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Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement

Ming Lei, Christophe Baehr

TL;DR

This work introduces a unified geometric-thermodynamic framework for neural networks by coupling parameter-space geometry to loss landscapes through a thermodynamically guided Ricci flow, enabling adaptive geometry and isometric knowledge embedding. It advances three pillars: (i) a coupled Ricci flow that preserves geometric structure while integrating loss gradients, (ii) geometric surgery for automatic topology control via curvature-driven modifications, and (iii) an AdS/CFT-inspired holographic duality that bounds entanglement and links neural dynamics to holographic physics. The framework yields concrete guarantees and practical gains: a convergence speedup of approximately 2.1×, a 63% reduction in Betti numbers indicating topological simplification, and an $O(N \log N)$ computational complexity compared to $O(N^2)$ baselines, with entanglement ratios remaining bounded by holographic limits. These results suggest a physically grounded, scalable approach to geometric deep learning with robust regularization and topological adaptability, opening avenues in quantum-classical learning, topological optimization, and physics-informed AI.

Abstract

This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem~\ref{thm:isometric}). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~\ref{thm:critical}) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~\ref{lem:surgery}). Third, we establish an AdS/CFT-type holographic duality (Theorem~\ref{thm:ads}) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1$\times$ convergence acceleration and 63\% topological simplification while maintaining $\mathcal{O}(N\log N)$ complexity, outperforming Riemannian baselines by 15.2\% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~\ref{thm:converge}) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement

TL;DR

This work introduces a unified geometric-thermodynamic framework for neural networks by coupling parameter-space geometry to loss landscapes through a thermodynamically guided Ricci flow, enabling adaptive geometry and isometric knowledge embedding. It advances three pillars: (i) a coupled Ricci flow that preserves geometric structure while integrating loss gradients, (ii) geometric surgery for automatic topology control via curvature-driven modifications, and (iii) an AdS/CFT-inspired holographic duality that bounds entanglement and links neural dynamics to holographic physics. The framework yields concrete guarantees and practical gains: a convergence speedup of approximately 2.1×, a 63% reduction in Betti numbers indicating topological simplification, and an computational complexity compared to baselines, with entanglement ratios remaining bounded by holographic limits. These results suggest a physically grounded, scalable approach to geometric deep learning with robust regularization and topological adaptability, opening avenues in quantum-classical learning, topological optimization, and physics-informed AI.

Abstract

This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations. First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, formally proved to preserve isometric knowledge embedding (Theorem~\ref{thm:isometric}). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~\ref{thm:critical}) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~\ref{lem:surgery}). Third, we establish an AdS/CFT-type holographic duality (Theorem~\ref{thm:ads}) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1 convergence acceleration and 63\% topological simplification while maintaining complexity, outperforming Riemannian baselines by 15.2\% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~\ref{thm:converge}) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

Paper Structure

This paper contains 49 sections, 16 theorems, 43 equations, 3 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Research Background
  2. Geometric Foundations of Deep Learning
  3. Curvature-Topology Interaction
  4. Quantum-Classical Interface
  5. Technical Challenges
  6. Related Work
  7. Literature Review
  8. Geometric Foundations in Machine Learning
  9. Ricci Flow Dynamics
  10. Holographic Neural Architectures
  11. Topological Optimization
  12. Comparative Analysis
  13. Research Motivation and Problem Formulation
  14. Geometric-Thermodynamic Coupling
  15. Curvature-Driven Phase Transitions
  16. ...and 34 more sections

Key Result

Theorem 1

The coupled system (eq:coupling)-(eq:ads) admits solutions iff:

Figures (3)

  • Figure 1: Topological simplification through curvature flow. Our method (blue) reduces topological complexity faster than static approaches.
  • Figure 2: Accelerated convergence via curvature-attention coupling (Algorithm \ref{['alg:meta']})
  • Figure 3: Computational complexity scaling: Our method achieves $\mathcal{O}(N\log N)$ time vs. baseline $\mathcal{O}(N^2)$, validating Theorem \ref{['thm:complexity']}.

Theorems & Definitions (34)

  • Theorem 1: Geometric-Topological-Physical Unification
  • proof
  • Definition 1: Coupled Ricci Flow
  • Theorem 2: Isometric Knowledge Embedding
  • proof
  • Theorem 3: Critical Learning Rate
  • proof
  • Lemma 1: Singularity Surgery
  • proof
  • Theorem 4: AdS/CFT Correspondence
  • ...and 24 more