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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2503.19867 |
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| _version_ | 1866913757697933312 |
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| author | Lei, Ming Baehr, Christophe |
| author_facet | Lei, Ming Baehr, Christophe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement Lei, Ming Baehr, Christophe Machine Learning Artificial Intelligence Signal Processing Geometric Topology Quantum Physics 68T05, 68T07, 68T27, 81V99, 37F40, I.2; K.3.2; F.4.1 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. |
| title | Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement |
| topic | Machine Learning Artificial Intelligence Signal Processing Geometric Topology Quantum Physics 68T05, 68T07, 68T27, 81V99, 37F40, I.2; K.3.2; F.4.1 |
| url | https://arxiv.org/abs/2503.19867 |