A Statistical Manifold Framework for Point Cloud Data
Yonghyeon Lee, Seungyeon Kim, Jinwon Choi, Frank Park
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-á-vis existing methods. Code is available at https://github.com/seungyeon-k/SMF-public.
BibTeX
@InProceedings{pmlr-v162-lee22d,
title = {A Statistical Manifold Framework for Point Cloud Data},
author = {Lee, Yonghyeon and Kim, Seungyeon and Choi, Jinwon and Park, Frank},
booktitle = {Proceedings of the 39th International Conference on Machine Learning},
pages = {12378--12402},
year = {2022},
editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
volume = {162},
series = {Proceedings of Machine Learning Research},
month = {17--23 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v162/lee22d/lee22d.pdf},
url = {https://proceedings.mlr.press/v162/lee22d.html},
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-á-vis existing methods. Code is available at https://github.com/seungyeon-k/SMF-public.}
}