Spectral Partitioning with Blends of Eigenvectors

TL;DR

This paper introduces new theoretical guarantees for spectral partitioning methods using blends of eigenvectors, providing practical stopping criteria and convergence assurances for data analysis tasks.

Contribution

It establishes pointwise convergence guarantees for linear combinations of eigenvectors and introduces a meaningful stopping criterion for spectral partitioning.

Findings

Derived eigenpairs for the Ring of Cliques model

Validated theoretical predictions with numerical experiments

Bridged linear algebra methods with convergence theory

Abstract

Many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy, which leads to the first meaningful stopping criterion for two way spectral partitioning. More generally, we provide pointwise convergence guarantees so that blends (linear combinations) of eigenvectors can be employed to solve data analysis problems with confidence in their accuracy. We demonstrate this theory on an accessible model problem, the Ring of Cliques, by deriving the relevant eigenpairs and comparing the predicted results to numerical solutions. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods.