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.

翻译：我们提出了一种新颖的回归框架，旨在对位于有限维或无限维球面 $\mathbb{S}$ 上的非线性响应进行建模。与传统的切空间回归（即将响应提升到切空间 $T_o \mathbb{S}$ 从而违背了球面内在距离）不同，我们的方法采用了一种内在的途径。我们通过一个截距项 $o \in \mathbb{S}$ 和一个线性预测函数 $f: \mathfrak{X} \to T_o \mathbb{S}$ 来对条件均值进行建模。这一公式将估计问题转化为在函数空间内寻找线性预测器，但使用的度量是基于球面几何而非标准的欧几里得距离。借助向量值再生核希尔伯特空间理论，我们的方法通过表示定理将无限维估计挑战简化为一个可处理的有限维问题，进而提出了一种高效的基于BFGS的估计算法。我们建立了收敛速率，分析了估计量的有限样本行为，并以密度回归的实际应用作为结论。该方法的完整实现已在R语言中提供。