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

TL;DR

This paper introduces a new concurrent algorithm for non-negative matrix factorization that provably converges to second-order stationary points, overcoming challenges posed by symmetry and non-Lipschitz gradients.

Contribution

It proposes a multiplicative weight update dynamics that avoids saddle points and leverages geometric insights, enabling parallel implementation.

Findings

The algorithm provably avoids saddle points.

It converges to second-order stationary points.

Supports parallel and scalable computation.

Abstract

Non-negative matrix factorization (NMF) is a fundamental non-convex optimization problem with numerous applications in Machine Learning (music analysis, document clustering, speech-source separation etc). Despite having received extensive study, it is poorly understood whether or not there exist natural algorithms that can provably converge to a local minimum. Part of the reason is because the objective is heavily symmetric and its gradient is not Lipschitz. In this paper we define a multiplicative weight update type dynamics (modification of the seminal Lee-Seung algorithm) that runs concurrently and provably avoids saddle points (first order stationary points that are not second order). Our techniques combine tools from dynamical systems such as stability and exploit the geometry of the NMF objective by reducing the standard NMF formulation over the non-negative orthant to a new formulation over (a scaled) simplex. An important advantage of our method is the use of concurrent updates, which permits implementations in parallel computing environments.