Global convergence of modified multiplicative updates for nonnegative matrix factorization
The paper focuses on slightly modified versions of the original multiplicative updates for nonnegative matrix factorization (NMF) proposed by \textit{D. D. Lee} and \textit{H. S. Seung} [``Algorithms for non-negative matrix factorization'', in: Advances in neural information processing systems 13, 556--562 (2001)]. One considers not only the Euclidian distance-based multiplicative update but also the I-divergence-based one. The first section is an introduction in the nature of the subject. The second section introduces briefly the NMF optimization problems and the multipicatie updates of Lee and Seung. In the third section, the modified multipicative updates based on the idea of \textit{N. Gillis} and \textit{F. Glineur} [``Nonnegative factorization and the maximum edge biclique problem'', Preprint, \url{arXiv:0810.4225}] are first introduced and then convergence theorems for updates are presented. Also, one proposed algorithms based on the modified multiplicative updates, proving that they always stop within a finite number of iterations. In the fourth section, the global convergence theorems in the previous section are proved using \textit{W. I. Zangwill}'s global convergence theorem [Nonlinear programming: A unified approach. Cliffs, NJ: Prentice-Hall Inc (1969; Zbl 0195.20804)], which is a fundamental result in optimization theory. The main conclusions are within the fifth section.
scientific article; zbMATH DE number 6353820
scientific article