## 3 Methods
### Auxiliary Classifier Generative Adversarial Networks  
Based on the idea of an adversarial game, [PERSON] et al. [21] pioneered the GAN model. The model consists of a generator (\(G\)) and a discriminator (\(D\)), where the input of the generator is the noise vector \(z\) and the output is the synthetic fake data \(X_{Synthetic}=G(z)\). The input of the discriminator is real data \(X_{Real}\) and synthetic data, and the output is true and false labels. As shown in Figure 1a, the training of the GAN model is divided into two stages. **Train the generator:** The parameters of the discriminator are fixed, and the synthetic data output by the generator is output by the discriminator to obtain the true and false discrimination loss. **Train the discriminator:** The generator parameters are fixed, and the synthetic data and real data are passed into the discriminator output to obtain the true and false discrimination loss. The two joint losses above are passed through the backpropagation algorithm, and the overall loss function is as follows:  
\[L(D,G)=E_{s\sim P_{\text{data}}}\left[\log D(s)\right]+E_{z\sim P_{z}}[\log(1- D(G(z)))] \tag{1}\]  
where, \(s\) is the discriminator input vector \(s\sim P_{\text{data}}\), \(P_{\text{data}}\) represent the real data distribution. \(z\) is the generator input vector \(z\sim P_{z}\), and \(P_{z}\) represent the noisy data distribution. \(E(\bullet)\) is the mathematical expectation. The optimization process of the model can also be simplified to the maximum and minimum function problem, and the formula is:  
\[\text{Goal }=\underset{G}{\text{argminmax}}L(G,D) \tag{2}\]  
The meaning of Formula (2) can be understood as follows. The optimization goal of the GAN model is that model \(D\), corresponding to the maximum loss function \(L(G,D)\), and model \(G\), corresponding to the minimum loss, are the trained GAN model.  
ACGAN is an improvement of GAN, which utilizes the introduction of label information to add an auxiliary classifier to the discriminator, thereby improving the training stability of the model and generating higher quality data samples, as shown in Figure 1b. Another advantage of ACGAN is that it only needs one trained model to synthesize the data of any label, reducing the time to repeatedly train the model for different label data.  
Specifically, the generator of ACGAN can be expressed as \(X_{Synthetic}=G(z,y)\), where \(y\) is a label vector, and the discriminator can not only judge the authenticity of the data but also the category of the data. During the training process, both the discriminator and the generator have two-part loss functions, where \(L_{\text{T-F}}\) is the true and false loss functions and \(L_{\text{class}}\) is the label loss function.  
\[L_{\text{T-F}}=E_{s\sim P_{\text{data}}}\left[\log D(s)\right]+E_{z\sim P(z)}[ \log(1-D(G(z)))] \tag{3}\]  
\[L_{\text{class}}\ =E_{s\sim P_{\text{data}}}\left[\log P(Y=y|S_{Real}\ )\right]+E_{z\sim P(z)}[\log P \Big{(}Y=y\Big{|}S_{Synthetic}\Big{)}\Big{]} \tag{4}\]  
The training process of the model is similar to that of GAN. First, the parameters of the discriminator are fixed, the generator is trained, and the loss function is \(L_{\text{T-F}}-L_{\text{class}}\). Second, the generator parameters are fixed, the discriminator is trained, and the loss function is \(L_{\text{T-F}}+L_{\text{class}}\).