Ricci Flow, Neural Networks, Nash Entropy, Ricci Yang-Mills flow

Hey readers,

It's Presidents day and Monday night. It's dark outside.. Interestingly, Around 40,000 reads on yesterdays blog. Thanks for continued readership. My memory jogs back to my 2013-14 Poincare Conjecture and Ricci Flow article followed by the lecture I gave at SDSU in 2017 followed by . One of the topics in the lecture apart from Convolutional Neural Networks, Generative Adversarial Networks etc. was Ricci Flow & Neural Networks. 

Ricci Flow is partial differential equation (PDE) for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.

Lot of research was inspired by this lecture including this paper -

1. RicciNets: Curvature-guided Pruning of High-performance Neural Networks Using Ricci Flow - https://arxiv.org/abs/2007.04216 (University of Cambridge)

This model combines the principle of Ricci curvature with ML to carry out neural architecture search. It successfully identifies salient computational paths, and demonstrates a reduction in computational cost for no degradation in baseline performance. It outperforms pruning via lowest-magnitude weights on randomly wired neural networks

2. Thoughts on the Consistency between Ricci Flow and Neural Network Behavior - https://arxiv.org/pdf/2111.08410.pdf (Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, China) 

In this paper, construct the linearly nearly Euclidean manifold as a background to observe the evolution of Ricci flow and the training of neural networks. It suggests that In view of the convergence and stability of linearly nearly Euclidean metrics against perturbations, we observe that the training of the neural network under the weakly approximated gradient flow is consistent with the evolution of the Ricci flow.

3. A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy (https://arxiv.org/pdf/2107.01419.pdf  UCSD  ) - 

A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy - Ricci flow with uniformly bounded Nash entropy must also have uniformly bounded ν-functional. Consequently, on such an ancient solution there are uniform logarithmic Sobolev and Sobolev inequalities.

4. This is the best paper I have seen so far -
Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network - https://www.mdpi.com/1099-4300/23/3/292

Since graph-structured data are inherently non-Euclidean, we seek to use a non-Euclidean mathematical tool, namely, Riemannian geometry, to analyze graphs (networks). In this paper, we present a novel model for semi-supervised learning called the Ricci curvature-based graph convolutional neural network, i.e., RCGCN. The aggregation pattern of RCGCN is inspired by that of GCN.. 

5. This is super awesome thesis guided by Abel Prize winner - Karen Uhlenbeck - 

Modified Ricci flow on a Principal Bundle - https://www.proquest.com/openview/9d0358256cc058132f31dc4b712b3ea9/1?pq-origsite=gscholar&cbl=18750 - 

Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow