Nonnegative matrix factorization with gradient vertex pursuit

Nonnegative Matrix Factorization (NMF), defined as factorizing a nonnegative matrix into two nonnegative factor matrices, is a particularly important problem in machine learning. Unfortunately, it is also ill-posed and NP-hard. We propose a fast, robust, and provably correct algorithm, namely Gradient Vertex Pursuit (GVP), for solving a well-defined instance of the problem which results in a unique solution: there exists a polytope, whose vertices consist of a few columns of the original matrix, covering the entire set of remaining columns. Our algorithm is greedy: it detects, at each iteration, a correct vertex until the entire polytope is identified. We evaluate the proposed algorithm on both synthetic and real hyperspectral data, and show its superior performance compared with other state-of-the-art greedy pursuit algorithms.