Rank Modifications of Semidefinite Matrices Associated with a Secant Update Formula
SIAM Journal on Matrix Analysis and Applications
Non-negative Matrix Factorization with Sparseness Constraints
The Journal of Machine Learning Research
Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
On the Complexity of Nonnegative Matrix Factorization
SIAM Journal on Optimization
Perturbation of Matrices and Nonnegative Rank with a View toward Statistical Models
SIAM Journal on Matrix Analysis and Applications
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Given any nonnegative matrix $A \in \mathbb{R}^{m \times n}$, it is always possible to express A as the sum of a series of nonnegative rank-one matrices. Among the many possible representations of A, the number of terms that contributes the shortest nonnegative rank-one series representation is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple procedure exists presently that is able to perform the nonnegative factorization. Based on the Wedderburn rank reduction formula, this paper proposes a heuristic approach to tackle this difficult problem numerically. Starting with A, the idea is to recurrently extrat, whenever possible, a rank-one nonnegative portion from the previous matrix while keeping the residual nonnegative and lowering its rank by one. With a slight modification for symmetry, the method can equally be applied to another important class of completely positive matrices. No convergence can be guaranteed, but repeated restart might help alleviate the difficulty. Extensive numerical testing seems to suggest that the proposed algorithm might serve as a first-step numerical means for exploring the intriguing problem of nonnegative rank factorization.