# Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds

## TL;DR

This paper introduces a kernel-based machine learning method to identify low-dimensional structures in complex stochastic systems, improving robustness and efficiency over previous approaches.

## Contribution

The authors develop a kernel embedding technique for transition manifolds, enhancing the mathematical framework for reaction coordinate computation in high-dimensional systems.

## Key findings

- Kernel embeddings preserve manifold structure under mild conditions.
- The new algorithm outperforms previous parametrization methods in robustness.
- Distortion bounds are derived for the kernel embedding process.

## Abstract

We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parametrization of a low-dimensional transition manifold in a certain function space. In this article, we enhance this approach by embedding and learning this transition manifold in a reproducing kernel Hilbert space, exploiting the favorable properties of kernel embeddings. Under mild assumptions on the kernel, the manifold structure is shown to be preserved under the embedding, and distortion bounds can be derived. This leads to a more robust and more efficient algorithm compared to previous parametrization approaches.