Numerical tolerance for spectral decompositions of random matrices

TL;DR

This paper introduces an optimal stopping criterion for spectral decompositions of random matrices, balancing numerical and statistical errors to improve efficiency without sacrificing accuracy, with practical validation on networks.

Contribution

It provides a theoretical framework for quantifying statistical error impact and proposes a new, optimal numerical tolerance for spectral decompositions of random matrices.

Findings

Optimal stopping criterion reduces computational cost.

Balancing errors maintains accuracy in spectral decompositions.

Practical benefits demonstrated on network data.

Abstract

We precisely quantify the impact of statistical error in the quality of a numerical approximation to a random matrix eigendecomposition, and under mild conditions, we use this to introduce an optimal numerical tolerance for residual error in spectral decompositions of random matrices. We demonstrate that terminating an eigendecomposition algorithm when the numerical error and statistical error are of the same order results in computational savings with no loss of accuracy. We also repair a flaw in a ubiquitous termination condition, one in wide employ in several computational linear algebra implementations. We illustrate the practical consequences of our stopping criterion with an analysis of simulated and real networks. Our theoretical results and real-data examples establish that the tradeoff between statistical and numerical error is of significant import for data science.