Topology, Convergence, and Reconstruction of Predictive States

TL;DR

This paper investigates the conditions for reliably reconstructing predictive states from time-series data, demonstrating convergence in the weak topology and exploring Hilbert space representations for high-memory processes.

Contribution

It introduces a framework for the convergence of predictive states from empirical data and links these states to reproducing kernel Hilbert spaces for complex processes.

Findings

Predictive states can be reconstructed from empirical samples in the weak topology.

Hilbert space representations facilitate the analysis of high-memory processes.

Connections to reproducing kernel Hilbert spaces enhance understanding of predictive state reconstruction.

Abstract

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.