Stability analysis of multiplicative update algorithms for non-negative matrix factorization

Roland Badeau (1) , Nancy Bertin (2) , Emmanuel Vincent (2)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 METISS - Speech and sound data modeling and processing

Multiplicative update algorithms have encountered a great success to solve optimization problems with non-negativity constraints, such as the famous non-negative matrix factorization (NMF) and its many variants. However, despite several years of research on the topic, the understanding of their convergence properties is still to be improved. In this paper, we show that Lyapunov's stability theory provides a very enlightening viewpoint on the problem. We prove the stability of supervised NMF and study the more difficult case of unsupervised NMF. Numerical simulations illustrate those theoretical results, and the convergence speed of NMF multiplicative updates is analyzed.

Mots clés: Optimization methods; non-negative matrix factorization; multiplicative update algorithms; stability; Lyapunov methods.; Lyapunov methods

Domaines: Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Roland Badeau, Nancy Bertin, Emmanuel Vincent. Stability analysis of multiplicative update algorithms for non-negative matrix factorization. International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2011, Prague, Czech Republic. 4 p. ⟨hal-00557789⟩