Samuel P. Loomis

Samuel P. Loomis, James P. Crutchfield

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).

Samuel P. Loomis, James P. Crutchfield

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.

S. Loomis, J. R. Mahoney, C. Aghamohammadi et al.

Given a classical channel---a stochastic map from inputs to outputs---the input can often be transformed to an intermediate variable that is informationally smaller than the input. The new channel accurately simulates the original but at a smaller transmission rate. Here, we examine this procedure when the intermediate variable is a quantum state. We determine when and how well quantum simulations of classical channels may improve upon the minimal rates of classical simulation. This inverts Holevo's original question of quantifying the capacity of quantum channels with classical resources. We also show that this problem is equivalent to another, involving the local generation of a distribution from common entanglement.