Diffusion-based kernel methods on Euclidean metric measure spaces

Diffusion-based kernel methods are commonly used for analyzing massive high dimensional datasets. These methods utilize a non-parametric approach to represent the data by using an affinity kernel that represents similarities, distances or correlations between data points. The kernel is based on a Markovian diffusion process, whose transition probabilities are determined by local distances between data points. Spectral analysis of this kernel provides a representation of the data, where Euclidean distances correspond to diffusion distances between data points. When the data lies on a low dimensional manifold, these diffusion distances encompass the geometry of the manifold. In this paper, we present a generalized approach for defining diffusion-based kernels by incorporating measure-based information, which represents the density or distribution of the data, together with its local distances. The generalized construction does not require an underlying manifold to provide a meaningful kernel interpretation but assumes a more relaxed assumption that the measure and its support are related to a locally low dimensional nature of the analyzed phenomena. This kernel is shown to satisfy the necessary spectral properties that are required in order to provide a low dimensional embedding of the data. The associated diffusion process is analyzed via its infinitesimal generator and the provided embedding is demonstrated in two geometric scenarios.