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

arXiv: https://arxiv.org/abs/1711.09598 · HTML: https://mlsanity.com/paper/1711.09598/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2017.

## Built on by (most cited first, 3 of 3)

- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Linearization and Identification of Multiple-Attractor Dynamical Systems through Laplacian Eigenmaps](https://mlsanity.com/paper/2202.09171/) (2022)
- [A Novel Unified Extended Matrix for Graph Signal Processing: Theory and Application](https://mlsanity.com/paper/2508.16633/) (2025)

## Abstract (arXiv)

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.