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149
Incremental Gradient on the Grassmannian for Online Foreground and Background Separation in Subsampled Video
 IN PROCEEDINGS OF THE 2012 IEEE CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION (CVPR
, 2012
"... It has recently been shown that only a small number of samples from a lowrank matrix are necessary to reconstruct the entire matrix. We bring this to bear on computer vision problems that utilize lowdimensional subspaces, demonstrating that subsampling can improve computation speed while still al ..."
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Cited by 36 (1 self)
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It has recently been shown that only a small number of samples from a lowrank matrix are necessary to reconstruct the entire matrix. We bring this to bear on computer vision problems that utilize lowdimensional subspaces, demonstrating that subsampling can improve computation speed while still allowing for accurate subspace learning. We present GRASTA, Grassmannian Robust Adaptive Subspace Tracking Algorithm, an online algorithm for robust subspace estimation from randomly subsampled data. We consider the specific application of background and foreground separation in video, and we assess GRASTA on separation accuracy and computation time. In one benchmark video example [16], GRASTA achieves a separation rate of 46.3 frames per second, even when run in MATLAB on a personal laptop.
Learning Incoherent Sparse and LowRank Patterns from Multiple Tasks
"... We consider the problem of learning incoherent sparse and lowrank patterns from multiple tasks. Our approach is based on a linear multitask learning formulation, in which the sparse and lowrank patterns are induced by a cardinality regularization term and a lowrank constraint, respectively. This f ..."
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Cited by 31 (7 self)
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We consider the problem of learning incoherent sparse and lowrank patterns from multiple tasks. Our approach is based on a linear multitask learning formulation, in which the sparse and lowrank patterns are induced by a cardinality regularization term and a lowrank constraint, respectively. This formulation is nonconvex; we convert it into its convex surrogate, which can be routinely solved via semidefinite programming for smallsize problems. We propose to employ the general projected gradient scheme to efficiently solve such a convex surrogate; however, in the optimization formulation, the objective function is nondifferentiable and the feasible domain is nontrivial. We present the procedures for computing the projected gradient and ensuring the global convergence of the projected gradient scheme. The computation of projected gradient involves a constrained optimization problem; we show that the optimal solution to such a problem can be obtained via solving an unconstrained optimization subproblem and an Euclidean projection subproblem. In addition, we present two projected gradient algorithms and discuss their rates of convergence. Experimental results on benchmark data sets demonstrate the effectiveness of the proposed multitask learning formulation and the efficiency of the proposed projected gradient algorithms.
Augmented Lagrangian alternating direction method for matrix separation based on lowrank factorization
, 2011
"... The matrix separation problem aims to separate a lowrank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclearnorm minimization models have been proposed for matrix separation and prov ..."
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Cited by 29 (2 self)
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The matrix separation problem aims to separate a lowrank matrix and a sparse matrix from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications. Nuclearnorm minimization models have been proposed for matrix separation and proved to yield exact separations under suitable conditions. These models, however, typically require the calculation of a full or partial singular value decomposition (SVD) at every iteration that can become increasingly costly as matrix dimensions and rank grow. To improve scalability, in this paper we propose and investigate an alternative approach based on solving a nonconvex, lowrank factorization model by an augmented Lagrangian alternating direction method. Numerical studies indicate that the effectiveness of the proposed model is limited to problems where the sparse matrix does not dominate the lowrank one in magnitude, though this limitation can be alleviated by certain data preprocessing techniques. On the other hand, extensive numerical results show that, within its applicability range, the proposed method in general has a much faster solution speed than nuclearnorm minimization algorithms, and often provides better recoverability.
A gradient descent algorithm on the grassman manifold for matrix completion
, 2009
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Dependent Hierarchical Beta Process for Image Interpolation and Denoising 1
"... A dependent hierarchical beta process (dHBP) is developed as a prior for data that may be represented in terms of a sparse set of latent features, with covariatedependent feature usage. The dHBP is applicable to general covariates and data models, imposing that signals with similar covariates are l ..."
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Cited by 24 (11 self)
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A dependent hierarchical beta process (dHBP) is developed as a prior for data that may be represented in terms of a sparse set of latent features, with covariatedependent feature usage. The dHBP is applicable to general covariates and data models, imposing that signals with similar covariates are likely to be manifested in terms of similar features. Coupling the dHBP with the Bernoulli process, and upon marginalizing out the dHBP, the model may be interpreted as a covariatedependent hierarchical Indian buffet process. As applications, we consider interpolation and denoising of an image, with covariates defined by the location of image patches within an image. Two types of noise models are considered: (i) typical white Gaussian noise; and (ii) spiky noise of arbitrary amplitude, distributed uniformly at random. In these examples, the features correspond to the atoms of a dictionary, learned based upon the data under test (without a priori training data). Stateoftheart performance is demonstrated, with efficient inference using hybrid Gibbs, MetropolisHastings and slice sampling.
