Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds
This work addresses dimensionality reduction for complex metastable systems, which is an incremental improvement over existing mathematical frameworks for reaction coordinate computation.
The authors tackled the problem of identifying low-dimensional geometry in high-dimensional multiscale stochastic systems by developing a kernel-based algorithm that embeds transition manifolds in a reproducing kernel Hilbert space, resulting in a more robust and efficient method compared to previous approaches.
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.