Point cloud analysis is still a challenging task due to the disorder and sparsity of samplings of their geometric structures from 3D sensors. In this paper, we introduce the homotopy equivalence relation (HER) to make the neural networks learn the data distribution from a high-dimension manifold. A shuffle operation is adopted to construct HER for its randomness and zero-parameter. In addition, inspired by prior works, we propose a local mutual information regularizer (LMIR) to cut off the trivial path that leads to a classification error from HER. LMIR utilizes mutual information to measure the distance between the original feature and HER transformed feature and learns common features in a contrastive learning scheme. Thus, we combine HER and LMIR to give our model the ability to learn non-Euclidean features from a high-dimension manifold. This is named the non-Euclidean feature learner. Furthermore, we propose a new heuristics and efficiency point sampling algorithm named ClusterFPS to obtain approximate uniform sampling but at faster speed. ClusterFPS uses a cluster algorithm to divide a point cloud into several clusters and deploy the farthest point sampling algorithm on each cluster in parallel. By combining the above methods, we propose a novel point cloud analysis neural network called PointShuffleNet (PSN), which shows great promise in point cloud classification and segmentation. Extensive experiments show that our PSN achieves state-of-the-art results on ModelNet40, ShapeNet and S3DIS with high efficiency. Theoretically, we provide mathematical analysis toward understanding of what the data distribution HER has developed and why LMIR can drop the trivial path by maximizing mutual information implicitly.