Discrete Eckart-Young Theorem for Integer Matrices
SIAM Journal on Matrix Analysis and Applications
Initialization of nonnegative matrix factorization with vertices of convex polytope
ICAISC'12 Proceedings of the 11th international conference on Artificial Intelligence and Soft Computing - Volume Part I
Nonnegative matrix factorizations performing object detection and localization
Applied Computational Intelligence and Soft Computing
Sparse and unique nonnegative matrix factorization through data preprocessing
The Journal of Machine Learning Research
Journal of Global Optimization
Nonnegative rank factorization--a heuristic approach via rank reduction
Numerical Algorithms
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In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with a known boundary, making it easier to trace the approximation procedure. In particular, the supporting hyperplane that separates a point from a disjoint polytope, a fact asserted by the Hahn-Banach theorem, can be calculated in finitely many steps. This approach leads to a convenient way of computing the proximity map which, in contrast to most existing algorithms where only an approximate map is used, finds the unique and global minimum per iteration. This paper sets up a theoretical framework, outlines a numerical algorithm, and suggests an effective implementation. Testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods.