Generalizing Kuramoto Models: Collective Motion on Lie Groups and Spheres in Machine Learning | HackerNoon

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Abstract and 1. Introduction

2. Some recent trends in theoretical ML

   2.1 Deep Learning via continuous-time controlled dynamical system

   2.2 Probabilistic modeling and inference in DL

   2.3 Deep Learning in non-Euclidean spaces

   2.4 Physics Informed ML

3. Kuramoto model

   3.1 Kuramoto models from the geometric point of view

   3.2 Hyperbolic geometry of Kuramoto ensembles

   3.3 Kuramoto models with several globally coupled sub-ensembles

4. Kuramoto models on higher-dimensional manifolds

   4.1 Non-Abelian Kuramoto models on Lie groups

   4.2 Kuramoto models on spheres

   4.3 Kuramoto models on spheres with several globally coupled sub-ensembles

   4.4 Kuramoto models as gradient flows

   4.5 Consensus algorithms on other manifolds

5. Directional statistics and swarms on manifolds for probabilistic modeling and inference on Riemannian manifolds

   5.1 Statistical models over circles and tori

   5.2 Statistical models over spheres

   5.3 Statistical models over hyperbolic spaces

   5.4 Statistical models over orthogonal groups, Grassmannians, homogeneous spaces

6. Swarms on manifolds for DL

   6.1 Training swarms on manifolds for supervised ML

   6.2 Swarms on manifolds and directional statistics in RL

   6.3 Swarms on manifolds and directional statistics for unsupervised ML

   6.4 Statistical models for the latent space

   6.5 Kuramoto models for learning (coupled) actions of Lie groups

   6.6 Grassmannian shallow and deep learning

   6.7 Ensembles of coupled oscillators in ML: Beyond Kuramoto models

7. Examples

   7.1 Wahba’s problem

   7.2 Linked robot’s arm (planar rotations)

   7.3 Linked robot’s arm (spatial rotations)

   7.4 Embedding multilayer complex networks (Learning coupled actions of Lorentz groups)

8. Conclusion and References

4.3 Kuramoto models on spheres with several globally coupled sub-ensembles

The above corollary asserts that (19) generates a trajectory in the D-product Q × · · · × Q. This trajectory depends on the initial distribution of all oscillators.

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Author:

(1) Vladimir Jacimovic, Faculty of Natural Sciences and Mathematics, University of Montenegro Cetinjski put bb., 81000 Podgorica Montenegro ([email protected]).

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This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.

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