Kuramoto Models as Gradient Flows: Langevin Dynamics and Hyperbolic Geometry

Table of Links

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

4.4 Kuramoto models as gradient flows

One favorable property of Kuramoto models is that many of them exhibit potential dynamics. As a rule, ensembles with identical oscillators and symmetric couplings are gradient flows (sometimes in more than one sense).

Underline that models on spheres, as well as on special orthogonal and unitary groups are gradient flows in the chordal metric on these manifolds [87]. On spheres, this is equivalent to the cosine metric.

Consensus algorithms on Grassmannian manifolds are gradient flows as well, we refer to [7] for an explanation.

The above observations further imply that by adding the noise in an appropriate way to the above systems we obtain various Langevin dynamics. For instance, (1) are Langevin dynamics for the potential

As already noted above, recent geometric investigations of Kuramoto models have shown that models on spheres (12) with global coupling induce gradient flows in hyperbolic balls [76]. In general, the question of hyperbolic gradient flows induced by Kuramoto models on spheres is still to be fully explored. In particular, it is interesting to examine conditions for the potential dynamics on hyperbolic multi-discs and multi-balls which are induced by the models with several sub-ensembles. The second interesting question is adding the noise in an appropriate way in order to obtain Langevin dynamics on hyperbolic balls.