Markov Chain Convergence in Kernel-Induced Geometry
Read the original on arXiv Statistics ML →The paper investigates how Markov chains converge within a geometry defined by Hilbertian embeddings of positive definite kernels, also known as Maximum Mean Discrepancy (MMD). By assuming a contraction in this kernel-induced geometry and a standard spectral gap, the authors derive non-asymptotic, high‑probability bounds on the empirical occupation measure’s convergence to the invariant measure. These bounds separate the influence of the spectral gap, which sets the effective sample size, from the kernel contraction, which appears only as a linear correction. The results are applied to one‑step prediction of dynamical systems from a single trajectory, yielding fast learning rates for a nonlinear least‑squares estimator that illustrate the interaction between kernel‑induced mixing and classical source and capacity conditions.
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