## 2 Methods
### Architecture of CycleGAN for Enhancing the Mandarin EL Speech
#### 2.1.3 Training Objectives  
During the training stage, the Mel-spectrogram parameters are used to train the transformation function. The training process is shown in Figure 3.  
The top part of Figure 3 contains the acoustic features (plot using praat) of source speech and target speech. The top part of Figure 3 shows the data flow mapping from the source to the target and the data flow mapping from the target to the source. The left bottom part of Figure 3 shows how the discriminator \(D_{Y}\) discriminates the generated \(\hat{Y}\) using the forward generator \(G_{X\to Y}\) of CycleGAN. The right bottom part of Figure 3 shows how the discriminator \(D_{X}\) discriminates the generated \(\hat{X}\) using the inverse generator \(F_{Y\to X}\) of CycleGAN. In CycleGAN shown as Figure 3, the mapping function \(G_{X\to Y}\) represents the forward generator; \(F_{Y\to X}\) represents the inverse generator; \(D_{X}\) represents the forward discriminator; \(D_{Y}\) represents the inverse discriminator; \(\hat{X}\) represents the predicted output of \(Y\) using \(F(Y)\); and \(\hat{Y}\) represents the predicted output of \(X\) using \(G(X)\). In particular, \(D_{X}\) is used to discriminate the \(X\) and \(F(Y)\); \(D_{Y}\) is used to discriminate the \(Y\) and \(G(X)\). Consistence loss and inverse consistence loss are both used to evaluate this model. The consistence loss is used to match the generated and target acoustic features. The inverse consistence loss is used to avoid the contradiction between mapping functions \(G\) and \(F\).  
The process of computing losses takes reference from [PERSON] et al. (2019). The adversarial loss \(L_{G}(G,D_{Y},X,Y)\) is computed as in Equation (8), and the inverse adversarial loss \(L_{F}(F,D_{X},Y,X)\) is computed as in Equation (9).  
\[L_{G}(G,D_{Y},X,Y)=E_{P_{Data}(Y)}(\text{InD}_{Y})+E_{P_{Data}(X)}(\text{In}(1-D _{Y}(G(X))) \tag{8}\]  
where \(P_{Data}(X)\) and \(P_{Data}(Y)\) represent the distribution probability of acoustic features for source and target speakers respectively; \(E\) represents the expected value of the acoustic feature distribution. Log (In) represents the logarithmic calculation. The smaller the loss is, the narrower the gap between the acoustic features of the converted and target speech.  
\[L_{F}(F,D_{X},Y,X)=E_{P_{Data}(X)}(\text{InD}_{X})+E_{P_{Data}(Y)}(\text{In}(1 -D_{X}(F(Y))) \tag{9}\]  
The final adversarial loss of CycleGAN equals to the mean of Equation (8) plus Equation (9). The final consistence loss \(L_{cyc\_con}(G,F)\) is computed as in Equation (10).  
\[L_{cyc\_con}(G,F)=E_{P_{Data}(X)}(\left\|F(G(X))-X\right\|_{1})+E_{P_{Data}(Y) }(\left\|G(F(Y))-Y\right\|_{1}) \tag{10}\]  
Figure 3: The data flow between the source space and the target space of CycleGAN.  
where \(\parallel\parallel_{1}\) represents the 1-norm. The loss of the whole model \(\mathrm{L_{full}}\) is computed as in Equation (11).  
\[\mathrm{L_{full}}=\mathrm{L_{G}}(\mathrm{G},\mathrm{D_{Y}},\mathrm{X},\mathrm{Y })+\mathrm{L_{F}}(\mathrm{F},\mathrm{D_{X}},\mathrm{Y},\mathrm{X})+\lambda \cdot\mathrm{L_{cyc_{con}}}(\mathrm{G},\mathrm{F}) \tag{11}\]  
where \(\lambda\) represents the control parameter used to balance the weights of the forward and inverse training process.  
We take reference from VC3 [27] to design the generator and discriminator of the proposed CycleGAN. We use the Adam algorithm for gradient descent [33; 34] and set the parameters \(\beta_{1}\) and \(\beta_{2}\) as 0.9 and 0.99 respectively.