## 2 Preliminaries
### Deep Convolutional Generative Adversarial Networks  
GAN models are widely employed across computer vision, natural language processing, and other fields. The primary structure of GANs comprises a generator, denoted as \(G\), and a discriminator, denoted as \(D\), as illustrated in Figure 2.  
The typical process of sample generation unfolds as follows. Initially, noise \(z\) is input into the generator \(G\) to produce synthetic data \(G(z)\), which resemble real data \(x\), with the intention of deceiving the discriminator. Subsequently, both the generated synthetic data and real samples are fed into the initial discriminator \(D\) until the discriminator becomes unable to distinguish between the genuine and synthetic data. It is commonly assumed that random noise \(z\) follows a Gaussian distribution. Ultimately, if the input data from the generator closely approximate the real data, the goal of learning the approximate distribution of genuine images is achieved.  
GANs are non-cooperative game-based models optimized through the maximization and minimization of probabilistic outputs. The generator and discriminator iteratively update their model parameters until reaching a Nash equilibrium state. This state refers to a condition where neither the generator nor the discriminator can improve their performance  
Figure 2: The basic structure of generative adversarial networks (GANs).  
by unilaterally changing their strategies. Specifically, this means that the generator produces samples that the discriminator cannot reliably distinguish from real data, while the discriminator maximizes its ability to differentiate between real and generated data given the current capabilities of the generator. This concept is fundamental to understanding the training dynamics of GANs. In this state, the generator is capable of producing samples akin to real data, while the discriminator struggles to accurately discern the authenticity of the generator's input data. This process can be formalized as a non-cooperative game that involves maximizing and minimizing the value function, denoted as \(V(D,G)\). The generator's objective is to maximize this value function, making it challenging for the discriminator to differentiate between samples generated by the generator and real data. Conversely, the discriminator aims to minimize the value function, enhancing its ability to distinguish between generator-produced samples and genuine data. The objective function is expressed as follows:  
\[\begin{split}\min_{G}\max_{D}V(D,G)&=\mathbb{E}_{x \sim p_{data}(x)}[\log D(x)]\\ &+\mathbb{E}_{z\sim p_{z}(z)}[\log(1-D(G(z)))],\end{split} \tag{1}\]  
where \(p_{data}(x)\) denotes the distribution of real data \(x\), \(P_{z}(z)\) indicates the prior distribution of the noise vector \(z\), and \(D(x)\) and \(D(G(z))\) represent the probabilities of real data and generated data, respectively.  
However, GANs can be unstable during training, leading to instances where the generator produces invalid outputs. A DCGAN [26] shares the same goal as traditional GANs. The primary distinction lies in the fact that a DCGAN utilizes multiple convolutional blocks to construct its generator and discriminator. Specifically, the generator employs a deconvolutional architecture, while the discriminator employs a convolutional architecture. The structures of the generator and discriminator of the DCGAN are illustrated in Figure 3.  
We can see that BN indicates a batch normalization operation, ensuring that its mean and unit variance are zero, thereby stabilizing learning. ReLU, leaky ReLU, and tanh represent three distinct activation functions that aim to expedite model learning and saturate the color space for comprehensive coverage. The aforementioned structure possesses a remarkable capability to generate high-quality images.  
Figure 3: The structures of the generator and discriminator of a DCGAN.