Rates of the Strong Uniform Consistency with Rates for Conditional U-Statistics Estimators with General Kernels on Manifolds

Abstract

\(U\)-statistics represent a fundamental class of statistics from modeling quantities of interest defined by multi-subject responses. \(U\)-statistics generalize the empirical mean of a random variable \(X\) to sums over every \(m\)-tuple of distinct observations of \(X\). Stute [103] introduced a class of so-called conditional \(U\)-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to:

$$r^{(k)}(\varphi,\tilde{\mathbf{t}}):=\mathbb{E}[\varphi(Y_{1},\ldots,Y_{k})|(X_{1},\ldots,X_{k})=\tilde{\mathbf{t}}]\quad\textrm{for}\quad\tilde{\mathbf{t}}=\left(\mathbf{t}_{1},\ldots,\mathbf{t}_{k}\right)\in\mathbb{R}^{dk}.$$

In the analysis of modern machine learning algorithms, sometimes we need to manipulate kernel estimation within the nonconventional setting with intricate kernels that might even be irregular and asymmetric. In this general setting, we obtain the strong uniform consistency result for the general kernel on Riemannian manifolds with Riemann integrable kernels for the conditional \(U\)-processes. We treat both cases when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are proved under some standard structural conditions on the classes of functions and some mild conditions on the model. Our findings are applied to the regression function, the set indexed conditional \(U\)-statistics, the generalized \(U\)-statistics, and the discrimination problem. The theoretical results established in this paper are (or will be) key tools for many further developments in manifold data analysis.