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.

翻译：我们提出利用Kuramoto模型（包括其高维推广）对非欧几里得数据集进行机器学习的思路。这些模型是描述抽象粒子（广义振子）在球面、齐性空间和李群上集体运动（群集动力学）的矩阵常微分方程组。自21世纪初以来，此类模型在统计物理和控制理论领域均受到广泛研究。它们为编码不同流形之间的映射提供了合适的框架，并能学习球面与双曲几何结构。此外，这类模型还可学习变换群（如特殊正交群、酉群和洛伦兹群）的耦合作用。我们还综述了为几何深度学习中的概率建模与推断提供适当统计模型的概率分布族。我们主张采用在粒子连续极限下不同Kuramoto模型中产生的统计模型。最便捷的概率分布族是对特定对称群作用保持不变的分布。