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67
Robust Principal Component Analysis?
, 2009
"... 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 co ..."
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Cited by 142 (6 self)
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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 ℓ1 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.
Matrix Completion with Noise
"... On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest ..."
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Cited by 82 (5 self)
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On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown lowrank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclearnorm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large lowrank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
"... ..."
A simpler approach to matrix completion
 the Journal of Machine Learning Research
"... This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candès and Recht [4], Candès and Tao [7], and Keshavan, Montanari, and Oh [18]. The reconstruction is accomplished by minim ..."
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Cited by 55 (3 self)
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This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candès and Recht [4], Candès and Tao [7], and Keshavan, Montanari, and Oh [18]. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular values, of the hidden matrix subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this assertion is short, self contained, and uses very elementary analysis. The novel techniques herein are based on recent work in quantum information theory.
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 45 (3 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,
Matrix Completion from Noisy Entries
"... Given a matrix M of lowrank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the ‘Netflix problem’) to structurefrommotion and positioning. We study a low ..."
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Cited by 44 (2 self)
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Given a matrix M of lowrank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the ‘Netflix problem’) to structurefrommotion and positioning. We study a low complexity algorithm introduced in [1], based on a combination of spectral techniques and manifold optimization, that we call here OPTSPACE. We prove performance guarantees that are orderoptimal in a number of circumstances. 1
Guaranteed rank minimization via singular value projection
 In NIPS 2010
, 2010
"... Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization under affine constraints (ARMP) and s ..."
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Cited by 39 (2 self)
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Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization under affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy a restricted isometry property (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newtonstep for our SVP framework to speedup the convergence with substantial empirical gains. Next, we address a practically important application of ARMP the problem of lowrank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold. However, we provide partial progress towards a proof of exact recovery for our algorithm by showing a more restricted isometry property and observe empirically that our algorithm recovers lowrank incoherent matrices from an almost optimal number of uniformly sampled entries. We also demonstrate empirically that our algorithms outperform existing methods, such as those of [5, 18, 14], for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes. In particular, results show that our SVPNewton method is significantly robust to noise and performs impressively on a more realistic powerlaw sampling scheme for the matrix completion problem. 1
Restricted strong convexity and (weighted) matrix completion: Optimal bounds with noise
, 2010
"... We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong ..."
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Cited by 33 (6 self)
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We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near lowrank matrices. Our results are based on measures of the “spikiness ” and “lowrankness ” of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an Mestimator that includes controls on both the rank and spikiness of the solution, and we establish nonasymptotic error bounds in weighted Frobenius norm for recovering matrices lying with ℓq“balls ” of bounded spikiness. Using informationtheoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.
Spectral Regularization Algorithms for Learning Large Incomplete Matrices
, 2009
"... We use convex relaxation techniques to provide a sequence of regularized lowrank solutions for largescale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the n ..."
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Cited by 31 (0 self)
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We use convex relaxation techniques to provide a sequence of regularized lowrank solutions for largescale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SoftImpute iteratively replaces the missing elements with those obtained from a softthresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a lowrank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity linear in the matrix dimensions. Our semidefiniteprogramming algorithm is readily scalable to large matrices: for example it can obtain a rank80 approximation of a 10 6 × 10 6 incomplete matrix with 10 5 observed entries in 2.5 hours, and can fit a rank 40 approximation to the full Netflix training set in 6.6 hours. Our methods show very good performance both in training and test error when compared to other competitive stateofthe art techniques. 1.
Tight Oracle Bounds for Lowrank Matrix Recovery from a Minimal Number of Random Measurements
, 2009
"... This paper presents several novel theoretical results regarding the recovery of a lowrank matrix from just a few measurements consisting of linear combinations of the matrix entries. We showthatproperlyconstrainednuclearnormminimizationstablyrecoversalowrankmatrix from a constant number of noisy ..."
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Cited by 25 (1 self)
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This paper presents several novel theoretical results regarding the recovery of a lowrank matrix from just a few measurements consisting of linear combinations of the matrix entries. We showthatproperlyconstrainednuclearnormminimizationstablyrecoversalowrankmatrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, the recovery error from noisy data is within a constant of three targets: 1) the minimax risk, 2) an ‘oracle ’ error that would be available if the column space of the matrix were known, and 3) a more adaptive ‘oracle ’ error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding lowrank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP) introduced in [6] for vectors, and in [22] for matrices.