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

Nicolas Brodu, James P. Crutchfield

TL;DR

This work presents kernel ε-machines, a principled framework that extends computational mechanics to continuous and heterogeneous data via RKHS embeddings of conditional distributions. By treating causal states as RKHS objects and employing diffusion-map techniques, it derives continuous-time Itô-diffusion dynamics on causal states and an RKHS evolution operator to predict observations in the original data space. The approach is validated on Even processes, uncountable causal-state processes, and thermally driven Lorenz attractors, demonstrating robustness to measurement and thermal noise and scalability to high-dimensional settings. The method offers a scalable, data-driven path to infer minimal, optimally predictive causal structures and perform long-horizon predictions, with potential extensions to spatiotemporal systems and stochastic thermodynamics.

Abstract

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.

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

TL;DR

This work presents kernel ε-machines, a principled framework that extends computational mechanics to continuous and heterogeneous data via RKHS embeddings of conditional distributions. By treating causal states as RKHS objects and employing diffusion-map techniques, it derives continuous-time Itô-diffusion dynamics on causal states and an RKHS evolution operator to predict observations in the original data space. The approach is validated on Even processes, uncountable causal-state processes, and thermally driven Lorenz attractors, demonstrating robustness to measurement and thermal noise and scalability to high-dimensional settings. The method offers a scalable, data-driven path to infer minimal, optimally predictive causal structures and perform long-horizon predictions, with potential extensions to spatiotemporal systems and stochastic thermodynamics.

Abstract

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 -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.

Paper Structure

This paper contains 30 sections, 40 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Computational Mechanics: A synopsis
  3. Processes
  4. Predictions
  5. Process Types
  6. Causal states
  7. Causal-State Dynamic for Discrete Sequences
  8. $\epsilon$-Machines
  9. Patterns Captured by $\epsilon$-Machines
  10. Constructing Causal States using Reproducing Kernels
  11. Distributions as Points in a Reproducing Kernel Hilbert Space
  12. Unconditional distributions
  13. Conditional distributions
  14. RKHS Causal States
  15. Dynamics of RKHS causal states
  16. ...and 15 more sections

Figures (11)

  • Figure 1: (Left) State-emitting Hidden Markov models: State transition probabilities $\Pr(s_i|s_j)$ are specified independently from the symbol-emission probabilities $q(a|s_i)$ and $q(b|s_j)$. (Right) $\epsilon$-Machines: Symbols are emitted on transitions and the (causal) states capture dependencies. Unfortunately, for state-emitting HMMs the number of hidden states is a poor proxy for structural complexity and is often a meta-parameter with low interpretability. Since $\epsilon$-machine is unique, so it directly represents a stochastic process' intrinsic properties, such as generated randomness (Shannon entropy rate) and structural complexity (memory).
  • Figure 2: Even Process state-transition diagram: An HMM that generates a binary process over outputs $v \in \{0,1\}$. Transitions are labeled with the symbol, followed by the probability to take this transition. The Even Process has infinite Markov order---emitted $1$s occur in even blocks (of arbitrary length) bounded by $0$s. The process is stationary when starting with state distribution $\Pr( \mathcal{S} = \sigma _0, \mathcal{S} = \sigma _1) = (2/3, 1/3)$. This HMM is an $\epsilon$-machine.
  • Figure 3: Even Process: Reconstructed-state coordinates $\psi_{1}$ on the first reduced-basis eigenvector $\Phi_{1}$, together with a graphical representation of the transitions inferred between the colored clusters.
  • Figure 4: Nonunifilar generative HMM mess3: Only transition probabilities are depicted; emitted symbols are described in the main text.
  • Figure 5: Projection of the mess3's causal states on the reduced basis $\left\{ \Phi_{1},\Phi_{2}\right\}$. The number of samples stacked on each point is indicated.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Definition 1: Stationarity
  • Definition 2: Conditional Stationarity
  • Definition 3: Predictive equivalence
  • Definition 4: Causal states Crut88a