The Wasserstein distance from optimal mass transport (OMT) is a powerful mathematical tool with numerous applications that provides a natural measure of the distance between two probability distributions. Several methods to incorporate OMT into widely used probabilistic models, such as Gaussian or Gaussian mixture, have been developed to enhance the capability of modeling complex multimodal densities of real datasets. However, very few studies have explored the OMT problems in a reproducing kernel Hilbert space (RKHS), wherein the kernel trick is utilized to avoid the need to explicitly map input data into a high-dimensional feature space. In the current study, we propose a Wasserstein-type metric to compute the distance between two Gaussian mixtures in a RKHS via the kernel trick, i.e., kernel Gaussian mixture models.

翻译：最优质量传输（OMT）中的Wasserstein距离是一种功能强大的数学工具，能自然度量两个概率分布间的距离，已广泛应用于多个领域。为增强真实数据集复杂多模态密度的建模能力，研究者开发了多种将OMT融入高斯模型及高斯混合模型等常用概率模型的方法。然而，在再生核希尔伯特空间（RKHS）中探讨OMT问题的研究仍十分有限——该空间通过核技巧避免将输入数据显式映射到高维特征空间。本研究提出了一种基于Wasserstein型的度量方法，利用核技巧在再生核希尔伯特空间中计算两个高斯混合模型（即核高斯混合模型）间的距离。