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

arXiv:2509.03758v3 Announce Type: replace 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 proposed method minimizes a total variation-type energy, thereby yielding a closed-form analytical approximation to the compressed sensing problem in the case where the forward operator is the identity. Finally, we present applications to sparse computational tomography reconstruction. Numerical experiments demonstrate that the proposed method achieves competitive reconstruction quality while significantly reducing computational time compared to classical total variation-based reconstruction methods.