## 2 Materials and Methods
### Squeeze-And-Excitation Modules  
Convolutional neural networks use the idea of local receptive fields to fuse channel information and spatial information to extract features containing information. The squeeze-and-excitation (SE) module proposed by [PERSON] et al. [25] can capture channel information. It is well applied in a variety of classification tasks [26; 27; 28; 29]. The SE module considers the relationships between feature channels and adds a focus mechanism for feature channels. This mechanism is particularly helpful for improving the recognition ability of the proposed model [19]. For fine-grained classification problems such as butterfly classification, the SE module can weaken the similarity problem between classes. By assigning different weights to different channels, the SE module enables the network to selectively enhance the characteristics of large amounts of information and suppress the useless characteristics. The structure of the SE module is shown in Figure 3.  
In order to achieve global information embedding, the global information is squeezed into the channel descriptor by using a global average pooling. The squeeze operation compresses each two-dimensional feature channel into a real number. The real number has a global acceptance field, and the size of the output matches the number of input feature channels. For feature maps \(\mathbf{U}=[\mathbf{u}_{1},\mathbf{u}_{2},\cdots,\mathbf{u}_{c}]\in\mathbb{R}^ {\mathbb{C}\times H\times W}\), where \(\mathbf{u}_{k}\in\mathbb{R}^{H\times W}\) is a feature map at the k-th channel. The k-th element of z (\(\mathbf{z}\in\mathbb{R}^{C}\)) can be obtained by using Equation (1).  
\[\mathrm{z}_{k}=\mathrm{F}_{sq}(\mathbf{u}_{k})=\frac{1}{W\times H}\sum_{i=1}^{ W}\sum_{j=1}^{H}\mathbf{u}_{k}(i,j) \tag{1}\]  
After the squeeze operation, the excitation operation is performed. The excitation operation relies on the information gathered in the squeeze operation to capture channel-wise dependencies. Weights are generated for each feature channel by a parameter w, which is used to explicitly model the correlation between functional channels. The weight of the excitation output after feature selection represents the importance of each feature channel. Then, multiplication is used to achieve channel-by-channel weighting of the previous features to complete the rescaling of the original features in the channel dimension.  
Figure 3: Squeeze-and-excitation module.  
Figure 2: Sample images of 10 categories in the Leeds Butterfly Dataset.  
The structure of the excitation operation includes \(1\times 1\) convolution layer, ReLU layer, \(1\times 1\) convolution layer and Sigmoid activations layer, in order. The results of the excitation operation can be obtained using Equation (2).  
\[\mathbf{s}=F_{\alpha}(\mathbf{z},\mathbf{W})=Sigmoid(\mathbf{W}_{2}\times ReLU (\mathbf{W}_{1}\mathbf{z})) \tag{2}\]  
Here, \(\mathbf{W}_{1}\in\mathbb{R}^{\frac{C}{r}\times C}\) and \(\mathbf{W}_{2}\in\mathbb{R}^{C\times\frac{C}{r}}\). Reduction ratio r = 16 [30].  
After the squeeze and excitation operations, according to Equation (3), the final output \(\mathbf{\tilde{x}}_{k}\) at the k-th channel is obtained by the rescaling of the feature map \(\mathbf{u}_{k}\) with the activations \(\mathbf{s}_{k}\).  
\[\mathbf{\tilde{x}}_{c}=F_{scale}(\mathbf{u}_{k},\mathbf{s}_{k})=\mathbf{s}_{k} \times\mathbf{u}_{k} \tag{3}\]  
Here, \(\mathbf{\tilde{X}}=\left[\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots,\mathbf{\tilde{x}}_{c}\right]\).