Exploring Predictive States via Cantor Embeddings and Wasserstein Distance

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).

Topology, Convergence, and Reconstruction of Predictive States

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.