Non-negative Matrix Factorization Using Bregman Monotone Operator Splitting

Abstract

A non-negative matrix factorization (NMF) algorithm based on Bregman monotone operator splitting (B-MOS) is proposed. Several commonly used NMF algorithms, such as the multiplicative update method, are often used in source separation for speech and image signals. To improve the convergence rate in the tail, applying the alternating direction method of multipliers (ADMM) is reported to be effective. However, a fixed step-size parameter has to be carefully chosen for fast and stable convergence. Our main idea to overcome this issue is to adaptively modify the variable space metric so that it matches the cost convexity. Besides this, selecting an appropriate MOS (e.g., Peaceman-Rachford splitting) instead of the Douglas-Rachford splitting used in the ADMM may effectively improve the convergence rate further. To realize these ideas w.r.t. adaptive metric modification and appropriate operator splitting selection, we apply B-MOS to the NMF problem and obtain a new NMF solver in this paper. Results of numerical experiments demonstrate that the proposed NMF solver with B-MOS improved the convergence rate in the tail.