[2509.03758] A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations

Computer Science > Machine Learning arXiv:2509.03758 (cs) [Submitted on 3 Sep 2025 (v1), last revised 4 Apr 2026 (this version, v3)] Title:A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations Authors:Alvaro Almeida Gomez View a PDF of the paper titled A Data-Driven Interpolation Method on Smooth Manifolds via Diffusion Processes and Voronoi Tessellations, by Alvaro Almeida Gomez View PDF Abstract:We propose a data-driven interpolation method for approximating real-valued functions on smooth manifolds, based on the Laplace--Beltrami operator and Voronoi tessellations. Given pointwise evaluations of a function, the method constructs a continuous extension over the manifold by exploiting diffusion processes and the intrinsic geometry of the data. The proposed approach is entirely data-driven and requires neither a training phase nor any preprocessing prior to inference. Furthermore, the computational complexity of the inference step scales linearly in the number of sample points, thereby providing substantial improvements in scalability and computational efficiency compared to classical data driven interpolation methods, including neural networks, radial basis function networks, and Gaussian process regression. We further show that the interpolant has vanishing gradient at the interpolation points and, with high probability as the number of samples increases, attenuates high-frequency components of the signal. Moreover, the ...