用图神经网络快速估算稀疏矩阵条件数，提速显著。
Estimating Condition Number with Graph Neural Networks
- 构建线性复杂度图特征，支持高效GNN部署。
- 预测矩阵逆的范数或整体条件数，无需显式求逆。
- 适用于任意范数，适合大规模稀疏矩阵计算场景。
本文提出一种基于图神经网络（GNN）的快速稀疏矩阵条件数估算方法。为提升GNN部署效率，设计了复杂度为O(nnz + n)的图特征构造方式，其中nnz为矩阵非零元素数量，n为矩阵维度。提出两种估算方案：一是分解条件数后预测计算开销较大的‖A⁻¹‖，避免显式形成逆矩阵；二是直接预测完整条件数κ，可扩展至任意范数。在1-范数和2-范数条件下进行大量实验，结果表明该方法相比传统数值估算方法有显著加速。相关代码已公开于https://github.com/inEXASCALE/sparse-kappa。
原文摘要 · Abstract (English)
In this paper, we propose a fast method for estimating the condition number of sparse matrices using graph neural networks (GNNs). For efficient deployment of GNNs, we introduce a graph feature construction with $\mathrm{O}(\mathrm{nnz} + n)$ complexity, where $\mathrm{nnz}$ is the number of non-zero elements in the matrix and $n$ denotes the matrix dimension. We propose two schemes for estimating the matrix condition number using GNNs; one follows by decomposing the condition number and predicts the relatively more computationally intensive part $\|\mathbf{A}^{-1}\|$, without explicitly forming the inverse, while the other is to predict the whole condition number $κ$. Our approach can be extended to an arbitrary norm. Extensive experiments are conducted for the estimation of the 1-norm and 2-norm condition numbers, which show that our method achieves a significant speedup over the traditional numerical estimation methods. Our software for GNN condition number estimator is made publicly available at https://github.com/inEXASCALE/sparse-kappa.
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