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92
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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Robust Recovery of Subspace Structures by LowRank Representation
"... In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method ter ..."
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Cited by 128 (24 self)
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In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed LowRank Representation (LRR), which seeks the lowestrank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.
Stable principal component pursuit
 In Proc. of International Symposium on Information Theory
, 2010
"... We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analys ..."
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Cited by 94 (3 self)
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We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant lowrank structure recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers lowrank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressed sensing and decomposing superpositions of multiple structured signals. 1
Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions
 ANNALS OF STATISTICS,40(2):1171
, 2013
"... We analyze a class of estimators based on convex relaxation for solving highdimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation X of the sum of an (approximately) low rank matrix � ⋆ with a second matrix Ɣ ⋆ endowed with a complementary for ..."
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Cited by 61 (8 self)
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We analyze a class of estimators based on convex relaxation for solving highdimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation X of the sum of an (approximately) low rank matrix � ⋆ with a second matrix Ɣ ⋆ endowed with a complementary form of lowdimensional structure; this setup includes many statistical models of interest, including factor analysis, multitask regression and robust covariance estimation. We derive a general theorem that bounds the Frobenius norm error for an estimate of the pair ( � ⋆,Ɣ ⋆ ) obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results use a “spikiness ” condition that is related to, but milder than, singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields nonasymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices � ⋆ that can be exactly or approximately low rank, and matrices Ɣ ⋆ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error. The sharpness of our nonasymptotic predictions is confirmed by numerical simulations.
Two proposals for robust PCA using semidefinite programming
, 2010
"... The performance of principal component analysis (PCA) suffers badly in the presence of outliers. This paper proposes two novel approaches for robust PCA based on semidefinite programming. The first method, maximum mean absolute deviation rounding (MDR), seeks directions of large spread in the data ..."
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Cited by 47 (2 self)
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The performance of principal component analysis (PCA) suffers badly in the presence of outliers. This paper proposes two novel approaches for robust PCA based on semidefinite programming. The first method, maximum mean absolute deviation rounding (MDR), seeks directions of large spread in the data while damping the effect of outliers. The second method produces a lowleverage decomposition (LLD) of the data that attempts to form a lowrank model for the data by separating out corrupted observations. This paper also presents efficient computational methods for solving these SDPs. Numerical experiments confirm the value of these new techniques.
Integrating LowRank and GroupSparse Structures for Robust MultiTask Learning
 In KDD
, 2011
"... Multitask learning (MTL) aims at improving the generalization performance by utilizing the intrinsic relationships among multiple related tasks. A key assumption in most MTL algorithms is that all tasks are related, which, however, may not be the case in many realworld applications. In this paper, ..."
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Cited by 30 (2 self)
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Multitask learning (MTL) aims at improving the generalization performance by utilizing the intrinsic relationships among multiple related tasks. A key assumption in most MTL algorithms is that all tasks are related, which, however, may not be the case in many realworld applications. In this paper, we propose a robust multitask learning (RMTL) algorithm which learns multiple tasks simultaneously as well as identifies the irrelevant (outlier) tasks. Specifically, the proposed RMTL algorithm captures the task relationships using a lowrank structure, and simultaneously identifies the outlier tasks using a groupsparse structure. The proposed RMTL algorithm is formulated as a nonsmooth convex (unconstrained) optimization problem. We propose to adopt the accelerated proximal method (APM) for solving such an optimization problem. The key component in APM is the computation of the proximal operator, which can be shown to admit an analytic solution. We also theoretically analyze the effectiveness of the RMTL algorithm. In particular, we derive a key property of the optimal solution to RMTL; moreover, based on this key property, we establish a theoretical bound for characterizing the learning performance of RMTL. Our experimental results on benchmark data sets demonstrate the effectiveness and efficiency of the proposed algorithm.
Recursive robust pca or recursive sparse recovery in large but structured noise
 in IEEE Intl. Symp. on Information Theory (ISIT
, 2013
"... This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more informati ..."
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Cited by 22 (17 self)
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This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact