Towards an Approximation Theory of Observable Operator Models
<|reference_start|>Towards an Approximation Theory of Observable Operator Models: Observable operator models (OOMs) offer a powerful framework for modelling stochastic processes, surpassing the traditional hidden Markov models (HMMs) in generality and efficiency. However, using OOMs to model infinite-dimensional processes poses significant theoretical challenges. This article explores a rigorous approach to developing an approximation theory for OOMs of infinite-dimensional processes. Building upon foundational work outlined in an unpublished tutorial [Jae98], an inner product structure on the space of future distributions is rigorously established and the continuity of observable operators with respect to the associated 2-norm is proven. The original theorem proven in this thesis describes a fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space. The presented findings lay the groundwork for future research in approximating observable operators of infinite-dimensional processes, while a remedy to the encountered obstacle is suggested.<|reference_end|>
@article{anyszka2024towards,
    title={Towards an Approximation Theory of Observable Operator Models},
    author={Wojciech Anyszka},
    journal={arXiv preprint arXiv:2404.12070},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.12070},
    primaryClass={math.PR cs.LG stat.ML}
}