  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).
