Point cloud is a prevalent format in representing 3D geometry. Regardless of the recent advances, unsupervised learning for 3D point clouds remains arduous for various tasks due to its unorganized and sparsely distributed nature. To address this challenge, we propose a geometry regularized point cloud autoencoder, aiming to preserve local geometry structure. In particular, based on the Mahalanobis distance, we propose a point cloud geometry metric counting the local statistics. It endeavors to maximize the posterior probability of the reconstruction conditioned on the input point cloud. Our eigenspace analysis reveals the adaptivity of the developed metric—it behaves differently given different local structures. Moreover, a coarse-to-fine training strategy by varying the metric granularity is applied, leading to our proposed geometry regularized point cloud autoencoder. By applying our proposal to several off-the-shelf point cloud autoencoders, we show an improved point cloud reconstruction quality. In addition, the superior representability of our learned features is also demonstrated via an object classification task.

Sliced Wasserstein distances preserve properties of classic Wasserstein distances while being more scalable for computation and estimation in high dimensions. The goal of this work is to quantify this scalability from three key aspects: (i) empirical convergence rates; (ii) robustness to data contamination; and (iii) efficient computational methods. For empirical convergence, we derive fast rates with explicit dependence of constants on dimension, subject to log-concavity of the population distributions. For robustness, we characterize minimax optimal, dimension-free robust estimation risks, and show an equivalence between robust sliced 1-Wasserstein estimation and robust mean estimation. This enables lifting statistical and algorithmic guarantees available for the latter to the sliced 1-Wasserstein setting. Moving on to computational aspects, we analyze the Monte Carlo estimator for the average-sliced distance, demonstrating that larger dimension can result in faster convergence of the numerical integration error. For the max-sliced distance, we focus on a subgradient-based local optimization algorithm that is frequently used in practice, albeit without formal guarantees, and establish an $O(\epsilon^{-4})$ computational complexity bound for it. Our theory is validated by numerical experiments, which altogether provide a comprehensive quantitative account of the scalability question.