# Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection

arXiv: https://arxiv.org/abs/2110.05649 · HTML: https://mlsanity.com/paper/2110.05649/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 11 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Learned Optimizers that Scale and Generalize](https://mlsanity.com/paper/1703.04813/) (2017)
- [Understanding and correcting pathologies in the training of learned optimizers](https://mlsanity.com/paper/1810.10180/) (2018)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020)
- [Learned D-AMP: Principled Neural Network based Compressive Image Recovery](https://mlsanity.com/paper/1704.06625/) (2017)
- [Deep Unfolded Robust PCA with Application to Clutter Suppression in Ultrasound](https://mlsanity.com/paper/1811.08252/) (2018)
- [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)
- [Training Stronger Baselines for Learning to Optimize](https://mlsanity.com/paper/2010.09089/) (2020)
- [Maximal Sparsity with Deep Networks?](https://mlsanity.com/paper/1605.01636/) (2016)
- [Accelerated Alternating Projections for Robust Principal Component Analysis](https://mlsanity.com/paper/1711.05519/) (2017)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation](https://mlsanity.com/paper/2010.07422/) (2020)
- [Nearly-optimal Robust Matrix Completion](https://mlsanity.com/paper/1606.07315/) (2016)
- [Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition](https://mlsanity.com/paper/2108.10448/) (2021)
- [A Deep-Unfolded Reference-Based RPCA Network For Video Foreground-Background Separation](https://mlsanity.com/paper/2010.00929/) (2020)

## Built on by (most cited first, 30 of 34)

- [ADBench: Anomaly Detection Benchmark](https://mlsanity.com/paper/2206.09426/) (2022)
- [LQ-LoRA: Low-rank Plus Quantized Matrix Decomposition for Efficient Language Model Finetuning](https://mlsanity.com/paper/2311.12023/) (2023)
- [RPCANet: Deep Unfolding RPCA Based Infrared Small Target Detection](https://mlsanity.com/paper/2311.00917/) (2023)
- [Hyperparameter Tuning is All You Need for LISTA](https://mlsanity.com/paper/2110.15900/) (2021)
- [Towards Constituting Mathematical Structures for Learning to Optimize](https://mlsanity.com/paper/2305.18577/) (2023)
- [RPCANet++: Deep Interpretable Robust PCA for Sparse Object Segmentation](https://mlsanity.com/paper/2508.04190/) (2025)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption](https://mlsanity.com/paper/2305.04080/) (2023)
- [Riemannian CUR Decompositions for Robust Principal Component Analysis](https://mlsanity.com/paper/2206.09042/) (2022)
- [Deep Unfolded Tensor Robust PCA with Self-supervised Learning](https://mlsanity.com/paper/2212.11346/) (2022)
- [Deep Unfolding: Recent Developments, Theory, and Design Guidelines](https://mlsanity.com/paper/2512.03768/) (2025)
- [LR-CSNet: Low-Rank Deep Unfolding Network for Image Compressive Sensing](https://mlsanity.com/paper/2212.09088/) (2022)
- [Specularity Factorization for Low-Light Enhancement](https://mlsanity.com/paper/2404.01998/) (2024)
- [A Comprehensive Review of Trends, Applications and Challenges In Out-of-Distribution Detection](https://mlsanity.com/paper/2209.12935/) (2022)
- [On the Robustness of Cross-Concentrated Sampling for Matrix Completion](https://mlsanity.com/paper/2401.15566/) (2024)
- [Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery](https://mlsanity.com/paper/2410.16826/) (2024)
- [Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2502.00463/) (2025)
- [Solving Inverse Problems with Model Mismatch using Untrained Neural Networks within Model-based Architectures](https://mlsanity.com/paper/2403.04847/) (2024)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [Deep Unrolling for Nonconvex Robust Principal Component Analysis](https://mlsanity.com/paper/2307.05893/) (2023)
- [Structured Sampling for Robust Euclidean Distance Geometry](https://mlsanity.com/paper/2412.10664/) (2024)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Working memory inspired hierarchical video decomposition with transformative representations](https://mlsanity.com/paper/2204.10105/) (2022)
- [FedRPCA: Enhancing Federated LoRA Aggregation Using Robust PCA](https://mlsanity.com/paper/2506.01194/) (2025)
- [A Deep-Unfolded Spatiotemporal RPCA Network For L+S Decomposition](https://mlsanity.com/paper/2211.03184/) (2022)
- [Generative Principal Component Regression via Variational Inference](https://mlsanity.com/paper/2409.02327/) (2024)
- [Adaptive Anomaly Detection in Network Flows with Low-Rank Tensor Decompositions and Deep Unrolling](https://mlsanity.com/paper/2409.11529/) (2024)
- [Nuclear Diffusion Models for Low-Rank Background Suppression in Videos](https://mlsanity.com/paper/2509.20886/) (2025)
- [Factor Decorrelation Enhanced Data Removal from Deep Predictive Models](https://mlsanity.com/paper/2509.23443/) (2025)
- [Robust Principal Component Completion](https://mlsanity.com/paper/2603.25132/) (2026)

## Abstract (arXiv)

Robust principal component analysis (RPCA) is a critical tool in modern machine learning, which detects outliers in the task of low-rank matrix reconstruction. In this paper, we propose a scalable and learnable non-convex approach for high-dimensional RPCA problems, which we call Learned Robust PCA (LRPCA). LRPCA is highly efficient, and its free parameters can be effectively learned to optimize via deep unfolding. Moreover, we extend deep unfolding from finite iterations to infinite iterations via a novel feedforward-recurrent-mixed neural network model. We establish the recovery guarantee of LRPCA under mild assumptions for RPCA. Numerical experiments show that LRPCA outperforms the state-of-the-art RPCA algorithms, such as ScaledGD and AltProj, on both synthetic datasets and real-world applications.