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Non-negative matrix factorization with sparseness constraints,”

by Patrik O Hoyer , Patrik Hoyer@helsinki , Fi - Journal of Machine Learning Research, , 2004
"... Abstract Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in parts-based representations. In this paper, we sho ..."
Abstract - Cited by 498 (0 self) - Add to MetaCart
Abstract Non-negative matrix factorization (NMF) is a recently developed technique for finding parts-based, linear representations of non-negative data. Although it has successfully been applied in several applications, it does not always result in parts-based representations. In this paper, we

Provable inductive matrix completion

by Prateek Jain, Inderjit S. Dhillon - 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 comple-tion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the prob ..."
Abstract - Cited by 3 (2 self) - Add to MetaCart
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 comple-tion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study

Matrix Completion with Noise

by Emmanuel J. Candès, Yaniv Plan
"... 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 ..."
Abstract - Cited by 255 (13 self) - Add to MetaCart
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

by Rudi Cilibrasi, Paul M. B. Vitányi - IEEE Transactions on Information Theory , 2005
"... Abstract—We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the l ..."
Abstract - Cited by 297 (25 self) - Add to MetaCart
-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 real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix

Linear invariant generation using non-linear constraint solving

by Michael A. Colón, Sriram Sankaranarayanan, Henny B. Sipma - IN COMPUTER AIDED VERIFICATION , 2003
"... We present a new method for the generation of linear invariants which reduces the problem to a non-linear constraint solving problem. Our method, based on Farkas' Lemma, synthesizes linear invariants by extracting non-linear constraints on the coefficients of a target invariant from a program. ..."
Abstract - Cited by 109 (14 self) - Add to MetaCart
. 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 non-linear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge

Completing any low-rank matrix, provably

by Yudong Chen , Srinadh Bhojanapalli , Sujay Sanghavi , Rachel Ward , Tong Zhang , Yudong Chen , Srinadh Bhojanapalli , Sujay Sanghavi , Bhojanapalli, Sanghavi Rachel Ward Chen , Ward - ArXiv:1306.2979 , 2013
"... Abstract Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint-known as incoherence-on its row and column spaces. In these cases, the subse ..."
Abstract - Cited by 1 (0 self) - Add to MetaCart
Abstract Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint-known as incoherence-on its row and column spaces. In these cases

Computing a nonnegative matrix factorization -- provably

by Sanjeev Arora, Rong Ge, Ravi Kannan, Ankur Moitra - 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 ..."
Abstract - Cited by 37 (4 self) - Add to MetaCart
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 Non-convex Stochastic Gradient Descent

by Chi Jin , Sham M Kakade , Praneeth Netrapalli
"... 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 non-convex stochastic gradient descent (SGD). After every observation, it performs a

Understanding alternating minimization for matrix completion

by Moritz Hardt - In Symposium on Foundations of Computer Science , 2014
"... Alternating minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank 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 ..."
Abstract - Cited by 16 (1 self) - Add to MetaCart
is iterative and non-convex in nature. We give a new algorithm based on alternating minimization that provably recovers an unknown low-rank 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 Non-Rigid Objects with Rank Constraints

by Lorenzo Torresani , Danny B. Yang, Eugene J. Alexander, Christoph Bregler , 2001
"... This paper presents a novel solution for flow-based tracking and 3D reconstruction of deforming objects in monocular image sequences. A non-rigid 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 ..."
Abstract - Cited by 159 (7 self) - Add to MetaCart
This paper presents a novel solution for flow-based tracking and 3D reconstruction of deforming objects in monocular image sequences. A non-rigid 3D object undergoing rotation and deformation can be effectively approximated using a linear combination of 3D basis shapes. This puts a bound
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