## 3 Method
### Overall Framework of ICSO
This search mechanism effectively leverages the individual experience of particles and the optimal experience of the population, heuristically guiding the population to search for blind spots on the periphery of the hypercube, thereby enhancing the global search capability of the operations 2 and 3). VC, which is an operation from the original CSO, is a crossover operation between two different dimensions of a particle, aimed at escaping stagnant dimensions to prevent particles from falling into local optimal solutions. The formula for VC is as follows:

\[D_{d3}=r\cdot D_{d1}+(1-r)\cdot D_{d2} \tag{15}\]

where \(r\) is the random number between \([0,1]\) and \(D_{di}\) is the \(i\) dimension of the particle. As with HC, only the better particles are retained to the next iteration. Since the number of iterations is directly proportional to the number of stagnant particles, the more iterations there are, the higher the value of \(P\), which in turn determines the operating probability of the VC. The formula is as follows:

\[P=item/\max(item) \tag{16}\]

where \(item\) is the current number of iterations and \(\max(item)\) is the total number of iterations. As shown in Figure 3, the process of VC is used to remove stagnant dimensions by performing crossover operation among different dimensions of an individual. Once a certain stagnant dimension of an individual escapes from a stuck value, it spreads rapidly through the whole population via the HC operation. Moreover, this also helps other stagnant dimensions to quickly escape local minima through the VC. It is the crisscross operation on both horizontal and vertical directions that makes ICSO have a unique global search ability for addressing multi-modal problems with many local minima. Figure 2: Procedure of improved HC. In the figure, \(X_{i}\) and \(X_{j}\) are two individuals generated through Gaussian noise. The diagram illustrates the process of generating offspring based on these two individuals, corresponding to the flowchart of Equation (14). Each iteration requires checking if the current individual's new position \(X_{i+1}\) is better than its historically best position \(\mathit{pbest}_{i}\). If the objective function at the current position is smaller, then it is updated as follows:

\[\mathit{pbest}_{i+1}= \begin{cases}X_{i+1},&\text{if }F(X_{i+1})<F(\mathit{pbest}_{i})\\ \mathit{pbest}_{i},&\text{otherwise}\end{cases}  \tag{17}\]

Similarly, the global best \(\mathit{gbest}\) is the best individual extremum among the entire population, representing the globally optimal solution within the current search space. During each iteration, it is necessary to find the optimal one among all individual' \(\mathit{gbest}\); that is

\[\mathit{gbest}=\arg\min_{i=1,\ldots,n}F(\mathit{pbest}_{i}) \tag{18}\]

## 4 Experimental Results and Analysis

In this section, ICSO is compared with some state-of-the-art algorithms to validate the quality and diversity of the generated images. In addition, we have integrated ICSO into StyleGAN3 and applied it in real industrial scenarios. ### Benchmark Dataset

The CIFAR-10 dataset contains 10 classes with a total of 60,000 32 \(\times\) 32 RGB images. The dataset contains the following classes: bird, cat, plane, boat, car, truck, dog, deer, horse, and frog, and each class contains 5000 training images and 1000 test images. The STL-10 dataset contains 10 classes with a total of 113,000 96 \(\times\) 96 RGB images. The training set has 5000 images, the test set has 8000, and the remaining 100,000 are unlabeled images. The STL-10 dataset is unique because it is suitable for semi-supervised learning and self-supervised learning research. The LSUN is a large-scale image dataset, comprising 10 scene categories, such as bedrooms, living rooms, and classrooms, and 20 object categories, totaling approximately 1 million labeled images with 512 \(\times\) 512 resolution. LSUN-64 refers to a subset with a resolution of 64 \(\times\) 64 in the LSUN dataset.