## 4 Experiments
### Experiment Details  
In this paper, when segmenting the benchmark images to superpixels, we equally used the SLIC algorithm and generated 75 superpixel regions. All baseline models initialize the positional vector of all nodes based on the RW algorithm. The vector dimension \(\mathbb{k}\) was set to the same value as the Chebyshev polynomial order \(K\) for ChebNet and was set to a fixed value of 4 for the GCN and GAT.  
The configuration of the baseline model is described as follows. Similar to all datasets, ChebNet has two graph convolutional layers consisting of aggregating the information for \(K\) hop nodes and the ReLU activation function and a graph max-pooling layer in order after passing the last convolution layer. In addition, it has two fully connected layers and has class prediction accuracy based on the softmax regression classifier.  
Unlike ChebNet, we designed a different GCN configuration for one-channel and three-channel datasets. The FashionMNIST dataset is passed two graph convolutional layers based on the \(MP\) and the readout layer defined as the mean function, graph max-pooling layer, and fully connected layer in order. Furthermore, its accuracy is based on the softmax regression classifier. The other RGB datasets are passed four graph convolutional layers, mean-readout, graph max-pooling, and three fully connected layers to obtain a result.  
The GAT, similar to all datasets, has two layers of graph convolution based on two multihead attention mechanisms and readout layers, defined by the sum function, and passes the multilayer perceptron as a fully connected layer to obtain the results.  
The learning procedures of the baseline models with the learnable positional embedding method are provided in Equations (24)-(26) in the order of the ChebNet, GCN, and GAT:  
\[\textit{Cheb}-\textit{LPE}:\begin{cases}h_{i}^{l+1}=ReLU\left(BN \left(\left[A_{1}^{l}\left[\begin{array}{c}h_{i}^{l}\\ p_{i}^{l}\end{array}\right],\left(\sum\limits_{k=1}^{K-1}A_{2}^{l}\left[ \begin{array}{c}h_{k}^{l}\\ p_{k}^{l}\end{array}\right]\right)\right]\right)\right)\\ \textit{e}_{ij}^{l+1}=ReLU\left(BN\left(\sum\limits_{k=1}^{K-1}(B_{2}^{l}e_{ ik}^{l})\right),\\ \textit{p}_{i}^{l+1}=\tanh(C_{1}^{l}p_{i}^{l}+\sum\
olimits_{j\in\mathcal{N}_{ i}}C_{2}^{l}p_{j}^{l})\end{cases} \tag{24}\]  
\[\textit{GCN}-\textit{LPE}:\begin{cases}h_{i}^{l+1}=ReLU(BN(A_{1}^{l}\left[ \begin{array}{c}h_{j}^{l}\\ p_{i}^{l}\end{array}\right]+\sum\
olimits_{j\in\mathcal{N}_{i}}A_{2}^{l}\left[ \begin{array}{c}h_{j}^{l}\\ p_{j}^{l}\end{array}\right])),\\ \textit{e}_{ij}^{l+1}=ReLU(BN(B_{1}^{l}h_{i}^{l}+B_{2}^{l}h_{j}^{l}+e_{ij}^{l}) ),\\ \textit{p}_{i}^{l+1}=\tanh(C_{1}^{l}p_{i}^{l}+\sum\
olimits_{j\in\mathcal{N}_{ i}}C_{2}^{l}p_{j}^{l})\end{cases}\end{cases} \tag{25}\]  
\[\textit{GAT}-\textit{LPE}:\begin{cases}e_{ij}=\overset{\rightarrow}{a}\left(A_ {1}^{l}\left[\begin{array}{c}h_{j}^{l}\\ p_{i}^{l}\end{array}\right],\ A_{2}^{l}\left[\begin{array}{c}h_{j}^{l}\\ p_{j}^{l}\end{array}\right]\right),\ \ \alpha_{ij}=Softmax(e_{ij})\\ \textit{h}_{i}^{l+1}=ELU(\frac{1}{K}\sum\limits_{j\in\mathcal{N}_{i}}\alpha_{ ij}^{k}A_{2}^{l}h_{j}^{l}),\\ e_{ij}^{l+1}=ReLU(BN(B_{1}^{l}h_{i}^{l}+B_{2}^{l}h_{j}^{l}+B_{3}^{l}e_{ij}^{l}) \\ \textit{p}_{i}^{l+1}=\tanh(C_{1}^{l}p_{i}^{l}+\sum\
olimits_{j\in\mathcal{N}_{ i}}C_{2}^{l}p_{j}^{l})\end{cases}\end{cases} \tag{26}\]  
where \(BN\) denotes batch normalization.  
We set the hyperparameters that the epoch was 100, the batch size was 64, the learning rate was 0.001, and the dropout rate was 0.5. For optimization, we used the stochastic gradient descent [47] with a momentum of 0.9 for ChebNet and Adam [48] with a weight decay of 0.95 for the GCN and GAT. In addition, the scale parameter (s) and the angular margin penalty (m) were set to 64 and 0.5 radian (approximately 29\({}^{\circ}\)), respectively. The weight of ArcFace was set to 0.2.