## 2 Sample Expansion Methodology Based on CVAE-Skegan
### Encoding network
\[\min_{G}L_{G}=-E_{G(z)\sim P_{g}}[D(G(z))] \tag{3}\]  
\[\max_{D}L_{D}=E_{x\sim P_{data}}[D(x)]-E_{G(z)\sim P_{g}}[D(G(z))]-\lambda L_{ GP} \tag{4}\]  
where \(G(z)\sim P_{g}\) represents that \(G(z)\) is sampled from the generated sample distribution \(P_{g}\), and \(\lambda\) is a tunable penalty parameter. The gradient penalty term GP is shown in Equation (5). \[ \begin{array}{l}L_{GP}=E_{y\sim P_{y}(y)}\Big{[}\big{(}\|\
abla yD(y)\|_{2}- 1\big{)}^{2}\Big{]}\\ y=\gamma x+(1-\gamma)G(z)\end{array}  \tag{5}\]  
Figure 2: Structure of the SK block. where \(y\) is a sample of random linear interpolation of x, \(G(z)\gamma\) is a random coefficient, and \(P_{y}\) indicates that the distributions of real and generated data are uniformly distributed along a straight line between pairs of sampling points. The generator loss function using least squares loss is shown in Equation (6), and the discriminator loss function is shown in Equation (7). \[L_{\text{G}}=\frac{1}{2}E_{G(z)\sim P_{g}}\Big{[}(D(G(z))-1)^{2}\Big{]} \tag{6}\]  
\[L_{\text{D}}=\frac{1}{2}E_{x\sim P_{data}}\Big{[}(D(X)-1)^{2}\Big{]}+\frac{1}{ 2}E_{G(z)\sim P_{g}}\Big{[}(D(G(z)))^{2}\Big{]} \tag{7}\]  
2 The different mutations are evaluated, and the optimal solution is determined by using the fitness function F to score different loss functions. The fitness function F is divided into two parts: quality fitness function \(F_{q}\) and diversity fitness function \(F_{d}\). The quality fitness score is the least squares fitness function. \[F_{q}=E_{z}\Big{[}(D(z)-1)^{2}\Big{]} \tag{8}\]  
The least squares fitness function is an unsaturated function, which means that the discriminator can promote generator optimization throughout the entire training phase, without allocating extremely high costs for generating fake samples. This can avoid gradient vanishing and pattern collapse. The diversity fitness function \(F_{d}\) measures the diversity of generated samples by utilizing the logarithmic gradient difference updated by the generator G and discriminator D.  
\[F_{d}=-\log\lVert\
abla_{D}\rVert+\log\lVert\
abla_{G}\rVert \tag{9}\]  
Thus, the final fitness function F is as follows:  
\[F=F_{q}+\gamma F_{d} \tag{10}\]  
where \(\gamma\geq 0\) balances two metrics. 3 The generator with a high adaptation score is used as the parent to move on to the next iteration until the network reaches the Nash equilibrium, and then the training ends. The above process can avoid being limited to one single loss function, which is beneficial for alleviating the problem of gradient vanishing. The fitness function can effectively point to a suitable loss function. The evolutionary process can serve as a universal framework for different inputs, reducing the need to optimize the hyper-parameters for each individual dataset, thus making CVAE-SKEGAN applicable to multiple working conditions. 3 Discriminator  
The inputs of the discriminator are the samples generated by the generator, the original real samples, and their corresponding label information. The discriminator is divided into two parts: true-false probability and classification discrimination. The input is a 28 \(\times\) 28 \(\times\) 1-dimensional feature sequence, which is processed in two parts: the true-false discrimination part simultaneously extracts features and reduces dimensions from generated samples and real samples to obtain a true-false probability output through Sigmoid activation function; after passing through two convolutional layers and flattening, the classification discrimination part obtains an output value of classification discrimination using Softmax activation function. In this function, the true-false discriminator part consists of convolutional structures and fully connected layers, and batch normalization operation is added after each convolutional layer. SKNets are also added to the true-false discrimination to improve the feature-extraction ability of the discriminator.