Frank-Wolfe Optimization for Symmetric-NMF under Simplicial Constraint

TL;DR

This paper introduces a simple, hyperparameter-free Frank-Wolfe algorithm for symmetric nonnegative matrix factorization under a simplicial constraint, with proven convergence and demonstrated effectiveness.

Contribution

Proposes a novel FW solver for symmetric NMF with simplicial constraints, providing convergence analysis and practical numerical validation.

Findings

Convergence rate of O(1/ε^2) to approximate KKT points.

Algorithm is simple to implement and hyperparameter-free.

Numerical results confirm effectiveness of the proposed method.

Abstract

Symmetric nonnegative matrix factorization has found abundant applications in various domains by providing a symmetric low-rank decomposition of nonnegative matrices. In this paper we propose a Frank-Wolfe (FW) solver to optimize the symmetric nonnegative matrix factorization problem under a simplicial constraint, which has recently been proposed for probabilistic clustering. Compared with existing solutions, this algorithm is simple to implement, and has no hyperparameters to be tuned. Building on the recent advances of FW algorithms in nonconvex optimization, we prove an $O(1/\varepsilon^2)$ convergence rate to $\varepsilon$-approximate KKT points, via a tight bound $\Theta(n^2)$ on the curvature constant, which matches the best known result in unconstrained nonconvex setting using gradient methods. Numerical results demonstrate the effectiveness of our algorithm. As a side contribution, we construct a simple nonsmooth convex problem where the FW algorithm fails to converge to the optimum. This result raises an interesting question about necessary conditions of the success of the FW algorithm on convex problems.