Geometric Meta-Learning via Coupled Ricci Flow: Unifying Knowledge Representation and Quantum Entanglement
This work addresses fundamental challenges in geometric deep learning by unifying knowledge representation with quantum entanglement concepts, though it appears to be a novel paradigm rather than incremental.
This paper tackles the problem of integrating geometric flows with deep learning by proposing a thermodynamically coupled Ricci flow that adapts parameter space geometry to loss landscape topology, resulting in 2.1× convergence acceleration and 15.2% improvement in few-shot accuracy over baselines.
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.