Estimating condition number with Graph Neural Networks

TL;DR

This paper introduces a fast, GNN-based method to estimate the condition number of large sparse matrices efficiently, avoiding costly full decompositions.

Contribution

It proposes a novel GNN feature engineering approach with linear complexity and two prediction schemes for condition number estimation, extending to arbitrary norms.

Findings

Achieves significant speedup over traditional methods

Effective for 1-norm and 2-norm condition number estimation

Extensible to arbitrary matrix norms

Abstract

For large sparse matrices, we almost never compute the condition number exactly because that would require computing the full SVD or full eigenvalue decompositionIn 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. One follows by decomposing the condition number and predicts the relatively more computationally intensive part $\|\mathbf{A}^{-1}\|$, while the other is to predict the whole condition number $\kappa$. Our approach can be extended to an arbitrary norm. 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 traditional numerical estimation methods.