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

TL;DR

This paper develops a game-theoretic framework for maximizing von Neumann entropy in data-driven contexts, providing a new foundation for spectral diversity measures in machine learning applications like kernel selection and matrix completion.

Contribution

It extends the maximum entropy principle to von Neumann entropy with a game-theoretic interpretation, unifying spectral diversity measures in machine learning.

Findings

Provides a robust interpretation of maximum VNE solutions under partial information.

Demonstrates applications in kernel selection and matrix completion.

Offers a unifying information-theoretic foundation for VNE-based 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.