# A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality

## TL;DR

This paper introduces a novel nonconvex splitting algorithm for symmetric nonnegative matrix factorization, guaranteeing convergence to KKT points, with proven convergence rates and conditions for optimality, applicable to data clustering and segmentation.

## Contribution

It presents a new nonconvex splitting method for SymNMF with convergence guarantees, parallel implementation, and conditions for global and local optimality.

## Key findings

- Algorithm converges quickly to a local minimum.
- Guarantees convergence to KKT points with a sublinear rate.
- Effective on synthetic and real datasets.

## Abstract

Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. The proposed algorithm is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem. Furthermore, it achieves a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in parallel. Further, sufficient conditions are provided which guarantee the global and local optimality of the obtained solutions. Extensive numerical results performed on both synthetic and real data sets suggest that the proposed algorithm converges quickly to a local minimum solution.