Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning

TL;DR

This paper introduces the use of Kuramoto models and their higher-dimensional variants for machine learning on non-Euclidean data, leveraging swarming dynamics on manifolds to encode geometric information and learn transformations.

Contribution

It presents a novel framework combining Kuramoto models with geometric deep learning, enabling learning over spheres, hyperbolic spaces, and Lie groups with symmetry-invariant probabilistic models.

Findings

Kuramoto models can encode maps between manifolds.

Models effectively learn transformations on spherical and hyperbolic geometries.

Statistical models invariant under symmetry groups are suitable for probabilistic inference.

Abstract

We propose the idea of using Kuramoto models (including their higher-dimensional generalizations) for machine learning over non-Euclidean data sets. These models are systems of matrix ODE's describing collective motions (swarming dynamics) of abstract particles (generalized oscillators) on spheres, homogeneous spaces and Lie groups. Such models have been extensively studied from the beginning of XXI century both in statistical physics and control theory. They provide a suitable framework for encoding maps between various manifolds and are capable of learning over spherical and hyperbolic geometries. In addition, they can learn coupled actions of transformation groups (such as special orthogonal, unitary and Lorentz groups). Furthermore, we overview families of probability distributions that provide appropriate statistical models for probabilistic modeling and inference in Geometric Deep Learning. We argue in favor of using statistical models which arise in different Kuramoto models in the continuum limit of particles. The most convenient families of probability distributions are those which are invariant with respect to actions of certain symmetry groups.