## 3 Method
### Overall Framework of ICSO
Normalization is applied to ensure fairness and consistency between the quality and diversity metrics, and the objective function is formulated in a minimization form:

\[\min_{G}\max_{D}V(D,G)=\mathbb{E}_{x\sim p_{data}(x)}[\log D(x)]+\mathbb{E}_{z \sim p_{z}(z)}[\log(1-D(G(z)))] \tag{1}\]3

where \(\lambda\) is a scalar parameter used to balance the contributions of \(F_{q}^{\prime}(z)\) and \(F_{d}^{\prime}(z)\). When \(\lambda\) is close to \(0\), the model focuses more on \(F_{d}^{\prime}(z)\), while when \(\lambda\) is close to \(1\), the model focuses more on \(F_{q}^{\prime}(z)\). \(\mu\) is another scalar parameter used to adjust the weight of the gradient penalty term \(GP(z)\). A smaller \(F(z)\) indicates that the evaluated individual has a better generative performance. ### Improved Crisscross Optimization

ICSO, an integration of PSO and CSO, primarily consists of improved HC and VC. The original CSO effectively speeds up the convergence ability of ICSO by running the HC and VC operations alternately. In addition, ICSO introduces the local optimal search mechanism of PSO, which helps ICSO to quickly search local optimal solution in its own neighborhood. Similarly, ICSO incorporates the global optimal search mechanism of PSO, guiding toward the direction of population optimization through information interaction between particles. This approach can accelerate the convergence of the population and avoid falling into local optimal solutions [17]. HC randomly selects two particles from the population to perform crossover operations between the same dimensions, aiming to promote diversity and enhance search capability among particles through information exchange. Unlike the original CSO, this operator incorporates principles from the PSO, introducing the local optimal solutions of individuals and the global optimal solution from population into the original CSO (as shown in Equation (14)). The perturbation, which consists of the local and global optimal solutions, can make the particle heuristic search and help the operator search towards the global optimal solution. By integrating the rapid convergence ability of the original CSO with the heuristic search strategy of PSO, this method not only accelerates the convergence speed of the population but also effectively avoids being trapped in local optimal solutions, thereby enhancing the efficiency of global search. The formula for HC is as follows:

\[ \begin{split} X_{i+1}=& X_{i}+r\cdot\big{(}X_{j}- X_{i}\big{)}\\ &+c_{1}\cdot(\textit{pbest}_{i}-X_{i})+c_{2}\cdot(\textit{gbest}-X _{i})\end{split}  \tag{14}\]

where \(r\) is the random number between \([0,1]\); \(c_{1}\) is the individual learning factor that controls the step size of the individual moving towards its _pbest_; \(c_{2}\) is the social learning factor that controls the step size of the individual moving towards the _gbest_. In the standard implementation of PSO, both \(c_{1}\) and \(c_{2}\) are commonly set to \(2\); \(X_{i}\) is the Gaussian noise of generation \(i\); \(\mathit{pbest}_{i}\) is the local optimal solution and \(\mathit{gbest}\) is the global optimal solution. The particles produced by the improved HC must be compared with their parent particles, and only the particles with better fitness can be retained in the VC operation. Figure 2 provides the schematic of a particle's trajectory of evolution in the procedure of improved HC. The operation 1 borrows from the original CSO. The main part of the search space is constructed with parent particles as the diagonal vertices of a hypercube. In particular, it is clear that to reduce unsearchable blind spots, the original CSO also searches the periphery of the hypercube with a lower probability. In this paper, however, we utilize the local optimal solutions and global optimal solutions from PSO as perturbations, replacing the aforementioned part of the original CSO.