Descent Methods for Nonnegative Matrix Factorization
TL;DR: By interpreting this method as a rank-one approximation of the residue matrix, it is proved that it \(converges\) and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.
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Abstract: In this paper, we present several descent methods that can be applied to nonnegative matrix factorization and we analyze a recently developed fast block coordinate method called Rank-one Residue Iteration (RRI). We also give a comparison of these different methods and show that the new block coordinate method has better properties in terms of approximation error and complexity. By interpreting this method as a rank-one approximation of the residue matrix, we prove that it \(converges\) and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.
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Citations
•Book
Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation
Andrzej Cichocki,Rafal Zdunek,Anh Huy Phan,Shun-ichi Amari +3 more
- 12 Oct 2009
TL;DR: This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF), including NMFs various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD).
2.2K
Nonnegative Matrix and Tensor Factorizations
TL;DR: A broad survey of models and efficient algorithms for nonnegative matrix factorization (NMF) can be found in this paper, where the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models.
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Scalable Coordinate Descent Approaches to Parallel Matrix Factorization for Recommender Systems
Hsiang-Fu Yu,Cho-Jui Hsieh,Si Si,Inderjit S. Dhillon +3 more
- 10 Dec 2012
TL;DR: It is shown that coordinate descent based methods have a more efficient update rule compared to ALS, and are faster and have more stable convergence than SGD, and it is empirically shown that CCD++ is much faster than ALS and SGD in both settings.
A Globally Convergent Algorithm for Nonconvex Optimization Based on Block Coordinate Update
Yangyang Xu,Wotao Yin +1 more
TL;DR: In this article, an algorithm for non-convex optimization with global convergence to a critical point has been proposed, where the variables of the underlying problem are either treated as one block or multiple disjoint blocks.
329
•Posted Content
A globally convergent algorithm for nonconvex optimization based on block coordinate update
Yangyang Xu,Wotao Yin +1 more
TL;DR: In this article, a generic nonconvex optimization formulation is proposed, and the convergence of the whole iterate sequence to a critical point is established along with a rate of convergence, and numerically demonstrate its efficiency.
References
Learning the parts of objects by non-negative matrix factorization
TL;DR: An algorithm for non-negative matrix factorization is demonstrated that is able to learn parts of faces and semantic features of text and is in contrast to other methods that learn holistic, not parts-based, representations.
14.2K
Learning parts of objects by non-negative matrix factorization
D. D. Lee
- 01 Jan 1999
TL;DR: In this article, non-negative matrix factorization is used to learn parts of faces and semantic features of text, which is in contrast to principal components analysis and vector quantization that learn holistic, not parts-based, representations.
9.6K
•Book
Solving least squares problems
Charles L. Lawson,Richard J. Hanson +1 more
- 01 Jun 1974
TL;DR: Since the lm function provides a lot of features it is rather complicated so it is going to instead use the function lsfit as a model, which computes only the coefficient estimates and the residuals.
8.3K
Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values†
Pentti Paatero,Unto Tapper +1 more
TL;DR: In this paper, a new variant of Factor Analysis (PMF) is described, where the problem is solved in the weighted least squares sense: G and F are determined so that the Frobenius norm of E divided (element-by-element) by σ is minimized.
5.9K