# Discovering Causal Structure with Reproducing-Kernel Hilbert Space ε-Machines

arXiv: https://arxiv.org/abs/2011.14821 · HTML: https://mlsanity.com/paper/2011.14821/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 4 of 4)

- [Topology, Convergence, and Reconstruction of Predictive States](https://mlsanity.com/paper/2109.09203/) (2021)
- [Exploring Predictive States via Cantor Embeddings and Wasserstein Distance](https://mlsanity.com/paper/2206.04198/) (2022)
- [Inferring Kernel ε-Machines: Discovering Structure in Complex Systems](https://mlsanity.com/paper/2410.01076/) (2024)
- [Score the Algebra, Not the Span: Dimension Reduction for Transfer Operator Models of Dynamical Systems](https://mlsanity.com/paper/2608.18918/) (2026)

## Abstract (arXiv)

We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation -- a finite- or infinite-state kernel ε-machine -- is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Plank equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably-infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally-driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high dimensional data.