# A Novel Equivariant Self-Supervised Vector Network for Three-Dimensional Point Clouds
## 4 Experiment
### Pose Change Estimation  
Our network shows excellent equivariant properties in the reconstructed task, so we set out to extend it to downstream tasks. We evaluate the pose change estimation ability of our pre-trained model on the ModelNet40 dataset. Our pose change estimation for the point cloud is mainly to predict the overall pose change direction of point clouds at the category level. When the point cloud is rotated, an isometric transformation is generated between the point cloud and the original state. This isometric rotation transformation can be expressed in the form of a \(3\times 3\) rotation matrix, which is denoted as \(S\in\mathbb{R}^{3\times 3}\). The previous  
Figure 5: Demonstration of the latent features of the network. The top of each row is the state of the input point cloud, and the bottom is the state of its corresponding latent feature, which can be seen to rotate with the rotation of the point cloud. The origin column is the point cloud without rotation, and the rest are the point clouds rotated by different angles, for example, 90\({}^{\circ}\)z means that the point cloud rotates 90 degrees around the z-axis. methods of pose estimation generally use quaternion to describe the rotation, which is more concise, but the description of rotation is too abstract to be expressed intuitively by images. While it is possible to convert it into a rotation matrix through formulaic calculations, this outcome does not align with our intended objective. We prefer to use the rotation matrix as an intuitive representation to express the pose change. The purpose of our fine-tuning head is therefore to directly predict this rotation matrix \(S\), allowing the network to learn the pose information of the point cloud and visualize the rotation at the same time. Since this task is complex, not a simple classification and segmentation, and it needs to maintain equivariant transmission, we use a one-dimensional convolutional layer and a vn-pooling layer to replace the original linear layer fine-tuning head. The final designed network will directly output the predicted matrix \(S^{\prime}\in\mathbb{R}^{3\times 3}\). This task is not a normal classification and segmentation task that uses label loss, so we design a new loss for this task:  
\[\mathcal{L}_{2}=\alpha\ d(S,S^{\prime})+\beta\ ||\ |S^{\prime}|-1\ || \tag{21}\]  
The function \(d\) computes the Euclidean distance between the corresponding vectors of the rotation matrix, \(\alpha\) and \(\beta\) are the coefficients used to adjust the weights, and the right-hand side is the matrix regularization term for the rotation of the output. If the determinant of the rotation matrix is not 1, that is, the matrix is not orthogonal, then the affine transformation applied to the point cloud by the rotation matrix is not an isometric change. Consequently, the point cloud may undergo stretching or compression, resulting in deformation. To address this issue, it becomes essential to impose specific regularization constraints on the rotation matrix. To measure the final effect, we use the cosine similarity. When the cosine similarity between \(S^{\prime}\) and each corresponding vector in \(S\) is larger, it means that each vector is closer in that direction, which also indicates that the prediction effect is better. Figure 6 is a visual presentation of the experiment. When the rotated point cloud is fed into the network, the network outputs its corresponding rotation matrix, which is the predicted pose change. We visualize this as the  
Figure 6: Gradual rotation of point clouds and their corresponding output results. This figure shows how the network outputs different poses for different angles of the same point cloud around the same axis (z-axis). The top row is the input point cloud \(P\) after applying the rotation matrix \(S\), and the bottom row is the predicted pose generated by the network based on the rotated point cloud \(P_{r}\), which is generated from the point cloud \(P\) after applying the rotation matrix \(S\).