Abstract: Point Cloud Sampling and Recovery (PCSR) is critical for massive real-time point cloud collection and processing since raw data usually requires large storage and computation. This paper addresses a fundamental problem in PCSR: How to downsample the dense point cloud with arbitrary scales while preserving the local topology of discarded points in a case-agnostic manner (i.e., without additional storage for point relationships)? We propose a novel Locally Invertible Embedding (PointLIE) framework to unify the point cloud sampling and upsampling into one single framework through bi-directional learning. Specifically, PointLIE decouples the local geometric relationships between discarded points from the sampled points by progressively encoding the neighboring offsets to a latent variable. Once the latent variable is forced to obey a pre-defined distribution in the forward sampling path, the recovery can be achieved effectively through inverse operations. Taking the recover-pleasing sampled points and a latent embedding randomly drawn from the specified distribution as inputs, PointLIE can theoretically guarantee the fidelity of reconstruction and outperform state-of-the-arts quantitatively and qualitatively.