Nonnegative Rank Factorization via Rank Reduction

Abstract. Any given nonnegative matrix A ∈ R can be expressed as the product A = UV for some nonnegative matrices U ∈ R and V ∈ R with k ≤ min{m, n}. The smallest k that makes this factorization possible 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 numerical procedure exists that can calculate the nonnegative factorization. This paper is the first to describe a heuristic approach to tackle this difficult problem. Based on the Wedderburn rank reduction formula, the idea is to recurrently extract, whenever possible, a rank-one nonnegative portion from the previous matrix, starting with A, 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. Numerical testing seems to suggest that the proposed algorithm, though still lacking in rigorous error analysis, might serve as an initial numerical means for analyzing the nonnegative rank factorization.