## 2 Proposed Method
### Domain Discriminator Based on Wasserstein Metric  
In the original GANs, a generator is defined to map the distribution of extracted noise samples \(\mathbb{P}_{z}\) in the low-dimensional latent space \(\mathbb{B}\) to the distribution of the generated images \(\mathbb{P}_{g}\) in the image space [21]. In order to train the generator, a discriminator is also defined in GANs. Through training, the discriminator considers the false output of the generator as a real sample \(\mathbb{P}_{r}\). The famous minimax game between the generator \(G(\cdot)\) and the discriminator \(D(\cdot)\) is formed. The optimization problem in GANs is expressed as follows [9; 25; 26]:  
\[\arg\min_{G}\max_{D}(\mathbb{P}_{r},\mathbb{P}_{g})=\mathbb{E}_{x\sim\mathbb{P }_{r}}\log(D(\mathbf{x}))+\mathbb{E}_{x\sim\mathbb{P}_{g}}\log(1-D(G(\mathbf{z}))) \tag{6}\]  
where \(D(x)\) represents the probability that comes from real data rather than noise. \(G(z)\) represents the mapping of the generator in the data space. Undoubtedly, original GANs have made remarkable strides in addressing domain adaptation challenges. Nonetheless, original GANs consistently exhibit instability throughout the training process. In [27], the authors show that minimizing the loss of the generator approximates minimizing the Jensen-Shanno Divergence between \(\mathbb{P}_{r}\) and \(\mathbb{P}_{g}\) when the discriminator is trained to near-optimality, while noting that when the overlap between the distributions can be ignored, the JSD will converge to a fixed value, at which point the gradient of the generator will be zero. To address this issue, the RASAN-W employs a Wasserstein Generative Adversarial Network (WGAN) as a domain discriminator. This approach is designed to minimize the differences in the global distribution across domains. Concurrently, it ensures that the feature distributions of the source domain are closely matched with those of the target domain. This solution is effective in tackling the challenge of gradient disappearance that the domain discriminator in DASAN often encounters during model training.  
It is pertinent to underscore that WGAN calculates the similarity between the source and target domain distributions by approximately fitting the Wasserstein distance, rather than the Jensen-Shannon Divergence or Kullback-Leibler Divergence. The Wasserstein metric function is given as follows:  
\[\text{Wass}(\mathbb{P}_{r},\mathbb{P}_{g})=\sup_{\|G_{d}(\cdot)\|_{L}\leq 1} \mathbb{E}_{x\sim\mathbb{P}_{r}}f(x)-\mathbb{E}_{x\sim\mathbb{P}_{g}}f(x) \tag{7}\]  
where sup denotes the supremum bound and \(L\) represents the 1-Lipschitz constraint, aimed at constraining the rate of parameter changes within the model, thus mitigating the risks of gradient explosion or vanishing during the training phase [28]. To ensure compliance with the 1-Lipschitz constraint, a gradient penalty term needs to be added:  
\[L_{penalty}=\Big{(}\Big{\|}\
abla_{z}G_{d}(\hat{z},\theta_{d})\Big{\|}-1 \Big{)}^{2} \tag{8}\]  
where \(\hat{z}=\varepsilon G_{\text{f}}(x)+(1-\varepsilon)G_{\text{f}}(y),\varepsilon \in[0,1)\). Combining Equations (6)-(8), the domain discriminator loss function based on Wasserstein metric is defined as follows:  
\[L_{w}(\theta_{f},\theta_{d})=\frac{1}{M}\sum_{i=1}^{M}[G_{d}(G_{f}(x_{i}^{s}; \theta_{f});\theta_{d})-G_{d}(G_{f}(x_{j}^{t};\theta_{f});\theta_{d})]-\delta L _{penalty} \tag{9}\]