# Exploring Predictive States via Cantor Embeddings and Wasserstein Distance

arXiv: https://arxiv.org/abs/2206.04198 · HTML: https://mlsanity.com/paper/2206.04198/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 7 of 7)

- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Hilbert Space Embeddings of Predictive State Representations](https://mlsanity.com/paper/1309.6819/) (2013)
- [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)
- [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)
- [Time Resolution Dependence of Information Measures for Spiking Neurons: Atoms, Scaling, and Universality](https://mlsanity.com/paper/1504.04756/) (2015)

## Built on by (most cited first, 1 of 1)

- [Inferring Kernel ε-Machines: Discovering Structure in Complex Systems](https://mlsanity.com/paper/2410.01076/) (2024)

## Abstract (arXiv)

Predictive states for stochastic processes are a nonparametric and interpretable construct with relevance across a multitude of modeling paradigms. Recent progress on the self-supervised reconstruction of predictive states from time-series data focused on the use of reproducing kernel Hilbert spaces. Here, we examine how Wasserstein distances may be used to detect predictive equivalences in symbolic data. We compute Wasserstein distances between distributions over sequences ("predictions"), using a finite-dimensional embedding of sequences based on the Cantor for the underlying geometry. We show that exploratory data analysis using the resulting geometry via hierarchical clustering and dimension reduction provides insight into the temporal structure of processes ranging from the relatively simple (e.g., finite-state hidden Markov models) to the very complex (e.g., infinite-state indexed grammars).