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120
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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Estimation of (near) lowrank matrices with noise and highdimensional scaling
"... We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Me ..."
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Cited by 103 (19 self)
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We study an instance of highdimensional statistical inference in which the goal is to use N noisy observations to estimate a matrix Θ ∗ ∈ R k×p that is assumed to be either exactly low rank, or “near ” lowrank, meaning that it can be wellapproximated by a matrix with low rank. We consider an Mestimator based on regularization by the traceornuclearnormovermatrices, andanalyze its performance under highdimensional scaling. We provide nonasymptotic bounds on the Frobenius norm error that hold for a generalclassofnoisyobservationmodels,and apply to both exactly lowrank and approximately lowrank matrices. We then illustrate their consequences for a number of specific learning models, including lowrank multivariate or multitask regression, system identification in vector autoregressive processes, and recovery of lowrank matrices from random projections. Simulations show excellent agreement with the highdimensional scaling of the error predicted by our theory. 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 86 (13 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.
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 45 (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
Lowrank matrix completion by riemannian optimization
 ANCHPMATHICSE, Mathematics Section, École Polytechnique Fédérale de
"... The matrix completion problem consists of finding or approximating a lowrank matrix based on a few samples of this matrix. We propose a novel algorithm for matrix completion that minimizes the least square distance on the sampling set over the Riemannian manifold of fixedrank matrices. The algorit ..."
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Cited by 39 (3 self)
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The matrix completion problem consists of finding or approximating a lowrank matrix based on a few samples of this matrix. We propose a novel algorithm for matrix completion that minimizes the least square distance on the sampling set over the Riemannian manifold of fixedrank matrices. The algorithm is an adaptation of classical nonlinear conjugate gradients, developed within the framework of retractionbased optimization on manifolds. We describe all the necessary objects from differential geometry necessary to perform optimization over this lowrank matrix manifold, seen as a submanifold embedded in the space of matrices. In particular, we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions. Additionally, we derive secondorder models that can be used in Newton’s method based on approximating the exponential map on this manifold to second order. Finally, we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations. The numerical experiments indicate that our approach scales very well for largescale problems and compares favorable with the stateoftheart, while outperforming most existing solvers. 1
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. 1.
Matrix estimation by universal singular value thresholding
, 2012
"... Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and ..."
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Cited by 25 (0 self)
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Abstract. Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Candès and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has ‘a little bit of structure’. Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation, and generalized Bradley–Terry models for pairwise comparison. 1.
Circlebased Recommendation in Online Social Networks
"... Online social network information promises to increase recommendation accuracy beyond the capabilities of purely rating/feedbackdriven recommender systems (RS). As to better serve users ’ activities across different domains, many online social networks now support a new feature of “Friends Circles” ..."
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Cited by 23 (0 self)
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Online social network information promises to increase recommendation accuracy beyond the capabilities of purely rating/feedbackdriven recommender systems (RS). As to better serve users ’ activities across different domains, many online social networks now support a new feature of “Friends Circles”, which refines the domainoblivious “Friends ” concept. RS should also benefit from domainspecific “Trust Circles”. Intuitively, a user may trust different subsets of friends regarding different domains. Unfortunately, in most existing multicategory rating datasets, a user’s social connections from all categories are mixed together. This paper presents an effort to develop circlebased RS. We focus on inferring categoryspecific social trust circles from available rating data combined with social network data. We outline several variants of weighting friends within circles based on their inferred expertise levels. Through experiments on publicly available data, we demonstrate that the proposed circlebased recommendation models can better utilize user’s social trust information, resulting in increased recommendation accuracy. 1.
Concentrationbased guarantees for lowrank matrix reconstruction
 24th Annual Conference on Learning Theory (COLT
, 2011
"... We consider the problem of approximately reconstructing a partiallyobserved, approximately lowrank matrix. This problem has received much attention lately, mostly using the tracenorm as a surrogate to the rank. Here we study lowrank matrix reconstruction using both the tracenorm, as well as the ..."
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Cited by 21 (6 self)
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We consider the problem of approximately reconstructing a partiallyobserved, approximately lowrank matrix. This problem has received much attention lately, mostly using the tracenorm as a surrogate to the rank. Here we study lowrank matrix reconstruction using both the tracenorm, as well as the lessstudied maxnorm, and present reconstruction guarantees based on existing analysis on the Rademacher complexity of the unit balls of these norms. We show how these are superior in several ways to recently published guarantees based on specialized analysis.
Online robust subspace tracking from partial information, Arxiv preprint arXiv:1109.3827
"... This paper presents GRASTA (Grassmannian Robust Adaptive Subspace Tracking Algorithm), an efficient and robust online algorithm for tracking subspaces from highly incomplete information. The algorithm uses a robust l1norm cost function in order to estimate and track nonstationary subspaces when t ..."
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Cited by 19 (1 self)
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This paper presents GRASTA (Grassmannian Robust Adaptive Subspace Tracking Algorithm), an efficient and robust online algorithm for tracking subspaces from highly incomplete information. The algorithm uses a robust l1norm cost function in order to estimate and track nonstationary subspaces when the streaming data vectors are corrupted with outliers. We apply GRASTA to the problems of robust matrix completion and realtime separation of background from foreground in video. In this second application, we show that GRASTA performs highquality separation of moving objects from background at exceptional speeds: In one popular benchmark video example [28], GRASTA achieves a rate of 57 frames per second, even when run in MATLAB on a personal laptop.