We introduce an intrinsic nonlinear regression framework for responses on a sphere of finite or infinite dimension, viewed as a Riemannian manifold. Unlike traditional tangent-space regressions, which lift responses to a common tangent space and thereby violate intrinsic spherical distances, our proposed method employs an intrinsic approach. We model the intrinsic conditional mean, fitting the responses with respect to the geodesic distance. We use a linear function space as a model space, which we compose with the Riemannian exponential map to obtain predictions on the sphere. Our approach generalises geodesic regression and parametric intrinsic regression models. Leveraging vector-valued reproducing kernel Hilbert space theory, our approach reduces the infinite-dimensional estimation problem to a finite-dimensional problem via a novel representer theorem, leading to an efficient BFGS-based estimation algorithm. We establish convergence rates and analyse the finite-sample behaviour of our estimator, concluding with an application to density regression. The full implementation is available in R.

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