PointOT: Interpretable Geometry-Inspired Point Cloud Generative Model via Optimal Transport

Point cloud generative models have aroused increasing concern for their realistic generation potentialities. However, most existing methods adopt deep-neural-network (DNN) models for continuous mapping. DNN usually induces mode collapse and mixture problems without clear interpretation. Consequently, in this paper, we design a geometry-inspired point cloud generative framework called PointOT. PointOT decouples the generative model into two separate sub-tasks: manifold learning of the point cloud and distribution transformation. Then, we propose corresponding instances according to the requirements in each sub-task, where they can be established by the point cloud auto-encoder (AE) and the semi-continuous optimal transportation (SCOT) mapping, respectively. In particular, the transportation map between the source and the target distributions is discrete rather than continuous in geometric view. The learned continuous shape model of the DNN point cloud does not conform with the discrete distribution transformation. Therefore, the proposed SCOT efficiently relieves these problems by connecting the continuous-to-discrete domain. Besides, we provide theoretical explanations from a geometric view and analyze the fundamental reason for mode collapse and mixture in point cloud generative models. The proposed SCOT algorithm without the DNN model is computationally efficient and makes the original black box semi-transparent. Final experiments validate the virtue of the proposed approach, including the designed decomposition framework and the rigorous theory.