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177
Robust principal component analysis?
 Journal of the ACM,
, 2011
"... Abstract This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank 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 lowrank and the ..."
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Cited by 569 (26 self)
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Abstract This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank 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 lowrank 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 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.
Synthesizing physically realistic human motion in lowdimensional, behaviorspecific spaces
 ACM Transactions on Graphics
, 2004
"... Optimization is an appealing way to compute the motion of an animated character because it allows the user to specify the desired motion in a sparse, intuitive way. The difficulty of solving this problem for complex characters such as humans is due in part to the high dimensionality of the search sp ..."
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Cited by 196 (13 self)
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Optimization is an appealing way to compute the motion of an animated character because it allows the user to specify the desired motion in a sparse, intuitive way. The difficulty of solving this problem for complex characters such as humans is due in part to the high dimensionality of the search space. The dimensionality is an artifact of the problem representation because most dynamic human behaviors are intrinsically low dimensional with, for example, legs and arms operating in a coordinated way. We describe a method that exploits this observation to create an optimization problem that is easier to solve. Our method utilizes an existing motion capture database to find a lowdimensional space that captures the properties of the desired behavior. We show that when the optimization problem is solved within this lowdimensional subspace, a sparse sketch can be used as an initial guess and full physics constraints can be enabled. We demonstrate the power of our approach with examples of forward, vertical, and turning jumps; with running and walking; and with several acrobatic flips.
RASL: Robust Alignment by Sparse and Lowrank Decomposition for Linearly Correlated Images
, 2010
"... This paper studies the problem of simultaneously aligning a batch of linearly correlated images despite gross corruption (such as occlusion). Our method seeks an optimal set of image domain transformations such that the matrix of transformed images can be decomposed as the sum of a sparse matrix of ..."
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Cited by 161 (6 self)
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This paper studies the problem of simultaneously aligning a batch of linearly correlated images despite gross corruption (such as occlusion). Our method seeks an optimal set of image domain transformations such that the matrix of transformed images can be decomposed as the sum of a sparse matrix of errors and a lowrank matrix of recovered aligned images. We reduce this extremely challenging optimization problem to a sequence of convex programs that minimize the sum of ℓ1norm and nuclear norm of the two component matrices, which can be efficiently solved by scalable convex optimization techniques with guaranteed fast convergence. We verify the efficacy of the proposed robust alignment algorithm with extensive experiments with both controlled and uncontrolled real data, demonstrating higher accuracy and efficiency than existing methods over a wide range of realistic misalignments and corruptions.
Robust principal component analysis: Exact recovery of corrupted lowrank matrices via convex optimization
 Advances in Neural Information Processing Systems 22
, 2009
"... The supplementary material to the NIPS version of this paper [4] contains a critical error, which was discovered several days before the conference. Unfortunately, it was too late to withdraw the paper from the proceedings. Fortunately, since that time, a correct analysis of the proposed convex prog ..."
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Cited by 149 (4 self)
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The supplementary material to the NIPS version of this paper [4] contains a critical error, which was discovered several days before the conference. Unfortunately, it was too late to withdraw the paper from the proceedings. Fortunately, since that time, a correct analysis of the proposed convex programming relaxation has been developed by Emmanuel Candes of Stanford University. That analysis is reported in a joint paper, Robust Principal Component Analysis? by Emmanuel Candes, Xiaodong Li, Yi Ma and John Wright,
Segmenting Motion Capture Data into Distinct Behaviors
 In Graphics Interface
, 2004
"... Much of the motion capture data used in animations, commercials, and video games is carefully segmented into distinct motions either at the time of capture or by hand after the capture session. As we move toward collecting more and longer motion sequences, however, automatic segmentation techniques ..."
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Cited by 137 (5 self)
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Much of the motion capture data used in animations, commercials, and video games is carefully segmented into distinct motions either at the time of capture or by hand after the capture session. As we move toward collecting more and longer motion sequences, however, automatic segmentation techniques will become important for processing the results in a reasonable time frame.
Robust Principal Component Analysis for Computer Vision
, 2001
"... Principal Component Analysis (PCA) has been widely used for the representation of shape, appearance, and motion. One drawback of typical PCA methods is that they are least squares estimation techniques and hence fail to account for "outliers" which are common in realistic training sets. In ..."
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Cited by 133 (3 self)
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Principal Component Analysis (PCA) has been widely used for the representation of shape, appearance, and motion. One drawback of typical PCA methods is that they are least squares estimation techniques and hence fail to account for "outliers" which are common in realistic training sets. In computer vision applications, outliers typically occur within a sample (image) due to pixels that are corrupted by noise, alignment errors, or occlusion. We review previous approaches for making PCA robust to outliers and present a new method that uses an intrasample outlier process to account for pixel outliers. We develop the theory of Robust Principal Component Analysis (RPCA) and describe a robust Mestimation algorithm for learning linear multivariate representations of high dimensional data such as images. Quantitative comparisons with traditional PCA and previous robust algorithms illustrate the benefits of RPCA when outliers are present. Details of the algorithm are described and a software implementation is being made publically available.
Robust PCA via outlier pursuit
, 2010
"... Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a wellknown, welldocumented sensitivity to outliers. Recent work has considered the setting w ..."
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Cited by 92 (9 self)
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Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a wellknown, welldocumented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimizationbased algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal lowdimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the lowdimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. 1
Robust L1 norm factorization in the presence of outliers and missing data by alternative convex programming
 IEEE CONF. COMPUTER VISION AND PATTERN RECOGNITION
, 2005
"... Matrix factorization has many applications in computer vision. Singular Value Decomposition (SVD) is the standard algorithm for factorization. When there are outliers and missing data, which often happen in real measurements, SVD is no longer applicable. For robustness Iteratively Reweighted Least ..."
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Cited by 87 (0 self)
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Matrix factorization has many applications in computer vision. Singular Value Decomposition (SVD) is the standard algorithm for factorization. When there are outliers and missing data, which often happen in real measurements, SVD is no longer applicable. For robustness Iteratively Reweighted Least Squares (IRLS) is often used for factorization by assigning a weight to each element in the measurements. Because it uses L2 norm, good initialization in IRLS is critical for success, but is nontrivial. In this paper, we formulate matrix factorization as a L1 norm minimization problem that is solved efficiently by alternative convex programming. Our formulation 1) is robust without requiring initial weighting, 2) handles missing data straightforwardly, and 3) provides a framework in which constraints and prior knowledge (if available) can be conveniently incorporated. In the experiments we apply our approach to factorizationbased structure from motion. It is shown that our approach achieves better results than other approaches (including IRLS) on both synthetic and real data.
Principal Component Analysis Based on L1norm Maximization
, 2008
"... A method of principal component analysis (PCA) based on a new L1norm optimization technique is proposed. Unlike conventional PCA, which is based on L2norm, the proposed method is robust to outliers because it utilizes the L1norm, which is less sensitive to outliers. It is invariant to rotations ..."
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Cited by 60 (6 self)
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A method of principal component analysis (PCA) based on a new L1norm optimization technique is proposed. Unlike conventional PCA, which is based on L2norm, the proposed method is robust to outliers because it utilizes the L1norm, which is less sensitive to outliers. It is invariant to rotations as well. The proposed L1norm optimization technique is intuitive, simple, and easy to implement. It is also proven to find a locally maximal solution. The proposed method is applied to several data sets and the performances are compared with those of other conventional methods.