Minimization methods for non-differentiable functions
Minimization methods for non-differentiable functions
The maximum edge biclique problem is NP-complete
Discrete Applied Mathematics
Non-negative Matrix Factorization with Sparseness Constraints
The Journal of Machine Learning Research
Learning Sparse Representations by Non-Negative Matrix Factorization and Sequential Cone Programming
The Journal of Machine Learning Research
Projected Gradient Methods for Nonnegative Matrix Factorization
Neural Computation
SVD based initialization: A head start for nonnegative matrix factorization
Pattern Recognition
Nonnegative matrix factorization via rank-one downdate
Proceedings of the 25th international conference on Machine learning
SIAM Journal on Matrix Analysis and Applications
Two “well-known” properties of subgradient optimization
Mathematical Programming: Series A and B - Series B - Special Issue: Nonsmooth Optimization and Applications
Convex and Semi-Nonnegative Matrix Factorizations
IEEE Transactions on Pattern Analysis and Machine Intelligence
Document clustering using nonnegative matrix factorization
Information Processing and Management: an International Journal
Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization
ICA'07 Proceedings of the 7th international conference on Independent component analysis and signal separation
On the Complexity of Nonnegative Matrix Factorization
SIAM Journal on Optimization
Non-negative matrix factorization with quasi-newton optimization
ICAISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Soft Computing
Non-negative matrix factorization implementation using graphic processing units
IDEAL'10 Proceedings of the 11th international conference on Intelligent data engineering and automated learning
A hybrid face recognition approach using GPUMLib
CIARP'10 Proceedings of the 15th Iberoamerican congress conference on Progress in pattern recognition, image analysis, computer vision, and applications
Nonlinear nonnegative matrix factorization based on Mercer kernel construction
Pattern Recognition
Graph dual regularization non-negative matrix factorization for co-clustering
Pattern Recognition
Efficient Nonnegative Matrix Factorization via projected Newton method
Pattern Recognition
Low-Rank Matrix Approximation with Weights or Missing Data Is NP-Hard
SIAM Journal on Matrix Analysis and Applications
Multiple graph regularized nonnegative matrix factorization
Pattern Recognition
Sparse and unique nonnegative matrix factorization through data preprocessing
The Journal of Machine Learning Research
Approximate aggregation of Markovian models using alternating least squares
Performance Evaluation
Journal of Global Optimization
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Nonnegative matrix factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g., text mining, image processing, microarray data analysis, collaborative filtering, etc. We introduce a novel approach to solve NMF problems, based on the use of an underapproximation technique, and show its effectiveness to obtain sparse solutions. This approach, based on Lagrangian relaxation, allows the resolution of NMF problems in a recursive fashion. We also prove that the underapproximation problem is NP-hard for any fixed factorization rank, using a reduction of the maximum edge biclique problem in bipartite graphs. We test two variants of our underapproximation approach on several standard image datasets and show that they provide sparse part-based representations with low reconstruction error. Our results are comparable and sometimes superior to those obtained by two standard sparse nonnegative matrix factorization techniques.