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

2025-03-25

AI Summary

This work addresses the geometric-topological mismatch between parameter-space geometry and loss-landscape topology in deep learning. Methodologically, it introduces a unified framework integrating geometric flow with deep learning: (i) a thermodynamically coupled Ricci flow dynamically reshapes the parameter manifold’s geometry to align with the loss topology; (ii) an AdS/CFT-type holographic duality links neural networks to conformal field theory, mapping training dynamics onto anti-de Sitter spacetime; (iii) a curvature blow-up mechanism enables singularity resolution driven by phase-transition thresholds; and (iv) a Lyapunov function is rigorously constructed by unifying Perelman entropy with Wasserstein gradient flows, guaranteeing exponential stability. Contributions include the first realization of isometric knowledge embedding and entanglement-entropy-regularized boundaries. Experiments demonstrate 2.1× faster convergence, 63% reduction in topological complexity, 15.2% higher few-shot accuracy over Riemannian baselines, while retaining O(N log N) computational complexity.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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~ ef{thm:isometric}). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~ ef{thm:critical}) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~ ef{lem:surgery}). Third, we establish an AdS/CFT-type holographic duality (Theorem~ ef{thm:ads}) between neural networks and conformal field theories, providing entanglement entropy bounds for regularization design. Experiments demonstrate 2.1$ imes$ convergence acceleration and 63% topological simplification while maintaining $mathcal{O}(Nlog N)$ complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~ ef{thm:converge}) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

Problem

• Unifying geometric flows with deep learning for knowledge representation
• Analyzing curvature blowup to automate singularity resolution
• Establishing holographic duality between neural networks and field theories

Innovation

• Thermodynamically coupled Ricci flow adapts geometry
• Phase transition thresholds enable singularity resolution
• AdS/CFT-type duality links networks to field theories