Robust principal component analysis?

2011/05/01 by Emmanuel J. Candès, Xiaodong Li, Yi Ma +1 · 6,898 citations
Computer Science · Engineering · Mathematics · #Algorithm #Artificial intelligence #Combinatorics #Component (thermodynamics) #Computer science #Eigenvalues and eigenvectors #Face (sociological concept) #Face and Expression Recognition #Fraction (chemistry) #Mathematical Inequalities and Applications #Mathematical optimization #Mathematics #Matrix (chemical analysis) #Matrix completion #Matrix norm #Norm (philosophy) #Pattern recognition (psychology) #Principal component analysis #Rank (graph theory) #Robust principal component analysis #Sparse PCA #Sparse and Compressive Sensing Techniques #Superposition principle

paper · doi:10.1145/1970392.1970395

published in Journal of the ACM 58(3), 1-37 (Association for Computing Machinery)

openalex publication_date 2011/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

This article is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit ; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the ℓ 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

Citations

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• PRISMA: PRoximal Iterative SMoothing Algorithm
• LCPNet: Latent Consistent Proximal Unfolding Network for Infrared Small Target Detection
• Reduced rank regression via adaptive nuclear norm penalization
• On the Subspace of Image Gradient Orientations
• Tensor Robust Principal Component Analysis: Better recovery with atomic norm regularization
• Clustering-Based Low-Rank Matrix Approximation for Medical Image Compression
• A Scalable, Adaptive and Sound Nonconvex Regularizer for Low-rank Matrix Completion
• Analyzing the Weighted Nuclear Norm Minimization and Nuclear Norm Minimization based on Group Sparse Representation
• Decentralized Dictionary Learning Over Time-Varying Digraphs
• Structured Matrix Recovery via the Generalized Dantzig Selector
• Adaptive Physics-Informed System Modeling with Control for Nonlinear Structural System Estimation
• Smooth optimization using global and local low-rank regularizers
• Low-dose spectral CT reconstruction using L0 image gradient and tensor dictionary
• Transfer Learning from an Auxiliary Discriminative Task for Unsupervised Anomaly Detection
• Provable Self-Representation Based Outlier Detection in a Union of Subspaces
• A sparse decomposition of low rank symmetric positive semi-definite matrices
• Diagonally-Dominant Principal Component Analysis
• Incorporating prior knowledge about structural constraints in model identification
• Robust Logistic Regression using Shift Parameters (Long Version)
• Tensor Completion Algorithms in Big Data Analytics
• Sketching Sparse Matrices
• Hedging parameter selection for basis pursuit
• Exact Tensor Completion from Sparsely Corrupted Observations via Convex Optimization
• Sparse Optimization on General Atomic Sets: Greedy and Forward-Backward Algorithms
• Recursive Sparse Recovery in Large but Structured Noise - Part 2
• Customizing First Person Image Through Desired Actions
• Dual Principal Component Pursuit: Probability Analysis and Efficient Algorithms
• Robust Principal Component Analysis: A Construction Error Minimization Perspective
• Sparse + Low Rank Decomposition of Annihilating Filter-based Hankel Matrix for Impulse Noise Removal
• SA-CNN: Dynamic Scene Classification using Convolutional Neural Networks
• Pain Intensity Estimation by a Self--Taught Selection of Histograms of Topographical Features
• Recovery of Coherent Data via Low-Rank Dictionary Pursuit
• Polar Alignment and Atomic Decomposition
• Analysis of the Optimization Landscapes for Overcomplete Representation Learning
• Compressed dynamic mode decomposition for background modeling
• Linear Disentangled Representation Learning for Facial Actions
• Efficient Algorithms for Robust and Stable Principal Component Pursuit Problems
• Phase diagram of matrix compressed sensing
• Online Optimization for Large-Scale Max-Norm Regularization
• Quaternion Nuclear Norms Over Frobenius Norms Minimization for Robust Matrix Completion
• Subject Information Extraction for Novelty Detection with Domain Shifts
• Robust Covariance Estimation for Approximate Factor Models
• Low-Rank Adaptation Redux for Large Models
• Joint Model and Data Sparsification via the Marginal Likelihood
• Discussion: Latent variable graphical model selection via convex optimization
• The Masked Matrix Separation Problem: A First Analysis
• Unveiling and Mitigating Adversarial Vulnerabilities in Iterative Optimizers
• Iterative Grassmannian optimization for robust image alignment
• Manipulating Machine Learning: Poisoning Attacks and Countermeasures for Regression Learning
• Compressing Complexity: A Critical Synthesis of Structural, Analytical, and Data-Driven Dimensionality Reduction in Dynamical Networks
• A salient dictionary learning framework for activity video summarization via key-frame extraction
• Robust Modifications of U-statistics and Applications to Covariance Estimation Problems
• Quantification of Morphological Features in Non-Contrast Ultrasound Microvasculature Imaging
• Fundamental Performance Limits for Ideal Decoders in High-Dimensional Linear Inverse Problems
• Target Localization With Jammer Removal Using Frequency Diverse Array
• Variable Smoothing for Weakly Convex Problems with Non-Euclidean Directions
• Tensor Laplacian Regularized Low-Rank Representation for Non-uniformly Distributed Data Subspace Clustering
