Diffusion Representations

TL;DR

This paper introduces a new data representation framework based on a measure-based diffusion kernel that preserves diffusion distances, is scalable, and invariant to scale, enhancing manifold learning and data analysis.

Contribution

It provides a closed-form decomposition of the measure-based diffusion kernel, enabling scalable, scale-invariant data representations without out-of-sample extension for stationary data.

Findings

Preserves pairwise diffusion distances independent of data size

Invariant to scale changes in the data

Effective on analytically generated datasets

Abstract

Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric structures in the data. Recently, it was suggested to replace the standard kernel by a measure-based kernel that incorporates information about the density of the data. Thus, the manifold assumption is replaced by a more general measure-based assumption. The measure-based diffusion kernel incorporates two separate independent representations. The first determines a measure that correlates with a density that represents normal behaviors and patterns in the data. The second consists of the analyzed multidimensional data points. In this paper, we present a representation framework for data analysis of datasets that is based on a closed-form decomposition of the measure-based kernel. The proposed representation preserves pairwise diffusion distances that does not depend on the data size while being invariant to scale. For a stationary data, no out-of-sample extension is needed for embedding newly arrived data points in the representation space. Several aspects of the presented methodology are demonstrated on analytically generated data.