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

• URL: http://arxiv.org/abs/2503.19867v1
• Date: Tue, 25 Mar 2025 17:32:31 GMT
• Title: Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement
• Authors: Ming Lei, Christophe Baehr,
• Abstract summary: This paper establishes a unified framework integrating geometric flows with deep learning through three fundamental innovations.<n>First, we propose a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology.<n>Second, we derive explicit phase transition thresholds and critical learning rates through curvature blowup analysis.<n>Third, we establish an AdS/CFT-type holographic duality (Theoremrefthm:ads) between neural networks and conformal field theories.
• Score: 7.410691988131121
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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