Estimations of singular functions of kernel cross-covariance operators

The constrained covariance (COCO) has been proposed for measuring dependence between random vectors. Kernel cross-covariance operators on reproducing kernel Hilbert spaces, as one of kernel methods which could extract nonlinear dependence, have attracted considerable attention. This paper establishes learning rates of some estimators associated with kernel cross-covariance. For kernel cross-covariance operators, we bound a weighted summation of squared estimation errors of empirical singular functions by 16 times of the estimation error of empirical cross-covariance. Our method actually applies in general setting, so that a new bound is obtained for perturbation of singular functions of Hilbert-Schmidt operators. It is much tighter than the classical result as the latter only bounds each error of singular function individually. This is interest in its own right. For normalized cross-covariance operator, we propose an estimator and obtain a learning rate. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The paper introduces the Constrained Covariance (COCO) for measuring dependence between random vectors, and focuses on kernel cross-covariance operators in reproducing kernel Hilbert spaces as a method to extract nonlinear dependence, establishing learning rates for associated estimators. It bounds the squared estimation errors of empirical singular functions in kernel cross-covariance operators and provides a new bound for perturbation of singular functions of Hilbert-Schmidt operators, which is tighter than classical results. A new estimator and learning rate is proposed for normalized cross-covariance operators.