A1337
Title: A theory of nonparametric covariance function estimation
Authors: Yoshikazu Terada - The University of Osaka; RIKEN (Japan) [presenting]
Atsutomo Yara - Graduate School of Engineering Science, Osaka University (Japan)
Abstract: Nonparametric covariance function estimation for functional data observed with noise at discrete locations on a $d$-dimensional domain is studied. Estimating the covariance function from discretely observed data is a challenging nonparametric problem, particularly in multidimensional settings, since the covariance function is defined on a product domain and thus suffers from the curse of dimensionality. This motivates the use of adaptive estimators, such as deep learning estimators. However, existing theoretical results are largely limited to estimators with explicit analytic representations, and the properties of general learning-based estimators remain poorly understood. An oracle inequality is established for a broad class of learning-based estimators that applies to both sparse and dense observation regimes in a unified manner, and convergence rates are derived for deep learning estimators over several classes of covariance functions. The resulting rates suggest that structural adaptation can mitigate the curse of dimensionality, similarly to classical nonparametric regression. Convergence rates of learning-based estimators are compared with several existing procedures. These results reveal a distinctive adaptivity-variance trade-off in covariance function estimation.
