Fast and Effective Computation of Generalized Symmetric Matrix Factorization

2026-03-19

AI Summary

This work proposes an averaged nonmonotone alternating update method (A-NAUM) to address the pervasive challenge of generalized symmetric matrix factorization, which is typically nonconvex, nonsmooth, and non-Lipschitz in machine learning and imaging sciences. By introducing auxiliary variable splitting and a symmetry-induced exact penalty term, the authors construct an equivalent relaxed merit function and, for the first time, integrate an averaged nonmonotone line search with Kurdyka–Łojasiewicz (KL) convergence analysis. Theoretically, the method guarantees global convergence of the iterative sequence to a critical point and provides explicit convergence rates. Numerical experiments on real-world datasets demonstrate its efficiency and robustness.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we study a nonconvex, nonsmooth, and non-Lipschitz generalized symmetric matrix factorization model that unifies a broad class of matrix factorization formulations arising in machine learning, image science, engineering, and related areas. We first establish two exactness properties. On the modeling side, we prove an exact penalty property showing that, under suitable conditions, the symmetry-inducing quadratic penalty enforces symmetry whenever the penalty parameter is sufficiently large but finite, thereby exactly recovering the associated symmetric formulation. On the algorithmic side, we introduce an auxiliary-variable splitting formulation and establish an exact relaxation relationship that rigorously links stationary points of the original objective function to those of a relaxed potential function. Building on these exactness properties, we propose an average-type nonmonotone alternating updating method (A-NAUM) based on the relaxed potential function. At each iteration, A-NAUM alternately updates the two factor blocks by (approximately) minimizing the potential function, while the auxiliary block is updated in closed form. To ensure the convergence and enhance practical performance, we further incorporate an average-type nonmonotone line search and show that it is well-defined under mild conditions. Moreover, based on the Kurdyka-Łojasiewicz property and its associated exponent, we establish global convergence of the entire sequence to a stationary point and derive convergence rate results. Finally, numerical experiments on real datasets demonstrate the efficiency of A-NAUM.

Problem

• generalized symmetric matrix factorization
• nonconvex optimization
• nonsmooth optimization
• matrix factorization
• symmetry constraints

Innovation

• generalized symmetric matrix factorization
• exact penalty property
• auxiliary-variable splitting
• nonmonotone alternating updating
• Kurdyka–Łojasiewicz convergence