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Non-negative matrix factorization (NMF) is a fundamental non-convexoptimization problem with numerous applications in Machine Learning (musicanalysis, document clustering, speech-source separation etc). Despite havingreceived extensive study, it is poorly understood whether or not there existnatural algorithms that can provably converge to a local minimum. Part of thereason is because the objective is heavily symmetric and its gradient is notLipschitz. In this paper we define a multiplicative weight update type dynamics(modification of the seminal Lee-Seung algorithm) that runs concurrently andprovably avoids saddle points (first order stationary points that are notsecond order). Our techniques combine tools from dynamical systems such asstability and exploit the geometry of the NMF objective by reducing thestandard NMF formulation over the non-negative orthant to a new formulationover (a scaled) simplex. An important advantage of our method is the use ofconcurrent updates, which permits implementations in parallel computingenvironments.

Convergence to Second-Order Stationarity for Non-negative Matrix Factorization: Provably and Concurrently