Data clustering based on quantum synchronization

Abstract

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.