Topology, Convergence, and Reconstruction of Predictive States
This work addresses theoretical foundations for inferring hidden structures in stochastic processes, relevant to statistical physics and machine learning, but it appears incremental as it builds on existing predictive equivalence concepts.
The paper tackles the problem of reliably reconstructing predictive states from time-series data, showing that convergence can be achieved from empirical samples in the weak topology of measures and that these states can be represented in Hilbert spaces, which is beneficial for high-memory processes.
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.