Dimensional reduction of gradient-like stochastic systems with multiplicative noise via Fokker-Planck diffusion maps

TL;DR

This paper introduces a graph-based method for reducing the dimensionality of stochastic systems with multiplicative noise, using Fokker-Planck diffusion maps to produce interpretable low-dimensional representations for complex data like single cell RNA sequencing.

Contribution

It presents a novel graph construction that captures the Fokker-Planck dynamics of stochastic systems with multiplicative noise, enabling effective dimensional reduction and analysis.

Findings

The method accurately captures the system's dynamics in low dimensions.

It provides a framework compatible with existing diffusion map techniques.

Application to single cell RNA data demonstrates practical utility.

Abstract

Dimensional reduction techniques have long been used to visualize the structure and geometry of high dimensional data. However, most widely used techniques are difficult to interpret due to nonlinearities and opaque optimization processes. Here we present a specific graph based construction for dimensionally reducing continuous stochastic systems with multiplicative noise moving under the influence of a potential. To achieve this, we present a specific graph construction which generates the Fokker-Planck equation of the stochastic system in the continuum limit. The eigenvectors and eigenvalues of the normalized graph Laplacian are used as a basis for the dimensional reduction and yield a low dimensional representation of the dynamics which can be used for downstream analysis such as spectral clustering. We focus on the use case of single cell RNA sequencing data and show how current diffusion map implementations popular in the single cell literature fit into this framework.