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

Abstract

We 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. The 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. Unlike 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 in closed 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. In contrast to many data-driven interpolation approaches, no additional offline computational stage is required before inference.