[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86360-en":3,"doc-seo-86360-105":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},86360,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations","A data-driven interpolation framework reconstructs real-valued functions on smooth manifolds from scattered pointwise observations. The approach blends a Gaussian Nadaraya–Watson kernel interpolant with a Voronoi-adaptive bandwidth derived solely from sampled-data geometry, producing an explicit closed-form construction without training, iterative optimization, preprocessing, or parameter tuning. The interpolant exactly matches observed values, enforces a vanishing intrinsic gradient at samples, and in the dense-sampling limit suppresses high-frequency oscillations via geometric regularization. It also connects to minimizing a discrete total-variation–type functional, enabling linear-time inference. Applied to sparse-angle computed tomography, it interpolates the sinogram prior to filtered backprojection for accurate reconstructions with lower compute time than total-variation baselines.","arXiv :2509 .03758v 5 [ cs .LG] 12 Jul 2026  \nA DATA-DRIVEN INTERPOLATION METHOD ON SMOOTH MANIFOLDS VIA DIFFUSION PROCESSES AND VORONOI  \nTESSELLATIONS  \nAlvaro Almeida Gomez  \nCentro de Modelamiento Matemtico  \nUniversidad de Chile  \nBeaucheff 851, Santiago, Chile  \n[alvaroalmeidagomez182@gmail.com](alvaroalmeidagomez182@gmail.com)  \nJuly 14, 2026  \nABSTRACT  \nWe propose a data-driven interpolation framework for reconstructing real-valued functions on smooth manifolds from scattered pointwise observations. The method combines a Gaussian Nadaraya–Watson kernel interpolant with a Voronoi-adaptive bandwidth determined entirely by the geometry of the sampled data, yielding an explicit closed-form construction that requires neither training, iterative optimization, preprocessing, nor parameter tuning.  \nThe proposed interpolant satisfies several theoretical properties. It reproduces the observed data exactly, enforces a vanishing intrinsic gradient at every sample point, and, in the dense-sampling limit, attenuates high-frequency oscillatory components through the geometric regularization induced by the adaptive bandwidth. Furthermore, the construction admits an interpretation in terms of minimizing a discrete total variation–type functional, establishing a natural connection with compressed sensing and sparsity-promoting regularization.  \nUnlike classical kernel interpolation methods employing a fixed global bandwidth, the proposed adaptive strategy automatically adjusts to the local sampling geometry through the Voronoi tessellation while preserving an explicit analytical formulation. Because the interpolant is available inclosed form, the overall computational cost is entirely determined by the inference stage: evaluating the interpolant at a query point requires only the computation of Gaussian kernel weights and their weighted combination, resulting in linear complexity with respect to the number of sample points.  \nIn contrast to many data-driven interpolation approaches, no additional offline computational stage is required before inference.  \nFinally, we demonstrate the practical performance of the proposed methodology in sparse-angle computed tomography reconstruction, where interpolation of the sinogram prior to filtered backprojection produces accurate reconstructions while substantially reducing the overall computational time compared with standard total variation–based reconstruction methods.  \nKeywords Diffusion Geometry ⋅ Voronoi Tessellations ⋅ Manifold Learning ⋅ Kernel Methods ⋅ Total Variation ⋅ Signal Attenuation ⋅ Sparse-View CT Reconstruction  \nInterpolation is a fundamental problem in applied mathematics, scientific computing, and machine learning, where the objective is to reconstruct an unknown function from a finite collection of observations. The subject has a long history, originating with the classical polynomial schemes of Lagrange and Newton and evolving, through the resolution of phenomena such as Runge’s instability for high-degree polynomial interpolants and the Whittaker–Shannon sampling theorem for bandlimited signals, into the modern theory of splines, radial basis functions, and finite element approximations [1, 2] . These methods furnish powerful and mathematically rigorous frameworks for function reconstruction and have been applied extensively in approximation theory, numerical analysis, and scientific computing. Nevertheless, their performance typically depends on assumptions regarding the regularity of the target function, the geometry  \nA PREPRINT-JULY 14, 2026  \nof the underlying domain, or the availability of suitable basis functions, and highly oscillatory signals often require dense sampling or high-dimensional approximation spaces to achieve satisfactory reconstruction accuracy.  \nRecent developments have extended classical Euclidean interpolation toward geometry-aware and data-adaptive methodologies. Scattered-data interpolation, kernel-based techniques, and Gaussian process model","cbCaij2EcyAFceBy","https://ap.wps.com/l/cbCaij2EcyAFceBy","pdf",3359977,5,1,39,"English","en",105,"# Abstract\n## Problem Setup\n## Method Overview\n## Theoretical Properties\n## Computational Complexity\n## Practical Application (Sparse-View CT)","[{\"question\":\"What problem does the proposed method address?\",\"answer\":\"It reconstructs real-valued functions on smooth manifolds from scattered pointwise observations.\"},{\"question\":\"How does the method construct the interpolant?\",\"answer\":\"It combines a Gaussian Nadaraya–Watson kernel with a Voronoi-adaptive bandwidth determined from the sampled data geometry, yielding an explicit closed-form formula.\"},{\"question\":\"What are the main theoretical and computational guarantees?\",\"answer\":\"The interpolant reproduces observed data exactly, enforces vanishing intrinsic gradients at sample points, and attenuates high-frequency oscillations in dense sampling, while inference is linear in the number of sample points and requires no offline 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problem does the proposed method address?","Question",{"text":76,"@type":77},"It reconstructs real-valued functions on smooth manifolds from scattered pointwise observations.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the method construct the interpolant?",{"text":81,"@type":77},"It combines a Gaussian Nadaraya–Watson kernel with a Voronoi-adaptive bandwidth determined from the sampled data geometry, yielding an explicit closed-form formula.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the main theoretical and computational guarantees?",{"text":85,"@type":77},"The interpolant reproduces observed data exactly, enforces vanishing intrinsic gradients at sample points, and attenuates high-frequency oscillations in dense sampling, while inference is linear in the number of sample points and requires no offline 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