A Projected Newton-type Algorithm for Nonnegative Matrix Factorization with Model Order Selection

Abstract

Nonnegative matrix factorization (NMF) has attracted considerable attention over the past few years as is met in many modern machine learning applications. NMF presents some inherent challenges when it comes both to its theoretical understanding and the task of devising efficient algorithmic tools. In this paper, we deal with an issue that is inherent in NMF, i.e., the a priori unawareness of the true nonnegative rank. To this end, a novel constrained NMF formulation is proposed. The main premise of the new formulation is to first assume an overestimate of the rank and then reduce it by imposing column sparsity jointly on the nonnegative matrix factors using proper penalization. Borrowing ideas from the block successive upper bound minimization framework, an alternating minimization strategy is followed, while inexact projected Newton-type updates are used in order to guarantee the descent direction of the cost function at each iteration. The effectiveness of the proposed approach is verified on simulated data and a real music signal decomposition experiment.