This paper investigates the convergence of Markov chains within a geometric framework defined by Hilbertian embeddings and positive definite kernels, specifically focusing on Maximum Mean Discrepancy (MMD). The research establishes non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure, contingent on a contraction assumption in the kernel-induced geometry and a spectral gap assumption. These bounds delineate the influence of each condition, with the spectral gap dictating the effective sample size and the kernel contraction contributing a linear correction. As a practical application, the study analyzes the one-step prediction of dynamical systems from a single trajectory, yielding rapid learning rates for a nonlinear least-squares estimator that underscore the interaction between mixing properties in the kernel-induced geometry and classical source/capacity conditions. AI
IMPACT Provides theoretical underpinnings for understanding complex system dynamics and estimation, potentially impacting future AI research in areas requiring robust convergence analysis.
RANK_REASON The cluster contains a single academic paper detailing theoretical research on Markov chains and kernel-induced geometry. [lever_c_demoted from research: ic=1 ai=0.7]
- arXiv
- dynamical systems
- empirical occupation measure
- Hilbertian embeddings
- invariant measure
- Markov chain
- Maximum Mean Discrepancy
- nonlinear least-squares estimator
- spectral gap
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