Estimating condition number with Graph Neural Networks

TL;DR

A fast method for estimating the condition number of sparse matrices using graph neural networks using graph neural networks (GNNs) achieves a significant speedup over the Hager-Higham and Lanczos methods.

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.

Figures (6)

• Figure 1: Pipeline for fast condition number estimation. The method consists of four stages: (1) input sparse matrix $\mathbf{A}$, (2) extract matrix-theoretic features, (3) model training using a multilayer perceptron, and (4) output the estimated condition number $\hat{\kappa}(\mathbf{A})$.
• Figure 2: Matrix condition number distribution for training set.
• Figure 3: Loss during training.
• Figure 4: Scalability analysis.
• Figure 5: Effect of non-zero element count on runtime.

Theorems & Definitions (1)

• Remark 1