector Data Description](https://mlsanity.com/paper/2104.14370/) (2021)
- [Universal Background Subtraction based on Arithmetic Distribution Neural Network](https://mlsanity.com/paper/2104.08390/) (2021)
- [Unlabeled Principal Component Analysis and Matrix Completion](https://mlsanity.com/paper/2101.09446/) (2021)
- [Robust Principal Component Analysis: A Median of Means Approach](https://mlsanity.com/paper/2102.03403/) (2021)
- [Less is Better: Recovering Intended-Feature Subspace to Robustify NLU Models](https://mlsanity.com/paper/2209.07879/) (2022)
- [Learning Temporal Distribution and Spatial Correlation Towards Universal Moving Object Segmentation](https://mlsanity.com/paper/2304.09949/) (2023)
- [Learning Spatial-Temporal Regularized Tensor Sparse RPCA for Background Subtraction](https://mlsanity.com/paper/2309.15576/) (2023)
- [FAST-PCA: A Fast and Exact Algorithm for Distributed Principal Component Analysis](https://mlsanity.com/paper/2108.12373/) (2021)
- [One Rank at a Time: Cascading Error Dynamics in Sequential Learning](https://mlsanity.com/paper/2505.22602/) (2025)
- [rSVDdpd: A Robust Scalable Video Surveillance Background Modelling Algorithm](https://mlsanity.com/paper/2109.10680/) (2021)
- [ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization](https://mlsanity.com/paper/2307.02745/) (2023)
- [Learning Low-dimensional Latent Dynamics from High-dimensional Observations: Non-asymptotics and Lower Bounds](https://mlsanity.com/paper/2405.06089/) (2024)
- [ALPCAH: Subspace Learning for Sample-wise Heteroscedastic Data](https://mlsanity.com/paper/2505.07272/) (2025)
- [Subspace Defense: Discarding Adversarial Perturbations by Learning a Subspace for Clean Signals](https://mlsanity.com/paper/2403.16176/) (2024)
- [FADE: A Dataset for Detecting Falling Objects around Buildings in Video](https://mlsanity.com/paper/2408.05750/) (2024)
- [FedRPCA: Enhancing Federated LoRA Aggregation Using Robust PCA](https://mlsanity.com/paper/2506.01194/) (2025)
- [PS-Net: Learned Partially Separable Model for Dynamic MR Imaging](https://mlsanity.com/paper/2205.04073/) (2022)
- [Byzantine-Resilient Federated PCA and Low Rank Column-wise Sensing](https://mlsanity.com/paper/2309.14512/) (2023)
- [CURATRON: Complete and Robust Preference Data for Rigorous Alignment of Large Language Models](https://mlsanity.com/paper/2403.02745/) (2024)
- [Tuning-Free Online Robust Principal Component Analysis through Implicit Regularization](https://mlsanity.com/paper/2409.07275/) (2024)
- [Computational Safety for Generative AI: A Hypothesis Testing Perspective](https://mlsanity.com/paper/2502.12445/) (2025)
- [Unsupervised Unfolded rPCA (U2-rPCA): Deep Interpretable Clutter Filtering for Ultrasound Microvascular Imaging](https://mlsanity.com/paper/2510.00660/) (2025)

## Abstract (arXiv)

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.