Statistics of Min-max Normalized Eigenvalues in Random Matrices

TL;DR

This paper studies the statistical properties of min-max normalized eigenvalues in random matrices, deriving new theoretical insights and verifying them through numerical experiments relevant to data normalization and matrix factorization.

Contribution

It introduces a new analysis of the distribution and residual errors of normalized eigenvalues in random matrices, extending previous models.

Findings

Effective distribution for normalized eigenvalues confirmed

Scaling law of the cumulative distribution derived

Residual error during matrix factorization characterized

Abstract

Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus, this study investigates the statistical properties of min-max normalized eigenvalues in random matrices. Previously, the effective distribution for such normalized eigenvalues has been proposed. In this study, we apply it to evaluate a scaling law of the cumulative distribution. Furthermore, we derive the residual error that arises during matrix factorization of random matrices. We conducted numerical experiments to verify these theoretical predictions.