A Statistical Manifold Framework for Point Cloud Data

[ Information ]

International Conference on Machine Learning, 2022.

[ Authors ]

Yonghyeon Lee*, Seungyeon Kim*, Jinwon Choi, and Frank C. Park (*: equal contribution)

[ Abstract ]

Many problems in machine learning involve data sets in which each data point is a point cloud in $\mathbb{R}^D$. A growing number of applications require a means of measuring not only distances between point clouds, but also angles, volumes, derivatives, and other more advanced concepts. To formulate and quantify these concepts in a coordinate-invariant way, we develop a Riemannian geometric framework for point cloud data. By interpreting each point in a point cloud as a sample drawn from some given underlying probability density, the space of point cloud data can be given the structure of a statistical manifold -- each point on this manifold represents a point cloud -- with the Fisher information metric acting as a natural Riemannian metric. Two autoencoder applications of our framework are presented: (i) smoothly deforming one 3D object into another via interpolation between the two corresponding point clouds; (ii) learning an optimal set of latent space coordinates for point cloud data that best preserves angles and distances, and thus produces a more discriminative representation space. Experiments with large-scale standard benchmark point cloud data show greatly improved classification accuracy vis-a-vis existing methods.