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840
Nonnegative matrix factorization with sparseness constraints,”
 Journal of Machine Learning Research,
, 2004
"... Abstract Nonnegative matrix factorization (NMF) is a recently developed technique for finding partsbased, linear representations of nonnegative data. Although it has successfully been applied in several applications, it does not always result in partsbased representations. In this paper, we sho ..."
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Cited by 498 (0 self)
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Abstract Nonnegative matrix factorization (NMF) is a recently developed technique for finding partsbased, linear representations of nonnegative data. Although it has successfully been applied in several applications, it does not always result in partsbased representations. In this paper, we
Provable inductive matrix completion
 CoRR
"... Consider a movie recommendation system where apart from the ratings information, side information such as user’s age or movie’s genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the prob ..."
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Cited by 3 (2 self)
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Consider a movie recommendation system where apart from the ratings information, side information such as user’s age or movie’s genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study
Matrix Completion with Noise
"... On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest ..."
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Cited by 255 (13 self)
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that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present
Clustering by compression
 IEEE Transactions on Information Theory
, 2005
"... Abstract—We present a new method for clustering based on compression. The method does not use subjectspecific features or background knowledge, and works as follows: First, we determine a parameterfree, universal, similarity distance, the normalized compression distance or NCD, computed from the l ..."
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Cited by 297 (25 self)
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developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorovcomplexity. We propose axioms to capture the realworld setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix
Linear invariant generation using nonlinear constraint solving
 IN COMPUTER AIDED VERIFICATION
, 2003
"... We present a new method for the generation of linear invariants which reduces the problem to a nonlinear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting nonlinear constraints on the coefficients of a target invariant from a program. ..."
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Cited by 109 (14 self)
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. These constraints guarantee that the linear invariant is inductive. We then apply existing techniques, including specialized quantifier elimination methods over the reals, to solve these nonlinear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge
Completing any lowrank matrix, provably
 ArXiv:1306.2979
, 2013
"... Abstract Matrix completion, i.e., the exact and provable recovery of a lowrank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraintknown as incoherenceon its row and column spaces. In these cases, the subse ..."
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Cited by 1 (0 self)
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Abstract Matrix completion, i.e., the exact and provable recovery of a lowrank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraintknown as incoherenceon its row and column spaces. In these cases
Computing a nonnegative matrix factorization  provably
 IN: PROCEEDINGS OF THE 44TH SYMPOSIUM ON THEORY OF COMPUTING, STOC ’12
, 2012
"... In the Nonnegative Matrix Factorization (NMF) problem we are given an n×m nonnegative matrix M and an integer r> 0. Our goal is to express M as AW where A and W are nonnegative matrices of size n×r and r×m respectively. In some applications, it makes sense to ask instead for the product AW to app ..."
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Cited by 37 (4 self)
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In the Nonnegative Matrix Factorization (NMF) problem we are given an n×m nonnegative matrix M and an integer r> 0. Our goal is to express M as AW where A and W are nonnegative matrices of size n×r and r×m respectively. In some applications, it makes sense to ask instead for the product AW
Provable Efficient Online Matrix Completion via Nonconvex Stochastic Gradient Descent
"... Abstract Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they pr ..."
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be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs nonconvex stochastic gradient descent (SGD). After every observation, it performs a
Understanding alternating minimization for matrix completion
 In Symposium on Foundations of Computer Science
, 2014
"... Alternating minimization is a widely used and empirically successful heuristic for matrix completion and related lowrank optimization problems. Theoretical guarantees for alternating minimization have been hard to come by and are still poorly understood. This is in part because the heuristic is ite ..."
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Cited by 16 (1 self)
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is iterative and nonconvex in nature. We give a new algorithm based on alternating minimization that provably recovers an unknown lowrank matrix from a random subsample of its entries under a standard incoherence assumption. Our results reduce the sample size requirements of the alternating minimization
Tracking and Modeling NonRigid Objects with Rank Constraints
, 2001
"... This paper presents a novel solution for flowbased tracking and 3D reconstruction of deforming objects in monocular image sequences. A nonrigid 3D object undergoing rotation and deformation can be effectively approximated using a linear combination of 3D basis shapes. This puts a bound on the rank ..."
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Cited by 159 (7 self)
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This paper presents a novel solution for flowbased tracking and 3D reconstruction of deforming objects in monocular image sequences. A nonrigid 3D object undergoing rotation and deformation can be effectively approximated using a linear combination of 3D basis shapes. This puts a bound
Results 1  10
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