Repository: Freie Universität Berlin, Math Department

Kernel autocovariance operators of stationary processes: Estimation and convergence

Mollenhauer, Mattes and Klus, Stefan and Schütte, Christof and Koltai, Péter (2020) Kernel autocovariance operators of stationary processes: Estimation and convergence. arXive . pp. 1-36. (Unpublished)

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Official URL: https://arxiv.org/abs/2004.00891

Abstract

We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and prove several asymptotic results as well as finite sample error bounds with a detailed analysis for the Gaussian kernel. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > BioComputing Group
ID Code:2604
Deposited By: Monika Drueck
Deposited On:17 Sep 2021 12:23
Last Modified:25 Oct 2021 20:37

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