Approximation of symmetric total variation on point clouds

TL;DR

This paper studies how to approximate symmetric Total Variation on point clouds using a discrete finite difference model, ensuring convergence to a weighted Total Variation as the sampling scales appropriately.

Contribution

It introduces a method for approximating symmetric Total Variation on point clouds with proven convergence guarantees under suitable scalings.

Findings

The approximation converges almost surely to the anisotropic weighted symmetric Total Variation.

Suitable scalings of point distributions are identified to ensure convergence.

The approach provides a rigorous link between discrete point cloud models and continuous Total Variation.

Abstract

The paper investigates the approximation of the symmetric Total Variation functional on graphs. Such an approximation is given in terms of a discrete and symmetric finite difference model defined on point clouds obtained by randomly sampling a reference probability measure. We identify suitable scalings of the point distribution that guarantee an almost surely $\Gamma$-convergence to an anisotropic weighted symmetric Total Variation.