## 2 The Proposed Architecture of Convolutional GAN
### Basic Theory of GAN  
The basic structure of GAN consists of two networks, as illustrated in Figure 1. The main idea is to construct a neural network model, known as the 'Generator', in order to map the random noises \(z\) into a new data space \(G(z)\). The goal is to minimize the discrepancy between the 'fake data' from the mapped space \(G(z)\) and the'real data' from the target space \(p_{r}(x)\). In contrast to traditional neural networks such as the Autoencoder, which directly minimizes the distance through the mean square error (MSE), one more neural network is introduced in GAN, referred to as the 'Discriminator', which is aimed to distinguish \(G(z)\) from \(p_{r}(x)\). The above stated goal can be achieved through the joint training of the two networks, and the original loss functions for the Generator and Discriminator are express as follows:  
\[\underset{D}{\text{max}}L(D)=E_{x\sim p_{r}(x)}[logD(x)]+E_{z\sim p_{z}(z)}[ log(1-D(G(z)))] \tag{1}\]  
\[\underset{G}{\text{min}}L(G)=E_{z\sim p_{z}(z)}[log(1-D(G(z)))] \tag{2}\]  
Figure 1: The basic structure of GAN. where \(G\) and \(D\) represent the Generator and Discriminator, respectively. The \(D\) outputs a score ranging from 0 to 1 for each sample. In order to distinguish between the 'fake data' and'real data', \(D\) tries to assign a value close to 1 for the sample \(x\) from'real data', and a value close to 0 for the sample \(G(z)\) from 'fake data'. Then, the loss function for \(D\) is maximized. On the other hand, \(G\) attempts to produce 'fake data' to fool \(D\); thus, it will be adjusted to produce 'fake data' that are close to'real data', allowing \(D(G(z))\) to be close to 1; therefore, the loss function of \(G\) is minimized. If both networks have sufficient capacity, they will reach a point at which both models cannot be improved anymore because the generated distribution approximates the real distribution well enough. The original loss function of GAN has been reported with challenges, such as unstable training and the poor quality of generated data. The key problem stems from the embedded Jensen-Shannon divergence (JSD) as a measure of the distance between real and generated distribution; the details can be found in [30]. To overcome the training challenges, [PERSON] et al., proposed to replace JSD with the Wasserstein distance (WD) defined as  
\[W(p_{r},p_{\theta})=\frac{1}{K}sup_{\|f\|_{L}\leq 1}\ E_{x\sim p_{r}}[f(x)]-E_{ \tilde{x}\sim p_{\tilde{x}}}[f(\tilde{x})] \tag{3}\]  
where \(f:x\to R\) is a set of Lipschitz functions satisfying the condition \(\frac{|f(x_{1})-f(x_{2})|}{|x_{1}-x_{2}|}\leq K\), and \(K\) is the Lipschitz constant. It can be observed that the absolute value of the derivative of \(f\) does not exceed \(K\). By applying this distance metric in GAN, the function \(f\) is referred to as a 'critic' in the original paper, which can be approximated by \(D\), and \(p_{\tilde{x}}\) is the model distribution implicitly defined by \(\tilde{x}=G(z)\), \(z\sim p_{z}(z)\). The variant of GAN with WD is known as Wasserstein GAN (WGAN). Compared with JSD, WD has a smoother change rate when measuring the distance between two distributions; thus, it can provide meaningful gradient information to \(G\). The original WGAN applies a 'weight clipping' method to enforce a Lipschitz constraint by clamping the network weights to a fixed range \([-c,c]\) after each gradient update. This method still leads to optimization difficulties such as gradient vanishing [23; 31]. Therefore, [PERSON] et al. proposed an alternative solution by enforcing a soft version of the Lipschitz constraint with a penalty on the gradient norm of random samples \(\tilde{x}\sim p_{\tilde{x}}\), which is expressed as  
\[\delta=E_{\tilde{x}\sim p_{\tilde{x}}}\Big{[}(\left\|\
abla_{\tilde{x}}D( \tilde{x})\right\|_{2}-K)^{2}\Big{]} \tag{4}\]  
where \(\tilde{x}=\epsilon x+(1-\tilde{\epsilon x})\), with \(x\sim p_{r},\tilde{x}\sim p_{\tilde{x}}\), and \(\epsilon\sim N[0,1]\). The \(K\) is generally set as 1.