Estimating condition number with Graph Neural Networks

2026-03-10

AI Summary

Accurate estimation of the condition number of sparse matrices is computationally expensive. This work proposes an efficient prediction method based on graph neural networks (GNNs) that enables rapid modeling of both the 1-norm and 2-norm condition numbers. By designing a feature engineering pipeline with linear complexity—O(nnz + n), where nnz denotes the number of nonzeros and n the matrix dimension—the approach supports two distinct prediction strategies. It achieves high accuracy while significantly outperforming classical methods such as the Hager–Higham and Lanczos algorithms. Extensive evaluation on multiple benchmark datasets demonstrates speedups of up to an order of magnitude, offering a scalable solution for condition number estimation in large-scale sparse systems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

In this paper, we propose a fast method for estimating the condition number of sparse matrices using graph neural networks (GNNs). To enable efficient training and inference of GNNs, our proposed feature engineering for GNNs achieves $\mathrm{O}(\mathrm{nnz} + n)$, where $\mathrm{nnz}$ is the number of non-zero elements in the matrix and $n$ denotes the matrix dimension. We propose two prediction schemes for estimating the matrix condition number using GNNs. The extensive experiments for the two schemes are conducted for 1-norm and 2-norm condition number estimation, which show that our method achieves a significant speedup over the Hager-Higham and Lanczos methods.

Problem

• condition number
• sparse matrices
• graph neural networks
• matrix estimation

Innovation

• Graph Neural Networks
• Condition Number Estimation
• Sparse Matrices
• Feature Engineering
• Computational Efficiency