• Interference Mitigation for FMCW Radar With Sparse and Low-Rank Hankel Matrix Decomposition
• Split Bregman Method for Sparse Inverse Covariance Estimation with Matrix Iteration Acceleration
• Complete Dictionary Learning via ℓ_p-norm Maximization
• On the Convergence of Projected-Gradient Methods with Low-Rank Projections for Smooth Convex Minimization over Trace-Norm Balls and Related Problems
• An alternating direction algorithm for matrix completion with nonnegative factors
• SILVar: Single Index Latent Variable Models
• Socializing the Semantic Gap
• Hybrid Subspace Learning for High-Dimensional Data
• VARS-fUSI: Variable Sampling for Fast and Efficient Functional Ultrasound Imaging using Neural Operators
• Robust and Efficient Subspace Segmentation via Least Squares Regression
• A Robust Principal Component Analysis for Outlier Identification in Messy Microcalorimeter Data
• Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering
• Self-paced Principal Component Analysis
• AltGDmin: Alternating GD and Minimization for Partly-Decoupled (Federated) Optimization
• Structured and Unstructured Outlier Identification for Robust PCA: A Fast Parameter Free Algorithm
• Fast dictionary learning from incomplete data
• Hierarchical Robust PCA for Scalable Data Quality Monitoring in Multi-level Aggregation Pipelines
• Study of Anomaly Detection Based on Randomized Subspace Methods in IP Networks
• Deep learning methods for solving linear inverse problems: Research directions and paradigms
• Randomized Robust Subspace Recovery and Outlier Detection for High Dimensional Data Matrices
• Urban Anomaly Analytics: Description, Detection, and Prediction
• Exponentially convergent stochastic k-PCA without variance reduction
• Robust and Scalable Column/Row Sampling from Corrupted Big Data
• Permutation-Invariant Subgraph Discovery
• Multiscale Shrinkage and Lévy Processes
• Estimating Principal Components under Adversarial Perturbations
• Natural Scene Character Recognition Using Robust PCA and Sparse Representation
• Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning
• Fast Robust PCA on Graphs
• Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization
• Attention Strongly Modulates Reliability of Neural Responses to Naturalistic Narrative Stimuli. [europepmc]
• Accelerating 3D-T 1ρ mapping of cartilage using compressed sensing with different sparse and low rank models. [europepmc]
• Compressed sensing acceleration of biexponential 3D-T 1ρ relaxation mapping of knee cartilage. [europepmc]
• Tuning characteristics of low-frequency EEG to positions and velocities in visuomotor and oculomotor tracking tasks. [europepmc]
• A Novel Sparse Compositional Technique Reveals Microbial Perturbations. [europepmc]
• Animal Scanner: Software for classifying humans, animals, and empty frames in camera trap images. [europepmc]
• The utility of multivariate outlier detection techniques for data quality evaluation in large studies: an application within the ONDRI project. [europepmc]
• Accelerated mono- and biexponential 3D-T1ρ relaxation mapping of knee cartilage using golden angle radial acquisitions and compressed sensing. [europepmc]
• Motion correction of chemical exchange saturation transfer MRI series using robust principal component analysis (RPCA) and PCA. [europepmc]
• Quantification of Morphological Features in Non-Contrast-Enhanced Ultrasound Microvasculature Imaging. [europepmc]
• Comprehensive Review of Vision-Based Fall Detection Systems. [europepmc]
• Data-driven cardiovascular flow modelling: examples and opportunities. [europepmc]
• Transcranial photobiomodulation and thermal stimulation induce distinct topographies of EEG alpha and beta power changes in healthy humans. [europepmc]
• MS-MDA: Multisource Marginal Distribution Adaptation for Cross-Subject and Cross-Session EEG Emotion Recognition. [europepmc]
• Synergies are minimally affected during emulation of cerebral palsy gait patterns. [europepmc]
• In vivo lensless microscopy via a phase mask generating diffraction patterns with high-contrast contours. [europepmc]
• Stability of sensorimotor network sculpts the dynamic repertoire of resting state over lifespan. [europepmc]
• EEG-Based Alzheimer's Disease Recognition Using Robust-PCA and LSTM Recurrent Neural Network. [europepmc]
• Neuromodulation of brain power topography and network topology by prefrontal transcranial photobiomodulation. [europepmc]
• Principal Component Pursuit for Pattern Identification in Environmental Mixtures. [europepmc]
• Multi-Step Extracellular Matrix Remodelling and Stiffening in the Development of Idiopathic Pulmonary Fibrosis. [europepmc]
• Whole-brain imaging of freely-moving zebrafish. [europepmc]
• Automatic Analysis of MRI Images for Early Prediction of Alzheimer's Disease Stages Based on Hybrid Features of CNN and Handcrafted Features. [europepmc]
• Quantitative 3D structural analysis of small colloidal assemblies under native conditions by liquid-cell fast electron tomography. [europepmc]
• Biosensors, Artificial Intelligence Biosensors, False Results and Novel Future Perspectives. [europepmc]

Related

• Robust Principal Component Analysis?
• Stable Principal Component Pursuit
• Efficient Algorithms for Robust and Stable Principal Component Pursuit Problems
• Informed Non-convex Robust Principal Component Analysis with Features
• Principal Component Pursuit with Reduced Linear Measurements