# Topology, Convergence, and Reconstruction of Predictive States

arXiv: https://arxiv.org/abs/2109.09203 · HTML: https://mlsanity.com/paper/2109.09203/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 19 Sep 2021.

## Builds on (most cited first, 9 of 9)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Hilbert Space Embeddings of Predictive State Representations](https://mlsanity.com/paper/1309.6819/) (2013)
- [Shannon Entropy Rate of Hidden Markov Processes](https://mlsanity.com/paper/2008.12886/) (2020)
- [Reproducing Kernel Hilbert Spaces Cannot Contain all Continuous Functions on a Compact Metric Space](https://mlsanity.com/paper/2002.03171/) (2020)
- [Bayesian Structural Inference for Hidden Processes](https://mlsanity.com/paper/1309.1392/) (2013)
- [DisCo: Physics-Based Unsupervised Discovery of Coherent Structures in Spatiotemporal Systems](https://mlsanity.com/paper/1909.11822/) (2019)
- [Discovering Causal Structure with Reproducing-Kernel Hilbert Space ε-Machines](https://mlsanity.com/paper/2011.14821/) (2020)
- [Local Causal States and Discrete Coherent Structures](https://mlsanity.com/paper/1801.00515/) (2018)
- [Spacetime Autoencoders Using Local Causal States](https://mlsanity.com/paper/2010.05451/) (2020)

## Built on by (most cited first, 5 of 5)

- [AI in a vat: Fundamental limits of efficient world modelling for agent sandboxing and interpretability](https://mlsanity.com/paper/2504.04608/) (2025)
- [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)
- [Agentic Information Theory: Ergodicity and Intrinsic Semantics of Information Processes](https://mlsanity.com/paper/2505.19275/) (2025)
- [Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures](https://mlsanity.com/paper/2507.07343/) (2025)

## Abstract (arXiv)

Predictive equivalence in discrete stochastic processes have been applied with great success to identify randomness and structure in statistical physics and chaotic dynamical systems and to inferring hidden Markov models. We examine the conditions under which they can be reliably reconstructed from time-series data, showing that convergence of predictive states can be achieved from empirical samples in the weak topology of measures. Moreover, predictive states may be represented in Hilbert spaces that replicate the weak topology. We mathematically explain how these representations are particularly beneficial when reconstructing high-memory processes and connect them to reproducing kernel Hilbert spaces.