Problem
Point-cloud learning often assumes that samples lie near an underlying manifold. Coordinates, pairwise distances and learned graph neighbourhoods capture part of this structure, but not the higher-order tangency information represented by differential forms in the smooth setting.
Method
Neural Point-Forms use Laplacian methods from diffusion geometry to construct a discrete comparison of differential forms. Learned matrices compare local geometric features across the point cloud.
The resulting layer is permutation invariant and works without a supplied tangent bundle. The paper also proves long-run consistency of the comparison matrices under standard assumptions on sampling, bandwidth, density and the manifold hypothesis.
Role and collaboration
I am the first author. The work was developed with Jacob Hume, Vincenzo Antonio Isoldi, Philipp Misof, Ekaterina S. Ivshina and Kelly Maggs.
Evaluation
The method is evaluated on synthetic data and biologically relevant point clouds, including tasks where labels depend on sampling density, manifold-like structure or response-relevant population geometry.