Statistics of Min-max Normalized Eigenvalues in Random Matrices

2025/12/17 by Nakada, Hyakka, Tanaka, Shu
#FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (cs.LG) #Statistical Mechanics (cond-mat.stat-mech) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2512.15427

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.

Citations

• 50 years of spin glass theory
• Random Matrices and complexity of Spin Glasses
• On the Distribution of the Roots of Certain Symmetric Matrices

Related

• Tail bounds for all eigenvalues of a sum of random matrices
• Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
• Extremal Eigenvalues of Random Kernel Matrices with Polynomial Scaling
• The largest eigenvalue of small rank perturbations of Hermitian random matrices
• Random matrices: Universality of local eigenvalue statistics