Statistics of Min-max Normalized Eigenvalues in Random Matrices

2025-12-17

AI Summary

This study systematically characterizes the statistical behavior of eigenvalues of random matrices after min-max normalization. Addressing the critical issue that normalization-induced perturbations distort spectral distributions, we establish, for the first time, the exact scaling law governing the cumulative distribution function of normalized eigenvalues and rigorously derive an analytical expression for the systematic residual error introduced by normalization in low-rank matrix decomposition. Methodologically, we integrate tools from random matrix theory, extreme-value statistics, and matrix perturbation analysis, corroborated by extensive numerical experiments. Results show that normalization significantly alters the asymptotic eigenvalue distribution—quantitatively captured by the derived scaling law—and that the residual error decays deterministically with increasing matrix dimension, revealing a quantifiable impact of preprocessing on spectral methods. This work provides the first theoretical benchmark for spectral analysis under data preprocessing.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Ethical and legal aspects of web-scale data analysis, uses and collection practices

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.

Problem

• Investigates statistical properties of min-max normalized eigenvalues in random matrices
• Applies effective distribution to evaluate scaling law of cumulative distribution
• Derives residual error from matrix factorization of random matrices

Innovation

• Min-max normalized eigenvalues statistical properties analysis
• Scaling law evaluation for cumulative distribution
• Derivation of residual error in matrix factorization