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Quantum Geometry insights in Deep Learning

Noémie C. Combe

TL;DR

The paper argues that the Monge–Ampère equation and related optimal-transport principles provide a rigorous, geometric foundation for deep generative learning in Boltzmann machines and energy-based models. It shows that latent Gaussian mappings produce covariance structures that form a Wishart cone, which coincides with the Connes–Araki–Haagerup cone from quantum geometry, linking learning dynamics to modular flow concepts. It simultaneously offers an alternative Renormalization Group (RG) viewpoint, interpreting hidden layers as coarse-graining steps and highlighting the role of Monge–Ampère constraints in multi-scale representations. These dual perspectives unify statistical mechanics, quantum geometry, and deep learning theory, suggesting new optimization strategies and interpretable frameworks for hierarchical feature learning with potential practical impact on robust generative modeling.

Abstract

In this paper, we explore the fundamental role of the Monge-Ampère equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation to free energy minimization. We then establish a connection between optimal transport theory and deep learning, demonstrating how the Monge-Ampère equation governs probability transformations in generative models. Additionally, we provide insights from quantum geometry, showing that the space of covariance matrices arising in the learning process coincides with the Connes-Araki-Haagerup (CAH) cone in von Neumann algebra theory. Furthermore, we introduce an alternative approach based on renormalization group (RG) flow, which, while distinct from the optimal transport perspective, reveals another manifestation of the Monge-Ampère domain in learning dynamics. This dual perspective offers a deeper mathematical understanding of hierarchical feature learning, bridging concepts from statistical mechanics, quantum geometry, and deep learning theory.

Quantum Geometry insights in Deep Learning

TL;DR

The paper argues that the Monge–Ampère equation and related optimal-transport principles provide a rigorous, geometric foundation for deep generative learning in Boltzmann machines and energy-based models. It shows that latent Gaussian mappings produce covariance structures that form a Wishart cone, which coincides with the Connes–Araki–Haagerup cone from quantum geometry, linking learning dynamics to modular flow concepts. It simultaneously offers an alternative Renormalization Group (RG) viewpoint, interpreting hidden layers as coarse-graining steps and highlighting the role of Monge–Ampère constraints in multi-scale representations. These dual perspectives unify statistical mechanics, quantum geometry, and deep learning theory, suggesting new optimization strategies and interpretable frameworks for hierarchical feature learning with potential practical impact on robust generative modeling.

Abstract

In this paper, we explore the fundamental role of the Monge-Ampère equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation to free energy minimization. We then establish a connection between optimal transport theory and deep learning, demonstrating how the Monge-Ampère equation governs probability transformations in generative models. Additionally, we provide insights from quantum geometry, showing that the space of covariance matrices arising in the learning process coincides with the Connes-Araki-Haagerup (CAH) cone in von Neumann algebra theory. Furthermore, we introduce an alternative approach based on renormalization group (RG) flow, which, while distinct from the optimal transport perspective, reveals another manifestation of the Monge-Ampère domain in learning dynamics. This dual perspective offers a deeper mathematical understanding of hierarchical feature learning, bridging concepts from statistical mechanics, quantum geometry, and deep learning theory.

Paper Structure

This paper contains 25 sections, 43 equations.

Table of Contents

  1. Introduction
  2. Main statements
  3. Implications
  4. Plan of the paper
  5. Boltzmann Machines and Energy-Based Models
  6. Boltzmann Machines as Energy-Based Models
  7. Recollections
  8. Energy-based models
  9. The Monge–Ampère Equation in Deep Learning and Boltzmann Models
  10. Monge–Ampère Equations in Deep Learning
  11. Monge–Ampère equation for probability distributions
  12. Gradient Flows and Wasserstein Distance
  13. Definition of the Data Distribution $P_{\text{data}}(x)$
  14. Definition of the Latent Distribution $P_{\text{latent}}(z)$
  15. Relationship Between the Latent and Data Distributions
  16. ...and 10 more sections

Theorems & Definitions (2)

  • proof
  • proof