## 3 Method
### Overall Framework of ICSO
These two attributes often conflict in practice: an excessive focus on quality may cause the model to generate samples that concentrate on only a few modes, lacking diversity, while an excessive emphasis on diversity may compromise the overall quality of the generated samples. The process of observing the average output value from the discriminator, when fed with images generated by the generator, can be considered a form of quality fitness function in the context of IE-GAN [15], as follows:

\[F_{q}=\mathbb{E}_{z\sim p_{z}}[D(G(z))] \tag{8}\]

where \(D\) is the discriminator, \(G\) is the generator, and \(z\) is random Gaussian noise. IE-GAN indirectly evaluates the distributional distance of the samples produced by the generator, so it is achieved by estimating the diversity through the mean absolute error (MAE) between samples. The formal definition of diversity fitness function is as follows:

\[F_{d}=\frac{1}{n_{e}}\sum_{i=1}^{n_{e}}\mathbb{E}_{z_{1},z_{i}\sim p_{z}}[ \|G(z_{1})-G(z_{i})\|_{1}] \tag{9}\]

where \(n_{e}\) refers to the number of times that each sample is compared with other samples. Generally, it is not appropriate to balance different objective functions only using Weighted Sum Approaches. \(F_{q}\) and \(F_{d}\) are two independent objective functions that sometimes differ by a factor of 10 or 100. Therefore, \(F_{q}\) and \(F_{d}\) of all individuals in the population are normalized, thus unifying the different fitnesses to an order of magnitude, wherein the min-max normalization is calculated by the following:

\[\min_{G}\max_{D}V(D,G)=\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))] \tag{1}\]0

\[\min_{G}\max_{D}V(D,G)=\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))] \tag{1}\]1

where \(\text{min}(x)\) and \(\text{max}(x)\) are used to obtain the minimum and maximum values of the data, respectively. Gradient-based regularization can stabilize the adversarial training of GAN and suppress the mode collapse of models [16]. The gradient penalty is a technology that applies constraints to the gradients of both the generator and the discriminator during training in order to make the training process more stable and smooth. This prevents gradient exploding or gradient vanishing between the generator and the discriminator, thereby improving the quality and diversity of the generated samples. Moreover, when the generator generates realistic samples, the discriminator does not reject the generated sample confidently (i.e., the discriminator updates with small gradient), and when the generator collapses to a small region, the discriminator subsequently labels collapsed points as fake with an obvious countermeasure (i.e., the discriminator updates with a big gradient). By constraining the gradient of discriminator, the generated samples tend to be dispersed enough to avoid mode collapse, and the formula is as follows:

\[\min_{G}\max_{D}V(D,G)=\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))] \tag{1}\]2

where \(\hat{x}\) is a sample drawn from the distribution \(\mathbb{P}_{\hat{x}}\); \((\left\|\
abla_{\hat{x}}D(\hat{x})\right\|_{2}-1)^{2}\) calculates the squared difference between the L2 norm of the gradient of the discriminator \(D\) with respect to \(\hat{x}\) and \(1\). The goal is to keep the gradient norm close to \(1\), which helps in satisfying the Lipschitz condition. A fitness function that integrates both the quality and diversity of generated images is designed and combined with GP to construct a multi-objective function, aiming to balance these two conflicting attributes.