## 3 The Proposed Hyper-CycleGAN
Mathematically, the loss function of WGAN-GP can be expressed as:  
\[ \begin{split}\mathcal{L}_{\text{WGAN-GP}}(\text{D}_{\text{S}})= \mathbb{E}_{\hat{s}\sim\text{P}_{\hat{s}}}[D(\hat{s})]-\mathbb{E}_{\hat{s}\sim \text{P}_{\hat{s}}}[D(\hat{s})]\\ +\lambda\mathbb{E}_{\hat{s}\sim\text{P}_{\hat{s}}}[\left(\|\
abla _{\hat{s}}\text{D}\left(\widetilde{\hat{s}}\right)\|_{2}-1\right)^{2}]\end{split}  \tag{5}\]  
Here, \(D_{\text{S}}\) still denotes the discriminator, \(P_{\hat{s}}\) is the distribution formed by a mixture of real and generated data, \(\lambda\) is the weight of the gradient penalty term, and \(\hat{s}\) is a sample sampled from \(P_{\hat{s}}\), a distribution formed by linear interpolation between real and generated data. The incorporation of a gradient penalty term in the Wasserstein generative adversarial network with gradient penalty (WGAN-GP) incentivizes the discriminator to acquire smoother and more stable decision boundaries throughout the training process. This augmentation leads to a substantial enhancement in the training efficiency of the model and elevates the quality of the generated samples. Hyper-CycleGAN classifier: Upon the generation of new hyperspectral data by the generator, our subsequent endeavor involved the development of a ResNet classifier to rigorously evaluate the quality and fidelity of the generated data. Figure 3 delineates the detailed architecture of our meticulously designed classifier, which was specifically crafted to examine the spectral signatures embedded within the synthesized hyperspectral dataset. At the heart of our hyperspectral classifier model resides a custom amalgamation of residual blocks (ResBlocks) and dense blocks (DenseBlocks), each meticulously engineered to navigate the intricacies and complexities inherent in hyperspectral data analysis. These architectural elements formed the foundation of our classifier, empowering it to adeptly capture and interpret the rich spectral information contained within each hyperspectral data sample. Within the ResBlocks, our classifier utilized the strength of residual connections to ensure unimpeded information flow and gradient propagation, thereby facilitating effective feature extraction and representation learning. The employment of these residual connections allowed our classifier to effectively counteract the vanishing gradient issue, a common challenge in deep neural networks, thus enhancing its capacity to detect subtle spectral variations and intricate patterns inherent in hyperspectral data. Complementing the ResBlocks, the DenseBlocks are distinguished by their dense connectivity patterns, which enhance feature reuse and enable the extraction of highly discriminative spectral features. Through this dense connectivity, our classifier was poised to harness the extensive spectral information available across different wavelengths, thereby enabling precise discrimination of subtle spectral nuances and accurate classification of hyperspectral data samples. Moreover, the architecture of our hyperspectral classifier was thoughtfully designed to include spectral-specific layers and operations, ensuring its suitability for tasks associated with hyperspectral data analysis. These spectral-specific adaptations permitted our classifier to effectively capture the unique spectral characteristics intrinsic to hyperspectral imagery, thereby augmenting its overall efficacy and robustness in hyperspectral data classification tasks. Hyper-CycleGAN training: Algorithm 2 shows the Hyper-CycleGAN training pseudocode. Our training was divided into two parts: generator-discriminator training and classifier training. In the generator-discriminator training part, two generators and two discriminators were trained simultaneously in an adversarial manner [37]. First, for \(\mathcal{L}_{\text{adv}}\) (Equation (1)), we replaced the negative log-likelihood objective with a least squares loss [38]. This loss was more stable during training and produced higher-quality results.