Global convergence of a modified HALS algorithm for nonnegative matrix factorization

Abstract

Hierarchical alternating least squares (HALS) algorithms are efficient computational methods for nonnegative matrix factorization (NMF). Given an initial solution, HALS algorithms update the solution block by block iteratively so that the error decreases monotonically. However, update rules in HALS algorithms are not well-defined. In addition, due to this problem, the convergence of the sequence of solutions to a stationary point cannot be proved theoretically. In this paper, we consider the HALS algorithm for the Frobenius norm-based NMF, and prove that a modified version has the global convergence property in the sense of Zangwill.