# A Novel Equivariant Self-Supervised Vector Network for Three-Dimensional Point Clouds
## 2 Related Work
### Equivariant Network  
Equivariant network research mainly focuses on two types of methods: steerable kernel-based methods that construct group convolutions, and vector network methods that use vector representations. In steerable kernel-based methods, [PERSON] et al. introduced networks using group equivariant convolutional [8; 48; 49; 50; 51] that learn the directional information in the input data by group convolutions, and mapped the 3D data to the sphere by applying spherical convolutions [52; 53]. By employing this approach, they could process 3D data in an equivariant manner. In vector-based networks, [PERSON] et al. target the CNN pooling operation to destroy the rotation and other characteristics of the network, design the capsule network [54; 55; 56; 57; 58], which uses capsules to store vector information, and design a dynamic routing mechanism so that direction information can be transmitted in the network. The equivariance embedded in the aforementioned networks is beneficial for processing the information of the 3D data. When it comes to tasks, they can be divided into the following two categories:  
**Rotation Invariant:** Rotation invariance is a desirable property for label-class tasks such as classification and segmentation, and many rotation-invariant architectures have been proposed to solve these problems [27; 28; 29; 30; 31]. For example, GC-Conv [29] relies on PCA-based multi-scale reference frames; SFCNN [30] is a method similar to multi-view, which maps the point cloud to a sphere and relies on spherical convolution to achieve the property of rotation invariance; ClusterNet [27] uses hierarchical clustering to explore the geometric structure of point clouds in hierarchical trees; and Deltaconv [31] employs a two-stream structure of a vector and scalar to capture the surface invariant information of point clouds, and makes the point cloud from all angles output the same feature through the construction of a basis vector.  
**Rotation Equivariant:** Rotation-equivariant deep learning frameworks [59; 60; 61] have mostly relied on the notion of steerable kernel-based convolution. For example, spherical CNNs [52; 53] convert the input point cloud into a spherical signal and use spherical harmonic filters that produce features in spherical space. However, these methods require a specific architecture to match the specially processed data which has good performance but also limits the applicability and adaptability of the network. The VNN [15] is a concise SO(3) equivariant framework with good accessibility and generality, where SO(3) denotes any rotation in 3D space. Conceptually, any standard point cloud processing network can be promoted to an SO(3) equivariant network while maximally retaining the original architecture.  
Although the above methods improve the network's ability to process rotated point clouds, most networks do not have expanded capabilities. Even some rotation-equivariant networks limit their tasks to rotation-invariant tasks like classification, and the complex structures of most networks make it difficult to adapt to different tasks. The method in this paper retains the vector information contained in the 3D data by constructing vector neurons and trains the network using self-supervised learning, making the network have excellent transferability and exploring both rotation-equivariant and rotation-invariant tasks.