# Express Construction for GANs from Latent Representation to Data Distribution
## 3 Problem Statement
We can first define the measures of them analogously, as follows. **Definition 4**. : _Let the probability measure \(\mu\) and \(\
u\) be defined on discrete sets, \(\mathcal{I}_{1}\) and \(\mathcal{J}_{1}\) denote the two disjoint sets of indices. Suppose \(\hat{X}_{1}=\{x_{i}\}_{i\in\mathcal{I}_{1}}\) and \(\hat{Y}_{1}=\{y_{j}\}_{i\in\mathcal{J}_{1}}\) are discrete subsets of \(\mathbb{R}^{d}\), and the cost function used to map the transportation from random noise to the latent representation is defined by \(c_{ij}^{1}=c_{1}(x_{i},y_{j})\), where \(c_{ij}^{1}\geqslant 0\) are positive real numbers. Further suppose the source measure \(\mu(x_{i})=\mu_{i}\), and the target measure \(\
u(y_{i})=\
u_{j}\). A transport plan \(\rho_{1}\) is a real function that takes values on \(\{(x_{i},y_{j})|\forall x_{i}\in\hat{\mathcal{X}}_{1},y_{j}\in\hat{\mathcal{Y}} _{1}\}\) such that \(\rho_{ij}^{1}=\rho_{1}(x_{i},y_{j})\geqslant 0\), \(\sum_{i\in\mathcal{I}}\rho_{ij}^{1}=\
u_{j}\) and \(\sum_{j\in\mathcal{J}}\rho_{ij}^{1}=\mu_{i}\). This is analogous to \(c_{ij}^{2}\) and \(\rho_{ij}^{2}\) defined by the transportation map from the latent representation to the data distribution, and \(c_{ij}^{3}\) and \(\rho_{ij}^{3}\) defined by the transportation map from random noise to the data distribution._  
As we know, the latent representation is a low-dimensional mapping related to the data distribution, thus we can assume that there is a linear correlation between \(c_{ij}^{1}\), \(c_{ij}^{2}\), and \(c_{ij}^{3}\), analogously for \(\rho_{ij}^{1}\), \(\rho_{ij}^{2}\), and \(\rho_{ij}^{3}\). Thus, we can say that:  
\[ \begin{cases}c_{ij}^{3}=k_{1}c_{ij}^{1}=k_{2}c_{ij}^{2}\\ \rho_{ij}^{3}=k_{1}\rho_{ij}^{1}=k_{2}\rho_{ij}^{2}\end{cases}  \tag{4}\]  
where \(k_{1},k_{2}\), \(\vec{k_{1}}\) and \(\vec{k_{2}}\) are positive numbers to estimate the relationship between them. The total transportation cost from random noise to the data distribution can also be the sum of two components approximately, the transportation cost from random noise to the latent representation and the one from the latent representation to the data distribution. This means \(k_{1}>1\) and \(k_{2}>1\). Besides, the transportation cost from random noise to the latent representation is significantly lower than the one from the latent representation to the data distribution since the former has a smaller dimension. Thus, we can say that \(k_{1}>k_{2}\). This indicates that \(k_{1}>k_{2}>1\) and \(\bar{k_{1}}>\bar{k_{2}}>1\). Therefore, the total transportation cost of the direct mapping defined as \(\mathcal{C}_{3}\) can be written as:  
\[ \begin{split}\mathcal{C}_{3}&=\sum_{i\in\mathcal{ I}_{3}}\sum_{j\in\mathcal{J}_{3}}c_{ij}^{3}\rho_{ij}^{3}\\ &=\sum_{i\in\mathcal{I}_{1}}\sum_{j\in\mathcal{J}_{1}}k_{1}c_{ij }^{1}\mathbb{I}_{1}\rho_{ij}^{1}\\ &=k_{1}\bar{k}_{1}\sum_{i\in\mathcal{I}_{1}}\sum_{j\in\mathcal{J} _{1}}c_{ij}^{1}\rho_{ij}^{1}\\ &=k_{1}\bar{k}_{1}\mathcal{C}_{1}\\ &=k_{2}\bar{k}_{2}C_{2}\end{split}  \tag{5}\]  
That is to say, the total transportation cost of the direct mapping from random noise to the data distribution is always larger than the one from random noise to the latent representation and the one from the latent representation to the data distribution, which indicates we can reduce the task complexity by Express Construction. Besides, with the decreasing of the task complex, the stability of the model will be further improved by searching for more transportation maps with less transportation cost.