Diffusion maps tailored to arbitrary non-degenerate Ito processes

TL;DR

This paper extends diffusion maps to handle arbitrary densities and general Ito processes, enabling broader applications like biased data analysis, dynamical systems, and flow field analysis.

Contribution

It introduces two generalizations of diffusion maps: one replacing the gradient of the sampling density, and another approximating generators of general Ito diffusions.

Findings

Enables diffusion maps to work with arbitrary densities.

Demonstrates applications in biased data and dynamical systems.

Shows effectiveness in flow field analysis and importance sampling.

Abstract

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.