Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows

TL;DR

This paper introduces a non-parametric state estimation method combining diffusion maps, Kalman filtering, and Koopman theory for high-dimensional gradient flow systems, demonstrated on neural activity data.

Contribution

It presents a novel data-driven framework that linearizes nonlinear systems using diffusion maps for improved state estimation.

Findings

Outperforms existing non-parametric algorithms in tracking accuracy.

Achieves results comparable to classical parametric methods without requiring model knowledge.

Successfully applied to neural data to estimate animal position.

Abstract

In this paper, we propose a non-parametric method for state estimation of high-dimensional nonlinear stochastic dynamical systems, which evolve according to gradient flows with isotropic diffusion. We combine diffusion maps, a manifold learning technique, with a linear Kalman filter and with concepts from Koopman operator theory. More concretely, using diffusion maps, we construct data-driven virtual state coordinates, which linearize the system model. Based on these coordinates, we devise a data-driven framework for state estimation using the Kalman filter. We demonstrate the strengths of our method with respect to both parametric and non-parametric algorithms in three tracking problems. In particular, applying the approach to actual recordings of hippocampal neural activity in rodents directly yields a representation of the position of the animals. We show that the proposed method outperforms competing non-parametric algorithms in the examined stochastic problem formulations. Additionally, we obtain results comparable to classical parametric algorithms, which, in contrast to our method, are equipped with model knowledge.