Optimality and Renormalization imply Statistical Laws

2025-02-22

AI Summary

This work addresses the pervasive statistical regularities in discrete information encoding. Methodologically, it introduces a unified modeling framework grounded in information-theoretic functional optimality and renormalization group (RG) principles—specifically, the first adaptation of continuous-domain RG techniques to discrete integer coding, yielding a scale-invariant, optimality-driven mechanism. Theoretically, it rigorously derives a generalized first-digit law that unifies Benford’s law (characterizing digit distribution bias) and Elias coding (establishing its asymptotic optimality). Empirically, the law is validated across diverse real-world datasets, achieving high-precision fits to both Benford’s distribution and Elias code-length distributions. By bridging experimental mathematics and coding theory, this work establishes a novel theoretical foundation and analytical paradigm for lossless compression, anomaly detection, and modeling of natural statistical laws.

Technology Category

• Machine Learning: Information Theory
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Benford's Law is an important instance of experimental mathematics that appears to constrain the information-theoretic behavior of numbers. Elias' encoding for integers is a remarkable approach to universality and optimality of codes. In the present analysis we seek to deduce a general law and its particular implications for these two cases from optimality and renormalization as applied to information-theoretical functionals. Both theoretical and experimental results corroborate our conclusions.

Problem

• Deduce general law from optimality and renormalization
• Analyze implications for Benford's Law
• Explore Elias' encoding universality and optimality

Innovation

• Applies optimality and renormalization principles
• Explores information-theoretical functionals
• Validates with theoretical and experimental data