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

arXiv: https://arxiv.org/abs/2503.19867 · HTML: https://mlsanity.com/paper/2503.19867/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 40 references that are in the map.
- First posted on arXiv 25 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems](https://mlsanity.com/paper/1610.06519/) (2016)
- [Fast, Provable Algorithms for Isotonic Regression in all ℓp-norms](https://mlsanity.com/paper/1507.00710/) (2015)

## Abstract (arXiv)

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~refthm:isometric). Second, we derive explicit phase transition thresholds and critical learning rates (Theorem~refthm:critical) through curvature blowup analysis, enabling automated singularity resolution via geometric surgery (Lemma~reflem:surgery). Third, we establish an AdS/CFT-type holographic duality (Theorem~refthm:ads) 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 O(Nlog N) complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Theoretically, we prove exponential stability (Theorem~refthm:converge) through a new Lyapunov function combining Perelman entropy with Wasserstein gradient flows, fundamentally advancing geometric deep learning.