The recent introduction of geometric partition entropy brought a new viewpoint to non-parametric entropy quantification that incorporated the impacts of informative outliers, but its original formulation was limited to the context of a one-dimensional state space. A generalized definition of geometric partition entropy is now provided for samples within a bounded (finite measure) region of a d-dimensional vector space. The basic definition invokes the concept of a Voronoi diagram, but the computational complexity and reliability of Voronoi diagrams in high dimension make estimation by direct theoretical computation unreasonable. This leads to the development of approximation schemes that enable estimation that is faster than current methods by orders of magnitude. The partition intersection ($\pi$) approximation, in particular, enables direct estimates of marginal entropy in any context resulting in an efficient and versatile mutual information estimator. This new measure-based paradigm for data driven information theory allows flexibility in the incorporation of geometry to vary the representation of outlier impact, which leads to a significant broadening in the applicability of established entropy-based concepts. The incorporation of informative outliers is illustrated through analysis of transient dynamics in the synchronization of coupled chaotic dynamical systems.

翻译：几何分割熵的最新引入为非参数熵量化带来了新视角，该方法纳入了信息性离群点的影响，但其原始公式仅限于一维状态空间。本文提出了d维向量空间有界（有限测度）区域内样本的广义化几何分割熵定义。基础定义援引了Voronoi图的概念，但高维Voronoi图的计算复杂性和可靠性使得直接理论计算估计变得不可行。这推动了近似方案的发展，该方案能以数量级优势超越现有方法的估计速度。特别是分割相交（$\pi$）近似法，能够在任何场景中直接估计边缘熵，从而形成高效且通用的互信息估计器。这种基于测度的数据驱动信息论新范式，在结合几何特征以调整离群点影响表征方面具有灵活性，显著拓展了基于熵的经典概念的适用范围。通过对耦合混沌动力系统同步过程中瞬态动力学的分析，阐明了信息性离群点的纳入机制。