Exploring predictive states via Cantor embeddings and Wasserstein distance

Area of Science:

• Stochastic processes
• Time-series analysis
• Machine learning

Background:

• Predictive states are key for understanding stochastic processes.
• Previous work used reproducing kernel Hilbert spaces for self-supervised reconstruction.
• New methods are needed for analyzing symbolic data and complex temporal structures.

Purpose of the Study:

• To investigate the use of Wasserstein distances for detecting predictive equivalences in symbolic data.
• To explore the application of Wasserstein distances in analyzing the temporal structure of stochastic processes.

Main Methods:

• Computed Wasserstein distances between sequence distributions.
• Employed a finite-dimensional embedding using the Cantor set for sequence geometry.
• Utilized hierarchical clustering and dimension reduction for exploratory data analysis.

Main Results:

• Successfully detected predictive equivalences in symbolic data using Wasserstein distances.
• Demonstrated that the resulting geometry provides insights into temporal structures.
• Applied the method to processes from finite-state hidden Markov models to infinite-state indexed grammars.

Conclusions:

• Wasserstein distances offer a novel nonparametric approach for analyzing symbolic time-series data.
• This method enhances the interpretability and discovery of temporal structures in stochastic processes.
• The approach is versatile, applicable to a wide range of process complexities.