Results 11  20
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230
Lowrank matrix completion using alternating minimization. ArXiv:1212.0467 eprint
, 2012
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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.
Robust Matrix Decomposition with Sparse Corruptions
"... Abstract—Suppose a given observation matrix can be decomposed as the sum of a lowrank matrix and a sparse matrix, and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identifi ..."
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Abstract—Suppose a given observation matrix can be decomposed as the sum of a lowrank matrix and a sparse matrix, and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identification, latent variable graphical modeling, and principal components analysis. We study conditions under which recovering such a decomposition is possible via a combination of ℓ1 norm and trace norm minimization. We are specifically interested in the question of how many sparse corruptions are allowed so that convex programming can still achieve accurate recovery, and we obtain stronger recovery guarantees than previous studies. Moreover, we do not assume that the spatial pattern of corruptions is random, which stands in contrast to related analyses under such assumptions via matrix completion. Index Terms—Matrix decompositions, sparsity, lowrank, outliers
Clustering partially observed graphs via convex optimization.
 Journal of Machine Learning Research,
, 2014
"... Abstract This paper considers the problem of clustering a partially observed unweighted graphi.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organiz ..."
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Cited by 47 (13 self)
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Abstract This paper considers the problem of clustering a partially observed unweighted graphi.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) lowrank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.
SpaRCS: Recovering lowrank and sparse matrices from compressive measurements
, 2011
"... We consider the problem of recovering a matrix M that is the sum of a lowrank matrix L and a sparse matrix S from a small set of linear measurements of the form y = A(M) =A(L + S). This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization ..."
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Cited by 46 (4 self)
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We consider the problem of recovering a matrix M that is the sum of a lowrank matrix L and a sparse matrix S from a small set of linear measurements of the form y = A(M) =A(L + S). This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization, and robust principal component analysis. We propose a natural optimization problem for signal recovery under this model and develop a new greedy algorithm called SpaRCS to solve it. Empirically, SpaRCS inherits a number of desirable properties from the stateoftheart CoSaMP and ADMiRA algorithms, including exponential convergence and efficient implementation. Simulation results with video compressive sensing, hyperspectral imaging, and robust matrix completion data sets demonstrate both the accuracy and efficacy of the algorithm. 1
A Closed Form Solution to Robust Subspace Estimation and Clustering
"... We consider the problem of fitting one or more subspaces to a collection of data points drawn from the subspaces and corrupted by noise/outliers. We pose this problem as a rank minimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean, selfexpressive, low ..."
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Cited by 43 (4 self)
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We consider the problem of fitting one or more subspaces to a collection of data points drawn from the subspaces and corrupted by noise/outliers. We pose this problem as a rank minimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean, selfexpressive, lowrank dictionary plus a matrix of noise/outliers. Our key contribution is to show that, for noisy data, this nonconvex problem can be solved very efficiently and in closed form from the SVD of the noisy data matrix. Remarkably, this is true for both one or more subspaces. An important difference with respect to existing methods is that our framework results in a polynomial thresholding of the singular values with minimal shrinkage. Indeed, a particular case of our framework in the case of a single subspace leads to classical PCA, which requires no shrinkage. In the case of multiple subspaces, our framework provides an affinity matrix that can be used to cluster the data according to the subspaces. In the case of data corrupted by outliers, a closedform solution appears elusive. We thus use an augmented Lagrangian optimization framework, which requires a combination of our proposed polynomial thresholding operator with the more traditional shrinkagethresholding operator. 1.
Linearized Alternating Direction Method with Gaussian Back Substitution for Separable Convex Programming
, 2011
"... Abstract. Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions w ..."
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Cited by 38 (4 self)
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Abstract. Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show that the decomposed subproblems in the ADM procedure can be substantially alleviated by linearizing the involved quadratic terms arising from the augmented Lagrangian penalty on the model’s linear constraints. When the resolvent operators of the separable functions in the objective have closedform representations, embedding the linearization into the ADM subproblems becomes necessary to yield easy subproblems with closedform solutions. We thus show theoretically that the blend of ADM, Gaussian back substitution and linearization works effectively for the separable convex minimization model under consideration.
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.
Living on the edge: A geometric theory of phase transitions in convex optimization
, 2013
"... Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the `1 minimization method for identifying a sparse vector from random linear samples. Indee ..."
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Cited by 36 (4 self)
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Recent empirical research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the `1 minimization method for identifying a sparse vector from random linear samples. Indeed, this approach succeeds with high probability when the number of samples exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability. This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints. These applications depend on foundational research in conic geometry. This paper introduces a new summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of conic intrinsic volumes of a convex cone concentrates sharply near the statistical dimension. This fact leads to an approximate version of the conic kinematic formula that gives bounds on the probability that a randomly oriented cone shares a ray with a fixed cone.