Strengthening Han's Fourier Entropy-Influence Inequality via an Information-Theoretic Proof

2025-12-02

AI Summary

This work investigates Han-type inequalities linking Fourier entropy and influences for Boolean functions. Addressing the limitation that existing inequalities apply only to Boolean-valued functions, we extend the inequality to all real-valued Boolean functions with unit L²-norm and provide a rigorous proof of its validity. Our method integrates information-theoretic techniques—particularly Shannon entropy analysis—with high-dimensional Fourier analytic tools, yielding a novel entropy-influence coupling estimation framework. Crucially, we optimize the inequality’s constants to $C_1 = C_2 = 1$, achieving theoretical optimality. This strengthened inequality uncovers a more fundamental structural relationship between Fourier entropy and influences, offering a sharper analytical tool for Boolean function theory, learning theory, and Boolean circuit complexity.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We strengthen Han's Fourier entropy-influence inequality $$ H[widehat{f}] leq C_{1}I(f) + C_{2}sum_{iin [n]}I_{i}(f)lnfrac{1}{I_{i}(f)} $$ originally proved for ${-1,1}$-valued Boolean functions with $C_{1}=3+2ln 2$ and $C_{2}=1$. We show, by a short information-theoretic proof, that it in fact holds with sharp constants $C_{1}=C_{2}=1$ for all real-valued Boolean functions of unit $L^{2}$-norm, thereby establishing the inequality as an elementary structural property of Shannon entropy and influence.

Problem

• Strengthen Han's Fourier entropy-influence inequality
• Provide an information-theoretic proof for sharp constants
• Establish inequality for real-valued Boolean functions

Innovation

• Sharp constants via information-theoretic proof
• Real-valued Boolean functions with unit L2-norm
• Strengthened Fourier entropy-influence inequality