Initialization of Nonnegative Matrix Factorization with Vertices of Convex Polytope

Nonnegative Matrix Factorization (NMF) is an emerging unsupervised learning technique that has already found many applications in machine learning and multivariate nonnegative data processing. NMF problems are usually solved with an alternating minimization of a given cost function, which leads to non-convex optimization. For this approach, an initialization for the factors to be estimated plays an essential role, not only for a fast convergence rate but also for selection of the desired local minima. If the observations are modeled by the exact factorization model (consistent data), NMF can be easily obtained by finding vertices of the convex polytope determined by the observed data projected on the probability simplex. For an inconsistent case, this model can be relaxed by approximating mean localizations of the vertices. In this paper, we discuss these issues and propose the initialization algorithm based on the analysis of a geometrical structure of the observed data. This approach is demonstrated to be robust, even for moderately noisy data.