# Inferring Kernel ε-Machines: Discovering Structure in Complex Systems

arXiv: https://arxiv.org/abs/2410.01076 · HTML: https://mlsanity.com/paper/2410.01076/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 1 Oct 2024.

## Builds on (most cited first, 11 of 11)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [Shannon Entropy Rate of Hidden Markov Processes](https://mlsanity.com/paper/2008.12886/) (2020)
- [Topology, Convergence, and Reconstruction of Predictive States](https://mlsanity.com/paper/2109.09203/) (2021)
- [Discovering Causal Structure with Reproducing-Kernel Hilbert Space ε-Machines](https://mlsanity.com/paper/2011.14821/) (2020)
- [On Principles of Emergent Organization](https://mlsanity.com/paper/2311.13749/) (2023)
- [A comparative study of non-deep learning, deep learning, and ensemble learning methods for sunspot number prediction](https://mlsanity.com/paper/2203.05757/) (2022)
- [Exploring Predictive States via Cantor Embeddings and Wasserstein Distance](https://mlsanity.com/paper/2206.04198/) (2022)

## Abstract (arXiv)

Previously, we showed that computational mechanic's causal states -- predictively-equivalent trajectory classes for a stochastic dynamical system -- can be cast into a reproducing kernel Hilbert space. The result is a widely-applicable method that infers causal structure directly from very different kinds of observations and systems. Here, we expand this method to explicitly introduce the causal diffusion components it produces. These encode the kernel causal-state estimates as a set of coordinates in a reduced dimension space. We show how each component extracts predictive features from data and demonstrate their application on four examples: first, a simple pendulum -- an exactly solvable system; second, a molecular-dynamic trajectory of n-butane -- a high-dimensional system with a well-studied energy landscape; third, the monthly sunspot sequence -- the longest-running available time series of direct observations; and fourth, multi-year observations of an active crop field -- a set of heterogeneous observations of the same ecosystem taken for over a decade. In this way, we demonstrate that the empirical kernel causal-states algorithm robustly discovers predictive structures for systems with widely varying dimensionality and stochasticity.