Infinite-Dimensional Spherical Kernel ridge Regression

Abstract

We introduce a novel regression framework designed to model non-linear responses situated on a sphere $\mathbb{S}$ of finite or infinite dimension. Unlike traditional tangent-space regressions, which lift responses to a tangent space $T_o \mathbb{S}$ and thereby violate intrinsic spherical distances, our proposed method employs an intrinsic approach. We model the conditional mean through an intercept $o \in \mathbb{S}$ and a linear predictor function $f: \mathfrak{X} \to T_o \mathbb{S}$. This formulation transforms the estimation problem into finding a linear predictor within a function space, but utilizing a metric defined by spherical geometry rather than standard Euclidean distance. Leveraging vector-valued reproducing kernel Hilbert space theory, our approach reduces the infinite-dimensional estimation challenge to a manageable finite-dimensional problem via the representer theorem, leading to an efficient BFGS-based estimation algorithm. We establish convergence rates and analyze the finite-sample behavior of our estimator, concluding with a practical application to density regression. The full implementation is available in R.