On the stability of multiplicative update algorithms. Application to non-negative matrix factorization Sur la stabilité des règles de mises à jour multiplicatives. Application à la factorisation en matrices positives

Multiplicative update algorithms have encountered a great success to solve optimization problems with nonnegativity constraints, such as the famous non-negative matrix factorization 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 exponential or asymptotic stability of the solutions to general optimization problems with non-negative constraints, and finally study the difficult case of non-negative matrix factorization.