Infinite-Dimensional Spherical Kernel ridge Regression
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arXiv
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| Format: | Preprint |
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2026
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| author | Matteo, Beatrice Stoecker, Almond Tavakoli, Shahin |
| author_facet | Matteo, Beatrice Stoecker, Almond Tavakoli, Shahin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite-Dimensional Spherical Kernel ridge Regression Matteo, Beatrice Stoecker, Almond Tavakoli, Shahin Methodology 62R10, 62R20, 62R30, 62J07, 46E22, 53C22, 62H11 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. |
| title | Infinite-Dimensional Spherical Kernel ridge Regression |
| topic | Methodology 62R10, 62R20, 62R30, 62J07, 46E22, 53C22, 62H11 |
| url | https://arxiv.org/abs/2606.00181 |