Difference-of-Convex Regularization for Graph Learning by Differentiable Programming

2026-08-12

AI Summary

This work addresses the computational expense and ill-conditioning of graph Laplacian pseudoinverse computation, which hinders scalability in large-scale graph learning. To overcome this, the authors propose a Difference-of-Convex Regularization (DCR) framework that approximates the spectral action of the pseudoinverse through regularized maximum likelihood estimation, thereby avoiding explicit matrix inversion. By introducing a dual representation, the method decouples pseudoinverse learning from instance-level inference, enabling efficient primal problem reconstruction. This study pioneers the integration of difference-of-convex optimization with differentiable programming in graph learning and provides theoretical guarantees on algorithmic convergence and the existence of a unique fixed point. Extensive experiments demonstrate that the proposed approach consistently outperforms conventional convex solvers and graph filtering baselines across diverse graph topologies.

Abstract

Laplacian-regularized minimization is fundamental in signal processing and machine learning, but is limited by the dense and ill-conditioned nature of the graph Laplacian pseudoinverse. While the Laplacian itself is sparse, its pseudoinverse is dense and often ill-conditioned, rendering direct computation impractical at scale. Moreover, pseudoinverse learning is more challenging than Laplacian learning. To address this challenge, this paper considers the setting where the graph Laplacian is given and proposes a Difference-of-Convex Regularizer (DCR) graph learning framework that approximates the spectral action of the Laplacian pseudoinverse without direct inversion via regularized Maximum Likelihood Estimation (MLE). By reformulating Laplacian-Regularized Nonnegative Least Squares (LR-NNLS) through a dual representation, DCR decouples pseudoinverse learning from instance-specific inference and enables efficient primal solution reconstruction via a differentiable dual-guided learning scheme. We establish theoretical guarantees on stability and the existence of a unique fixed point for DCR algorithm. Numerical experiments demonstrate improved performance over convex solvers and graph filtering baselines and robust performance across diverse graph topologies.

Problem

• graph learning
• Laplacian pseudoinverse
• ill-conditioned
• dense matrix
• scalability

Innovation

• Difference-of-Convex Regularization
• Graph Learning
• Differentiable Programming
• Laplacian Pseudoinverse
• Dual-Guided Learning