There exists a specific class of methods for data clustering problem inspired by synchronization of coupled oscillators. This approach requires an extension of the classical Kuramoto model to higher dimensions. In this paper, we propose a novel method based on so-called non-Abelian Kuramoto models. These models provide a natural extension of the classical Kuramoto model to the case of abstract particles (called Kuramoto–Lohe oscillators) evolving on matrix Lie groups U(n). We focus on the particular case \(n=2\), yielding the system of matrix ODE’s on SU(2) with the group manifold \(S^3\). This choice implies restriction on the dimension of multivariate data: in our simulations we investigate data sets where data are represented as vectors in \({\mathbb {R}}^k\), with \(k \le 6\). In our approach each object corresponds to one Kuramoto–Lohe oscillator on \(S^3\) and the data are encoded into matrices of their intrinsic frequencies. We assume global (all-to-all) coupling, which allows to greatly reduce computational cost. One important advantage of this approach is that it can be naturally adapted to clustering of multivariate functional data. We present the simulation results for several illustrative data sets.

中文翻译：

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基于量子同步的数据聚类

对于由耦合振荡器的同步引起的数据聚类问题，存在一类特定的方法。这种方法需要将经典的仓本模型扩展到更高的维度。在本文中，我们提出了一种基于所谓的非阿贝尔Kuramoto模型的新方法。这些模型为经典的仓本模型提供了自然扩展，扩展为在矩阵李群U（n）上演化的抽象粒子（称为仓本-罗氏振荡器）。我们专注于特定情况\（n = 2 \），在SU（2）上产生具有群流形\（S ^ 3 \）的矩阵ODE的系统。这种选择意味着对多元数据维度的限制：在我们的模拟中，我们研究数据集，其中数据用\（{\ mathbb {R}} ^ k \）表示为矢量，其中\（k \ le 6 \）。在我们的方法中，每个对象对应于\（S ^ 3 \）上的一个Kuramoto-Lohe振荡器，并且数据被编码为其固有频率的矩阵。我们假设使用全局（所有）耦合，这可以大大降低计算成本。这种方法的一个重要优点是，它可以自然地适用于多元功能数据的聚类。我们提供了几个示例性数据集的仿真结果。