Optimal Transport for Kernel Gaussian Mixture Models
This work addresses a gap in optimal transport applications for kernel-based probabilistic models, but it appears incremental as it extends existing OMT methods to a specific kernel setting without broad new breakthroughs.
The authors tackled the problem of computing distances between Gaussian mixture models in a reproducing kernel Hilbert space (RKHS) by proposing a Wasserstein-type metric using the kernel trick, resulting in a method that avoids explicit high-dimensional mapping.
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.