Paper · Mar 25, 2025

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

Ming Lei, Christophe Baehr

cs.LGarXiv:2503.19867
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As on arXiv

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 ). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem ) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma ). Third, we establish an AdS/CFT-type holographic duality (Theorem ) 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 ) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.

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Submitted
Mar 25, 2025
Subjects
cs.LG, cs.AI, eess.SP, math.GT, quant-ph

Comments: 9 pages, submitted to IEEE PAMI

Ricci flowSingularitiesSurgeryPerelman entropyStability & uniquenessDiscrete & graph Ricci

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)6Perelman 2003 Dynamically Stable Poincaré Embeddings for Neural Manifolds (Jun Chen, Yuang Liu, Xiangrui Zhao et al., 2021)8Chen 2021 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)4Zhu 2023 Deep Learning as Ricci Flow (Anthony Baptista, Alessandro Barp, Tapabrata Chakraborti et al., 2024)1Baptista 2024 A modified Ricci flow on arbitrary weighted graph (Jicheng Ma and Yunyan Yang, 2024)5Ma 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)7Rafik 2024 Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts (Mauricio Angel, 2025)2Angel 2025 On the singularity formation of gauge fields coupled to Ricci flow (Andoni Royo Abrego, 2026)3Abrego 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Deep Learning as Ricci Flow

    Anthony Baptista, Alessandro Barp, Tapabrata Chakraborti et al.

    Topics Discrete & graph Ricci · Shared terms “deep learning”, “neural networks”, “topological simplification”

  2. Extensions and Applications of Bredon's Trick in Geometric and Topological Contexts

    Mauricio Angel

    Topics Discrete & graph Ricci, Singularities, Stability & uniqueness · Shared terms “unifying”, “neural networks”, “providing”

  3. On the singularity formation of gauge fields coupled to Ricci flow

    Andoni Royo Abrego

    Topics Perelman entropy, Singularities

  4. The existence of inversive distance circle packing on polyhedral surface

    Xiang Zhu

    Topics Surgery, Discrete & graph Ricci, Singularities

  5. A modified Ricci flow on arbitrary weighted graph

    Jicheng Ma and Yunyan Yang

    Topics Surgery, Discrete & graph Ricci, Stability & uniqueness

  6. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon

    Topics Surgery, Discrete & graph Ricci

  7. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy · Shared terms “integrating”, “complexity”, “Lyapunov”

  8. Dynamically Stable Poincaré Embeddings for Neural Manifolds

    Jun Chen, Yuang Liu, Xiangrui Zhao et al.

    Topics Discrete & graph Ricci · Shared terms “neural networks”, “learning”, “dynamically”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).