Markov Chain Convergence in Kernel-Induced Geometry
Abstract
We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.
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BibTeX
@misc{kachaiev2026markov,
title = {{Markov Chain Convergence in Kernel-Induced Geometry}},
author = {Oleksii Kachaiev and Luc Brogat-Motte and Silvia Villa and Lorenzo Rosasco},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13916},
eprint = {2610.13916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13916},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Markov Chain Convergence in Kernel-Induced Geometry
AU - Oleksii Kachaiev
AU - Luc Brogat-Motte
AU - Silvia Villa
AU - Lorenzo Rosasco
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13916
UR - https://arxiv.org/abs/2610.13916
AB - We study the convergence of Markov chains in the geometry induced by Hilbertian embeddings associated with positive definite kernels, also called Maximum Mean Discrepancy (MMD). Under a contraction assumption in the kernel-induced geometry, and a standard spectral gap assumption, we establish non-asymptotic, high-probability bounds for the convergence of the empirical occupation measure to the invariant measure. The obtained bounds separate the roles of the two conditions: the spectral gap determines the effective sample size in the leading term, while the kernel contraction enters only through a linear correction. As an application, we study one-step prediction of dynamical systems from a single trajectory, deriving fast learning rates for a nonlinear least-squares estimator. The rates highlight the interplay between mixing properties in the kernel-induced geometry and the classical source and capacity conditions used in the analysis of least-squares.
ER -