# Bootstrapped SSL CycleGAN for Asymmetric Domain Transfer
## 3 Proposed Approach
### Avoiding the Overfitting by SSL Strategy  
Firstly, we employ the semi supervised learning (SSL) strategy based on the simple \(\|\cdot\|_{1}\) norm error term, which is computed over all pairs or elements in the provided small set of labeled or paired training samples \((x_{i}^{p},y_{i}^{p})\) that represent the same thing in the involved scarce \(\mathcal{X}\) and a fully observable domain \(\mathcal{Y}\), respectively.  
Thus, despite an unfavourable situation in which the number of unlabeled training samples in domain \(\mathcal{X}\) is considered as significantly lower than the number of available unlabeled observations in the domain \(\mathcal{Y}\), i.e., \(|\mathcal{X}|\ll|\mathcal{Y}|\), it is assumed that there still exists a small amount of labeled training pairs \((x_{i}^{p},y_{i}^{p})\) that are possible to be reliably matched into pairs based on their ground-truth labels, and thus provide the basis for the SSL strategy.  
Based on the valuable information contained in this small set of \(M_{SSL}\) paired observations, i.e., \(\mathcal{P}_{data}=\{(x_{i}^{p},y_{i}^{p})|i=1,\dots,M_{SSL}\}\), it is possible to define an additional SSL optimization term \(\mathcal{L}_{SSL}\), which will enforce that the learned CycleGAN mappings in Equations (2) and (3) respect the fact that \(x_{i}^{p}\in\mathcal{X}\) and \(y_{i}^{p}\in\mathcal{Y}\) are paired, i.e., have the same label. This is illustrated in Figure 2, where the expected closeness of the samples \(x_{p}\) and \(y_{p}\) after transformation of \(x_{p}\) by the learned mapping \(G\) is depicted.  
As the result of previous considerations, the closeness of the \"transformed\" \(G(x_{i}^{p})\) and the \"true\" labeled samples \(y_{i}^{p}\) during the SSL phase of the proposed training procedure can be described by the following optimization objective:  
\[\mathcal{L}_{SSL}(G_{X\to Y},G_{Y\to X})=\frac{1}{m}\sum_{i=1}^{m}\big{[}\;\|G _{X\to Y}(x_{i}^{p})-y_{i}^{p}\|_{1}\;+\;\|G_{Y\to X}(y_{i}^{p})-x_{i}^{p}\| _{1}\;\big{]}, \tag{11}\]  
which evaluates and imposes the corresponding closeness over a predefined amount of \(M_{SSL}\) available paired examples in both directions, from \(\mathcal{X}\) into \(\mathcal{Y}\), as well to the opposite side, from \(\mathcal{Y}\) into \(\mathcal{X}\), Figures 1 and 2.  
Thus, by applying the described SSL procedure and the introduced \(\mathcal{L}_{SSL}\) objective, we prevent discriminator \(D_{\mathcal{X}}\) in the scarce domain \(\mathcal{X}\) to overfit during the initial iterations of the learning process. In the standard unsupervised learning setting, characteristic for any CycleGAN model, overfitting of discriminator \(D_{\mathcal{X}}\) would mostly be inevitable, due  
Figure 2: SSL loss \(\mathcal{L}_{SSL}\), Equation (11), which enforces closeness of the “transformed” \(g_{i}^{p}=G(x_{i}^{p})\) and the “true” \(y_{i}^{p}\) over a small set of paired, i.e., labeled samples \(\mathcal{P}_{data}=\{(x_{i}^{p},y_{i}^{p})|i=1,\dots,M_{SSL}\}\) from a scarce domain \(\mathcal{X}\) and a fully observable \(\mathcal{Y}\). The same also holds in the opposite direction, for \(F(y_{i}^{p})\).  
to assumed limited number of available unlabeled training samples in the scarce domain, i.e., adopted assumption that \(|\mathcal{X}|\ll|\mathcal{Y}|\).  
By combining Equation (11) with the standard CycleGAN objective presented in Equation (10), the full SSL objective is made of:  
\[\mathcal{L}(G_{X\to Y},G_{Y\to X},D_{X},D_{Y})= \mathcal{L}_{adv}(G_{X\to Y},D_{Y})\ +\ \mathcal{L}_{adv}(G_{Y\to X},D_{X})\] \[+\ \lambda_{cyc}\ \mathcal{L}_{cyc}(G_{X\to Y},G_{Y\to X})\] \[+\ \lambda_{SSL}\ \mathcal{L}_{SSL}(G_{X\to Y},G_{Y\to X}), \tag{12}\]  
where \(\lambda_{SSL}\) is the corresponding regularization parameter.