## 2 Theoretical Method Introduction  
### Deep Convolutional Generative Adversarial Network  
The Generative Adversarial Network is a machine learning architecture proposed by [PERSON] in 2014, and it is one of the most promising methods of unsupervised learning in recent years. Due to its outstanding data generative capabilities, Generative Adversarial Networks have been fully applied in many fields, especially in image processing, text generation, audio processing, data enhancement, and sample prediction. The main idea of GAN is derived from game theory, in which a discriminative model and a generative model are set up to compete with each other to improve the learning effect and obtain high-dimensional and complex data distribution of real samples. Its network structure is shown in Figure 1:  
The main structure of GAN includes a generator and a discriminator [37; 38]. The generator is used to generate real samples, and the discriminator is used to determine real samples and fake samples. The generator \(G\) inputs a random noise vector \(z\) (usually uniform or normal distribution) and maps it to a new multi-dimensional data space to achieve a generated fake sample \(G(z)\). The discriminator \(D\) is used to perform classification and to calculate the probability that the testing sample is a real sample. In the training, the generator \(G\) tries to learn the data distribution of the real sample to generate a fake sample, and the discriminator \(D\) tries to distinguish the generated fake sample from the real sample. The objective function of the generative adversarial network is as follows:  
\[\underset{G}{\text{minimax}}V(G,D)=E_{x\sim P_{data}(x)}[\log D(x)]+E_{z\sim P _{z}(z)}[\log(1-D(G(z)))] \tag{1}\]  
During the training process of the GAN network, the objective function of \(D(x)\) is:  
\[\underset{D}{\text{max}}V(D,G)=E_{x\sim P_{data}(x)}[\log(D(x))]+E_{z\sim P_{z }(z)}[\log(1-D(G(z)))] \tag{2}\]  
Figure 1: GAN network structure diagram. The parameters are optimized by Equation (2) to improve \(D(x)\). The objective function of \(G\) is:  
\[\underset{G}{\text{min}}V(D,G)=E_{z\sim P_{z}(z)}[\log(1-D(G(z)))] \tag{3}\]  
Among them, \(x\) represents the data sample, \(P_{z}(z)\) represents the data distribution of the input noise, \(G(z)\) represents the sample generated by the input noise, \(D(x)\) represents the probability that sample \(x\) determined by the discriminator is the real sample instead of the generated sample. \(E_{x\sim P_{data}(x)}[\log D(x)]\) represents the probability of \(D(x)\) obtaining real data; \(E_{x\sim P_{z}(z)}[\log(1-D(G(z)))]\) represents the probability of \(D(x)\) discriminating real data. During the training process, alternate iterations are performed. Firstly, the network \(G\) is fixed, and the discriminator network \(D\) is trained so that the labels of the training samples are classified correctly with the greatest probability, with maximized \(\log D(x)\) and \(\log(1-D(G(z)))\). Then, the network \(D\) is fixed, and the network \(G\) is trained to minimize \(\log(1-D(G(z)))\). The training of the generator network puts \(D(G(z))\) close to 1, so that the generator loss will be the smallest. The goal of training the discriminator network is to distinguish the real data and the generated data; that is, the discriminator output of the real data is expected to be close to 1, and the output of the generated data \(D(G(z))\) is close to 0. Finally, the training reaches a balance, and \(G\) can estimate the distribution of sample data. The focus of GAN is to find the Nash equilibrium point, and the goal of the gradient descent method in the neural network is to find the smallest loss function, so in model training, there may be situations where the model cannot converge or the convergence is unstable. Moreover, there may be a pattern collapse phenomenon in GAN; that is, the generated data sample pattern is singular, and it does not cover all categories of real data. Therefore, how to ensure stable training of GAN has always been a difficult topic [33; 39].