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105
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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SOLVING A LOWRANK FACTORIZATION MODEL FOR MATRIX COMPLETION BY A NONLINEAR SUCCESSIVE OVERRELAXATION ALGORITHM
"... Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes an ..."
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Cited by 58 (8 self)
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Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving largescale problems, we propose a lowrank factorization model and construct a nonlinear successive overrelaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Convergence of this nonlinear SOR algorithm is analyzed. Numerical results show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclearnorm minimization algorithms. Key words. Matrix Completion, alternating minimization, nonlinear GS method, nonlinear SOR method AMS subject classifications. 65K05, 90C06, 93C41, 68Q32
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 54 (2 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
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 51 (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.
Parallel stochastic gradient algorithms for largescale matrix completion
 Mathematical Programming Computation
, 2013
"... This paper develops Jellyfish, an algorithm for solving dataprocessing problems with matrixvalued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and leastsquares problems regularized by the nuclear norm or ..."
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Cited by 47 (4 self)
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This paper develops Jellyfish, an algorithm for solving dataprocessing problems with matrixvalued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and leastsquares problems regularized by the nuclear norm or γ2norm. Jellyfish implements a projected incremental gradient method with a biased, random ordering of the increments. This biased ordering allows for a parallel implementation that admits a speedup nearly proportional to the number of processors. On largescale matrix completion tasks, Jellyfish is orders of magnitude more efficient than existing codes. For example, on the Netflix Prize data set, prior art computes rating predictions in approximately 4 hours, while Jellyfish solves the same problem in under 3 minutes on a 12 core workstation.
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 33 (3 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
Collaborative filtering in a nonuniform world: Learning with the weighted trace norm
, 2010
"... We show that matrix completion with tracenorm regularization can be significantly hurt when entries of the matrix are sampled nonuniformly, but that a properly weighted version of the tracenorm regularizer works well with nonuniform sampling. We show that the weighted tracenorm regularization i ..."
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Cited by 30 (5 self)
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We show that matrix completion with tracenorm regularization can be significantly hurt when entries of the matrix are sampled nonuniformly, but that a properly weighted version of the tracenorm regularizer works well with nonuniform sampling. We show that the weighted tracenorm regularization indeed yields significant gains on the highly nonuniformly sampled Netflix dataset.
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 22 (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. 1.
A gradient descent algorithm on the grassman manifold for matrix completion
, 2009
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FAST GLOBAL CONVERGENCE OF GRADIENT METHODS FOR HIGHDIMENSIONAL STATISTICAL RECOVERY
 SUBMITTED TO THE ANNALS OF STATISTICS
, 2012
"... Many statistical Mestimators are based on convex optimization problems formed by the combination of a datadependent loss function with a normbased regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a highdi ..."
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Cited by 19 (1 self)
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Many statistical Mestimators are based on convex optimization problems formed by the combination of a datadependent loss function with a normbased regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a highdimensional framework that allows the ambient dimension d to grow with (and possibly exceed) the sample size n. Our theory identifies conditions under which projected gradient descent enjoys globally linear convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter θ ∗ and an optimal solution ̂ θ. By establishing these conditions with high probability for numerous statistical models, our analysis applies to a wide range of Mestimators, including sparse linear regression using Lasso; group Lasso for block sparsity; loglinear models with regularization; lowrank matrix recovery using nuclear norm regularization; and matrix decomposition