# A Novel Equivariant Self-Supervised Vector Network for Three-Dimensional Point Clouds
## 4 Experiment
### Reconstruction  
The goal of pre-training in this paper is the reconstruction task, which can realize the reconstruction of incomplete point cloud data. The datasets adopted for this task are ShapeNet and point clouds sampled from the HumanBody dataset. For ShapeNet, we use an FPS operation to unify all point cloud models into 1024 points and divide them into 64 patches with 32 points in each patch, which are used as input. For point clouds sampled from the HumanBody dataset, we divide 4096 points into 256 patches with 32 points in each patch. Before the data are fed into the network, we randomly rotate it to ensure that each sample has a different rotation state each time. The network of the reconstruction is a complete network structure, including the equivariant vector layers at the beginning and the subsequent autoencoder module. In the end, all tokens are mapped and reconstructed into the ground truth shape through a simple MLP layer, and the output is \(P_{pre}\in\mathbb{R}^{mn\times k\times 3}\), which is finally combined with the input visible point cloud patches to form a complete point cloud. Since our input is a randomly rotated point cloud, our reconstruction target should be a point cloud that is consistent with the rotated state of the input to verify the equivariance of our network. To demonstrate the effectiveness of our approach, we visualize [6] our reconstruction results on the ShapeNet dataset in Figure 3 and human point clouds in Figure 4. As we can see in Figure 3, the reconstruction effect of our network is very excellent. It can not only well reconstruct the point cloud under different rotation states, but also yield a reconstructed point cloud with less noise, and the reconstruction result is very simple and fits the original input. As for the human point clouds, we construct a clean human point cloud dataset from the HumanBody dataset. Our network also performs very well on complex human point clouds; there is no excessive adhesion, and the human limbs are relatively distinct. The network is pre-trained with a 60% mask rate, but it is also able to reconstruct the input with various masked ratios. This high generalization and strong robustness verify that our model can better learn advanced latent features and can be used as a good pre-trained model for downstream tasks. Meanwhile, we also study the latent features of our network. The MAE has been active in networks of various data types with excellent performance since its advent, but its performance in tasks of rotation equivariance and rotation invariance has not been well explored. And, because the neural network is a black box, its internal representation has always had a fuzzy existence that is difficult to express. We verify the interpretability and equivariance by visualizing the latent features generated by the encoder module. As depicted in Figure 5, we visualize the latent feature \(z(\theta)\) in our network. Remarkably, we observe that \(z(\theta)\) exhibits rotation congruent with the input as the data undergo rotation. This indicates that, after using vector neurons, our network possesses rotation equivariance,  
Figure 4: Rendering of the reconstruction effect of our network on human point clouds. The human point clouds are trained in the same way as ShapeNet. These human point clouds were obtained by sampling on the mesh model of the HumanBody dataset. Since the initial pose in the original HumanBody dataset is confusing (i.e., the pose of the point cloud below the mesh is the origin pose), we manually adjusted the mesh to show its rough pose and appearance. The training in the figure uses a 60% mask rate. Figure 3: Rendering of the reconstruction effect of our network on ShapeNet. The network will reconstruct the visible point clouds under different rotation states. During training, the network only learns the non-rotated points, but, in the test, the network can also reconstruct the rotated point clouds under z and SO(3) conditions well. The training in the figure uses a 60% mask rate.