## 3 The Proposed Hyper-CycleGAN
In particular, for the GAN loss \(\mathcal{L}_{\text{adv}}(\text{G}_{\text{S}\rightarrow\text{T}},\text{D}_{ \text{T}})\), we trained [PERSON] to minimize \(\mathbb{E}_{\text{t}\sim\text{T}(\text{t})}[\text{logD}_{\text{T}}(\text{t})]\) and trained [PERSON] to minimize \(\mathbb{E}_{\text{S}\sim\text{P}_{\text{S}}(\text{s})}[\text{log}(1-\text{D} _{\text{T}}(\text{G}_{\text{S}\rightarrow\text{T}}(\text{s})))]\). Secondly, to mitigate model oscillations and optimize training for hyperspectral data synthesis, we incorporated a strategy proposed in [39]. This strategy involved updating the discriminator using a history of previously generated hyperspectral images rather than the most recent ones. An image buffer was maintained to store the 60,000 previously generated hyperspectral samples. Moreover, we introduced label smoothing during the training process to enhance model robustness and generalization capability, particularly in the context of hyperspectral data. Label smoothing involves adjusting the target labels for real and fake samples to encourage more calibrated and confident predictions from the discriminator. This regularization technique prevents the discriminator from becoming overly confident  
Figure 3: Hyper-CycleGAN classifier architecture. in its predictions and fosters the generation of diverse and realistic hyperspectral outputs, essential for accurate and reliable hyperspectral data synthesis. 1. Input: - Source domain hyperspectral data S and target domain hyperspectral data T - Models: \(\text{G}_{\text{S}\sim\text{T}}\), \(\text{G}_{\text{T}\to\text{S}}\), \(\text{D}_{\text{S}}\), \(\text{D}_{\text{T}}\), hyperspectral classifier f - Hyperparameters: \(\text{Agp}\), Acycle, training iterations N, classifier training iterations M 2. Output: - Trained models: \(\text{G}_{\text{S}\sim\text{T}}\), \(\text{G}_{\text{T}\to\text{S}}\), \(\text{D}_{\text{S}}\), \(\text{D}_{\text{T}}\), f 3. procedure TRAIN HYPER-CYCLEGAN
4. **for n = 1 to N do //** Main training loop
5. **Train the discriminators and generators:**
6. Draw a minibatch of samples \(\{\text{s}^{\text{(i)}},\ldots,\text{s}^{\text{(m)}}\}\) from domain S
7. Draw a minibatch of samples \(\{\text{t}^{\text{(i)}},\ldots,\text{t}^{\text{(n)}}\}\) from domain T
8. Compute the discriminator loss on real images: \[\text{J}_{\text{real}}^{\text{(D)}}=\frac{1}{\text{m}}\sum_{i=1}^{\text{m}} \Bigl{(}\text{D}_{\text{S}}\bigl{(}\text{s}^{\text{(i)}}\bigr{)}-1\Bigr{)}^{2 }+\frac{1}{\text{n}}\sum_{j=1}^{\text{n}}\Bigl{(}\text{D}_{\text{T}}\bigl{(} \text{t}^{\text{(i)}}\bigr{)}-1\Bigr{)}^{2}\]
9. Compute the discriminator loss on fake images: \[\text{J}_{\text{fake}}^{\text{(D)}}=\frac{1}{\text{m}}\sum_{i=1}^{\text{m}} \bigl{(}\text{D}_{\text{T}}\Bigl{(}\text{G}_{\text{S}\sim\text{T}}\Bigl{(} \text{s}^{\text{(i)}}\Bigr{)}\Bigr{)}^{2}+\frac{1}{\text{n}}\sum_{j=1}^{\text{ n}}\Bigl{(}\text{D}_{\text{S}}\Bigl{(}\text{G}_{\text{T}\to\text{S}}\bigl{(} \text{t}^{\text{(i)}}\bigr{)}\Bigr{)}^{2}\Bigr{)}\]
10. Compute the WGAN-GP loss term: \[\text{J}_{\text{SP}}^{\text{(D)}}=\lambda_{\text{SP}}\sum_{k=1}^{\text{K}} \bigl{(}\parallel\
abla_{\text{D}}\text{s}^{\text{(K)}}\parallel_{2}-1\bigr{)} ^{2}+\lambda_{\text{SP}}\sum_{k=1}^{\text{K}}\bigl{(}\parallel\
abla_{\text{ Dt}}\text{t}^{\text{(k)}}\parallel_{2}-1\bigr{)}^{2}\]
11. Compute the total discriminator loss with WGAN-GP: \[\text{J}_{\text{total}}^{\text{(D)}}=\text{J}_{\text{real}}^{\text{(D)}}+ \text{J}_{\text{fake}}^{\text{(D)}}+\text{J}_{\text{gp}}^{\text{(D)}}\]
12. Update the discriminators
13. Compute the T\(\rightarrow\)S generator loss: \[\text{J}(\text{G}_{\text{T}\to\text{S}})=\frac{1}{\text{n}}\sum_{j=1}^{ \text{n}}\Bigl{(}\text{D}_{\text{S}}\Bigl{(}\text{G}_{\text{T}\to \text{S}}\Bigl{(}\text{t}^{\text{(i)}}\Bigr{)}\Bigr{)}-1\Bigr{)}^{2}+\lambda_{ \text{cycle}}\text{J}_{\text{cycle}}\bigl{(}\text{T}\to\text{S}\bigr{)}\]
14.