Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization

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 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 HahnBanach 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 algorithm 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 better low rank nonnegative matrix approximation in fewer steps than conventional methods. AMS subject cl...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Moody T. Chu, Matthew M. Lin
• Keywords: Nonnegative Matrix Factorization, Polytope, Probability Simplex, SIAMSC 2008