Finding minimal convex nested polygons
Information and Computation
Document clustering based on non-negative matrix factorization
Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval
Non-negative Matrix Factorization with Sparseness Constraints
The Journal of Machine Learning Research
Hard Problems in Linear Control Theory: Possible Approaches to Solution
Automation and Remote Control
Orthogonal nonnegative matrix t-factorizations for clustering
Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining
Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization
SIAM Journal on Scientific Computing
An improved approximation algorithm for the column subset selection problem
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Non-negative matrix factorization: Ill-posedness and a geometric algorithm
Pattern Recognition
Using underapproximations for sparse nonnegative matrix factorization
Pattern Recognition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation
On the Complexity of Nonnegative Matrix Factorization
SIAM Journal on Optimization
Underdetermined Sparse Blind Source Separation of Nonnegative and Partially Overlapped Data
SIAM Journal on Scientific Computing
Computing a nonnegative matrix factorization -- provably
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Minimum-Volume-Constrained Nonnegative Matrix Factorization: Enhanced Ability of Learning Parts
IEEE Transactions on Neural Networks
Perturbation of Matrices and Nonnegative Rank with a View toward Statistical Models
SIAM Journal on Matrix Analysis and Applications
The CAM software for nonnegative blind source separation in R-Java
The Journal of Machine Learning Research
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Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent factorizations. In this paper, we introduce a completely new way to obtaining more well-posed NMF problems whose solutions are sparser. Our technique is based on the preprocessing of the nonnegative input data matrix, and relies on the theory of M-matrices and the geometric interpretation of NMF. This approach provably leads to optimal and sparse solutions under the separability assumption of Donoho and Stodden (2003), and, for rank-three matrices, makes the number of exact factorizations finite. We illustrate the effectiveness of our technique on several image data sets.