From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties

2025-07-15

AI Summary

The selection of divergence measures in machine learning lacks rigorous theoretical guidance, particularly regarding their behavior in high-dimensional spaces. Method: Drawing on dynamical systems theory, we systematically reconstruct the mathematical properties and physical interpretations of high-dimensional divergences by unifying information geometry, statistical physics, and probabilistic modeling. We analyze classical measures—including relative entropy and KL divergence—revealing novel characteristics such as stability under high-dimensional perturbations, curvature sensitivity, and gradient dynamics. We then propose a principled divergence design criterion grounded in dynamical systems principles, yielding an interpretable and optimization-friendly unified theoretical framework. Contribution/Results: Empirical evaluation demonstrates substantial improvements across distribution matching, generative modeling, and model robustness tasks. Our framework establishes a new paradigm bridging physical priors with statistical learning, offering both theoretical insight and practical utility for AI system design.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback-Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence.

Problem

• Selecting optimal divergence measures for machine learning models
• Exploring kinetic theory-based divergences for AI applications
• Comparing divergence properties across physics and ML domains

Innovation

• High-dimensional divergence measures from kinetic theory
• Comparative review of divergence measures
• Applications in machine learning and AI