A Statistical Manifold Framework for Point Cloud Data

A large class of problems in machine learning involve data sets in which each data point is a point cloud in $\mathbb{R}^D$. The reason that most machine learning algorithms designed for point cloud data tend to be ad hoc, and difficult to measure their performance in a uniform and quantitative way, can be traced to the lack of a rigorous mathematical characterization of this space of point cloud data. The primary contribution of this paper is a Riemannian geometric structure for point cloud data. By interpreting the point cloud data as a set of samples from some underlying probability distribution, the set of point cloud data can be given the structure of a statistical manifold, with the Fisher information metric acting as a natural Riemannian metric; this structure then leads to, e.g., distance metrics, volume forms, and other coordinate-invariant, geometrically well-defined measures needed for applications. The only requirement on the part of the user is the choice of a meaningful underlying probability distribution, which is more intuitive and natural to make than what is required in existing ad hoc formulations. Two autoencoder case studies involving point cloud data are presented to demonstrate the advantages of our statistical manifold framework: (i) interpolating between two 3D point cloud data sets to smoothly deform one object into another; (ii) transforming the latent coordinates into another with less distortion. Experiments with synthetic and large-scale standard benchmark point cloud data show more natural and intuitive shape evolutions, and improved classification accuracy for linear SVM vis-\`{a}-vis existing methods.