[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83683-en":3,"doc-seo-83683-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83683,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Kernel-Based Learning of Manifold-to-Manifold Maps from Scattered Data","This paper develops and analyzes new kernel-based approaches to approximate manifold-to-manifold maps using only scattered data. It begins with kernel-based approximation for scalar functions on manifolds, establishing a new sampling inequality and Sobolev-space error estimates. The resulting framework is then combined with a closest point projection to reconstruct manifold-valued mappings. Theoretical error bounds are derived and verified through numerical examples.","Kernel-based learning of manifold-to-manifold maps from  \nscattered data  \narXiv :2607 .02789v1 [math .NA] 2 Jul 2026  \nDaniel Fischer Department of Mathematics University of Bayreuth 95440 Bayreuth Germany  \n[daniel.fischer@uni-bayreuth.de](daniel.fischer@uni-bayreuth.de)  \nHolger Wendland∗  \nDepartment of Mathematics University of Bayreuth 95440 Bayreuth Germany  \n[holger.wendland@uni-bayreuth.de](holger.wendland@uni-bayreuth.de)  \nJuly 7, 2026  \nAbstract  \nWe describe and analyze new methods for approximating manifold-to-manifold maps using only scattered data information. To this end, we first study kernel-based approximation methods for scalar-valued functions defined on a manifold and derive a new sampling inequality and error estimates for functions from Sobolev spaces. After that, these methods are combined with a closest point projection to reconstruct manifold-to-manifold maps. The new methods are analyzed and error estimates are derived. Finally, numerical examples are given to verify the theoretical findings.  \nKeywords: kernel-based learning, scattered data, manifold-valued functions  \n1 Introduction  \nInterpolation and approximation with positive definite kernels provides a powerful framework for reconstructing a multivariate function f : Ω → R from known function values at a finite set of data sites scattered throughout the domain of definition Ω ⊆ Rd. Since these methods were first proposed, they have been generalized in various directions. On the one hand, numerous techniques have been developed for defining positive definite kernels on more specific domains of definition, such as smooth manifolds M, thereby enabling the reconstruction of functions f : M → R (see, for example, [15, 22, 20, 6]) . On the other hand, the theory of positive definite kernels and the associated reproducing kernel Hilbert spaces has been extended from the scalar-valued setting to vector- and, more generally, Hilbert space-valued functions, thus allowing the approximation of mappings f : Ω → W, where W is a Hilbert space [30, 21, 5, 11] .  \nMotivated by applications in numerous scientific and engineering domains, recent years have seen a growing interest in approximation methods for functions f : Ω → N, whose codomain N possesses the structure of a smooth manifold rather than that of a vector space. Such mappings arise, for  \n∗ The work of HW has been supported by the DFG under project no. 514588180.  \ninstance, in parametric model order reduction [38], where the codomain typically is the Stiefel manifold St(n, r) of column orthogonal matrices in Rn ×r or the Grassmann manifold Gr(k, n) of all k-dimensional subspaces of an n dimensional linear space, in interpolation of diffusion tensor images created from magnetic resonance imaging [26], where the codomain is the manifold of symmetric positive definite matrices, in the context of Cosserat-type material models [25], where N = R 3 × SO(3), or in crystallographic texture analysis [17], where the codomain can be seen as quotient SO(3)/S with some finite symmetry group S.  \nMany of the proposed manifold-valued approximation techniques have already been analyzed mathematically and can be broadly categorized into three classes:  \n1. Projection-based methods [9, 13, 17], where first an interpolant is constructed in the ambient space of an embedded submanifold, and the resulting values are subsequently projected onto the manifold by means of a suitable mapping;  \n2. Tangent space methods (Push-Interpolate-Pull methods) [36, 37, 39], where the given data are first mapped to a linear space-typically the tangent space at a chosen base point-followed by classical vector-valued interpolation and finally mapping back the reconstructed values to the manifold via an inverse transformation;  \n3. Riemannian mean-based methods [12, 14], where the evaluation of the interpolant involves computing a weighted mean of function values in N through nonlinear optimization.  \nThere also exist several techniques ","cbCaimRUT87LhmzS","https://ap.wps.com/l/cbCaimRUT87LhmzS","pdf",919073,5,1,27,"English","en",105,"# Introduction\n## Overview of kernel-based interpolation\n## Manifold-valued approximation approaches\n## Problem setting and contributions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper reconstructs manifold-to-manifold (manifold-valued) maps from values given only at scattered data sites on a source manifold.\"},{\"question\":\"How are kernel methods used in the proposed approach?\",\"answer\":\"Kernel-based approximation is first developed for scalar functions on manifolds, then adapted to the manifold-to-manifold setting via projection to ensure the interpolant takes values in the target manifold.\"},{\"question\":\"What theoretical results and validation are provided?\",\"answer\":\"A new sampling inequality and Sobolev-space error estimates are derived for the scalar kernel approximation stage, and corresponding error estimates are obtained for the manifold-to-manifold reconstruction, supported by numerical 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problem does the paper address?","Question",{"text":76,"@type":77},"The paper reconstructs manifold-to-manifold (manifold-valued) maps from values given only at scattered data sites on a source manifold.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are kernel methods used in the proposed approach?",{"text":81,"@type":77},"Kernel-based approximation is first developed for scalar functions on manifolds, then adapted to the manifold-to-manifold setting via projection to ensure the interpolant takes values in the target manifold.",{"name":83,"@type":74,"acceptedAnswer":84},"What theoretical results and validation are provided?",{"text":85,"@type":77},"A new sampling inequality and Sobolev-space error estimates are derived for the scalar kernel approximation stage, and corresponding error estimates are obtained for the manifold-to-manifold reconstruction, supported by numerical 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