## 2 Proposed Methodology
### Wasserstein Generative Adversarial Network  
As outlined in Section 2.2.3, the limitations for GAN arise from the generator receiving meaningless gradients. To provide the generator with meaningful gradients, employing an alternative loss function is essential. WGAN uses Earth Mover's Distance (EMD) or Wasserstein distance as loss functions, which can be expressed as Equation (4).  
\[\mathrm{W}\big{(}\mathbb{P}_{\mathrm{r}},\mathbb{P}_{\mathrm{g}}\big{)}=\inf_{ \gamma\in\Pi(\mathbb{P}_{\mathrm{r}},\mathbb{P}_{\mathrm{g}})}\mathbb{E}_{( \mathrm{x},\mathrm{y})\sim\gamma}[\|\mathrm{x}-\mathrm{y}\|] \tag{4}\]  
Here, \(\Pi(\mathbb{P}_{\mathrm{r}},\mathbb{P}_{\mathrm{g}})\) represents all sets of the joint distributions \(\gamma(\mathrm{x},\mathrm{y})\), which have \(\mathbb{P}_{\mathrm{r}}\), \(\mathbb{P}_{\mathrm{g}}\) as marginal individually. The intuitive definition of the Wasserstein distance means the quantity of mass that must be transported from \(\mathrm{x}\) to \(\mathrm{y}\) for the distribution of the generated data \(\mathbb{P}_{\mathrm{g}}\) to align with the distribution of real data \(\mathbb{P}_{\mathrm{r}}\).  
However, it is impossible to apply the Wasserstein distance in Equation (4) directly as a loss function because it is almost impossible to find all the joint distributions and minimum values. Due to this restriction, Kantorovich-Rubinstein duality is introduced, which constrains the objective function by using the 1-Lipshitz method [56]. In this context, since it is difficult to identify the exact function we want to constrain, the objective function is approximated using a neural network. After applying the Kantorovich-Rubinstein duality and 1-Lipshitz method, the loss function using Wasserstein distance turns into Equation (5).  
\[\mathrm{Loss\ function}=\max_{\mathrm{w}\in\mathrm{W}}\mathbb{E}_{\mathrm{x} \sim\mathbb{P}_{\mathrm{r}}}[\mathrm{f}_{\mathrm{w}}(\mathrm{x})]-\mathbb{E}_ {\mathrm{z}\sim\mathrm{p}(\mathrm{z})}[\mathrm{f}_{\mathrm{w}}(\mathrm{g}_{ \mathrm{\Theta}}(\mathrm{z}))] \tag{5}\]  
Here, \(\mathrm{f}_{\mathrm{w}}\) denotes a function approximated by the neural network that has variables of \(\mathrm{w}\), and similarly, \(\mathrm{g}_{\mathrm{\Theta}}\) is a function that involves variables of \(\mathrm{\theta}\). For brevity of the study, detailed mathematical formulae of the Kantorovich-Rubinstein duality and Lipshitz method are omitted.  
The gradients for updating WGAN must be restricted because it does not use the sigmoid function for the last layer of the discriminator. This could potentially lead to unbounded output values and destabilizing gradients during the training process. For this purpose, the 'weight clipping' technique is used to restrict the scale of the gradients between \(-0.01\) and \(0.01\). Because of this mathematical difference between the discriminator in the conventional GAN and the discriminator in WGAN, the latter one is named the 'critic'. Through this overall process in WGAN, the generator can obtain proper gradients, and the training generator and critic can be balanced. Furthermore, through generating meaningful gradients, the stability of the training process can be improved.