Matrix completion for multilabel image classification
 In NIPS
"... Abstract Recently, image categorization has been an active research topic due to the urgent need to retrieve and browse digital images via semantic keywords. This paper formulates image categorization as a multilabel classification problem using recent advances in matrix completion. Under this set ..."
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Cited by 22 (3 self)
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Abstract Recently, image categorization has been an active research topic due to the urgent need to retrieve and browse digital images via semantic keywords. This paper formulates image categorization as a multilabel classification problem using recent advances in matrix completion. Under this setting, classification of testing data is posed as a problem of completing unknown label entries on a data matrix that concatenates training and testing features with training labels. We propose two convex algorithms for matrix completion based on a Rank Minimization criterion specifically tailored to visual data, and prove its convergence properties. A major advantage of our approach w.r.t. standard discriminative classification methods for image categorization is its robustness to outliers, background noise and partial occlusions both in the feature and label space. Experimental validation on several datasets shows how our method outperforms stateoftheart algorithms, while effectively capturing semantic concepts of classes.
Robust Visual Domain Adaptation with LowRank Reconstruction
"... Visual domain adaptation addresses the problem of adapting the sample distribution of the source domain to the target domain, where the recognition task is intended but the data distributions are different. In this paper, we present a lowrank reconstruction method to reduce the domain distribution ..."
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Cited by 20 (0 self)
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Visual domain adaptation addresses the problem of adapting the sample distribution of the source domain to the target domain, where the recognition task is intended but the data distributions are different. In this paper, we present a lowrank reconstruction method to reduce the domain distribution disparity. Specifically, we transform the visual samples in the source domain into an intermediate representation such that each transformed source sample can be linearly reconstructed by the samples of the target domain. Unlike the existing work, our method captures the intrinsic relatedness of the source samples during the adaptation process while uncovering the noises and outliers in the source domain that cannot be adapted, making it more robust than previous methods. We formulate our problem as a constrained nuclear norm and ℓ2,1 norm minimization objective and then adopt the Augmented Lagrange Multiplier (ALM) method for the optimization. Extensive experiments on various visual adaptation tasks show that the proposed method consistently and significantly beats the stateoftheart domain adaptation methods. 1.
Fast algorithms for recovering a corrupted lowrank matrix
"... This paper studies algorithms for solving the problem of recovering a lowrank matrix with a fraction of its entries arbitrarily corrupted. This problem can be viewed as a robust version of classical PCA, and arises in a number of application domains, including image processing, web data ranking, a ..."
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Cited by 18 (5 self)
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This paper studies algorithms for solving the problem of recovering a lowrank matrix with a fraction of its entries arbitrarily corrupted. This problem can be viewed as a robust version of classical PCA, and arises in a number of application domains, including image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, it can be exactly solved via a convex programming surrogate that combines nuclear norm minimization and ℓ 1norm minimization. This paper develops and compares two complementary approaches for solving this convex program. The first is an accelerated proximal gradient algorithm directly applied to the primal; while the second is a gradient algorithm applied to the dual problem. Both are several orders of magnitude faster than the previous stateoftheart algorithm for this problem, which was based on iterative thresholding. Simulations demonstrate the performance improvement that can be obtained via these two algorithms, and clarify their relative merits.
An Alternating Direction Algorithm for Matrix Completion with Nonnegative Factors
"... Abstract. This paper introduces a novel algorithm for the nonnegative matrix factorization and completion problem, which aims to find nonnegative matrices X and Y from a subset of entries of a nonnegative matrix M so that XY approximates M. This problem is closely related to the two existing problem ..."
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Cited by 17 (3 self)
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Abstract. This paper introduces a novel algorithm for the nonnegative matrix factorization and completion problem, which aims to find nonnegative matrices X and Y from a subset of entries of a nonnegative matrix M so that XY approximates M. This problem is closely related to the two existing problems: nonnegative matrix factorization and lowrank matrix completion, in the sense that it kills two birds with one stone. As it takes advantages of both nonnegativity and low rank, its results can be superior than those of the two problems alone. Our algorithm is applied to minimizing a nonconvex constrained leastsquares formulation and is based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties and numerical simulation results are presented. Compared to a recent algorithm for nonnegative random matrix factorization, the proposed algorithm yields comparable factorization through accessing only half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the results of the proposed algorithm have overall better qualities than those of two recent algorithms for matrix completion.