The Maximum von Neumann Entropy Principle: Theory and Applications in Machine Learning

2026-02-02

AI Summary

This work addresses the lack of decision-theoretic and game-theoretic foundations for the maximum entropy principle based on von Neumann entropy (VNE) in data-driven settings, which has limited its theoretical grounding and application as a spectral-domain diversity measure. Building upon the maximum entropy minimax framework introduced by Grünwald and Dawid, we extend this approach to VNE, thereby providing the first game-theoretic justification for maximum-VNE solutions over density matrices and trace-normalized positive semidefinite operators. This establishes the maximum VNE principle as a unified information-theoretic foundation for the least biased inference in the spectral domain. The proposed framework is successfully applied to multi-embedding kernel representation selection and kernel matrix completion under partial observations, significantly enhancing model robustness and generalization performance.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Von Neumann entropy (VNE) is a fundamental quantity in quantum information theory and has recently been adopted in machine learning as a spectral measure of diversity for kernel matrices and kernel covariance operators. While maximizing VNE under constraints is well known in quantum settings, a principled analogue of the classical maximum entropy framework, particularly its decision theoretic and game theoretic interpretation, has not been explicitly developed for VNE in data driven contexts. In this paper, we extend the minimax formulation of the maximum entropy principle due to Gr\"unwald and Dawid to the setting of von Neumann entropy, providing a game-theoretic justification for VNE maximization over density matrices and trace-normalized positive semidefinite operators. This perspective yields a robust interpretation of maximum VNE solutions under partial information and clarifies their role as least committed inferences in spectral domains. We then illustrate how the resulting Maximum VNE principle applies to modern machine learning problems by considering two representative applications, selecting a kernel representation from multiple normalized embeddings via kernel-based VNE maximization, and completing kernel matrices from partially observed entries. These examples demonstrate how the proposed framework offers a unifying information-theoretic foundation for VNE-based methods in kernel learning.

Problem

• Von Neumann Entropy
• Maximum Entropy Principle
• Kernel Learning
• Density Matrices
• Partial Information

Innovation

• von Neumann entropy
• maximum entropy principle
• minimax formulation
• kernel learning
• density matrices