## 2 Problem Formulation
### Diffusion Probabilistic Model  
The diffusion model comprises a diffusion process that perturbs the data with noise and a reverse process that transforms the noisy data generated by the diffusion process back into the original data. This framework is modeled by the Markov chain. The former is typically designed to transform the data distribution into a standard Gaussian, while the latter learns a transition kernel parameterized by a deep neural network to invert the former process. Given the distribution of ground-truth data \(x_{0}\sim q(x_{0})\), the diffusion process generates a sequence of random variables \(x_{1},\ x_{2},\ \ldots,x_{T}\) with a transition kernel \(q(x_{t}|x_{t-1})\). By utilizing the chain rule of probabilities along with the Markov property, the  
Figure 3: Overview of wireless communication based on generative AI. joint distribution of \(x_{1},\ x_{2},\ \ldots,x_{T}\) conditioned on \(x_{0}\) can be represented as \(q(x_{1},\ldots,x_{T}|x_{0})\), which can be expressed as follows:  
\[q(x_{1},\ldots,x_{T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1}). \tag{5}\]  
In diffusion models, the transition kernel is explicitly designed to gradually transform the data distribution \(q(x_{0})\) into a tractable prior distribution. The typical design of the transition kernel involves Gaussian perturbation, which is given below:  
\[q(x_{t}|x_{t-1})=\mathcal{N}\Big{(}x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t} \mathbf{I}\Big{)}, \tag{6}\]  
where \(\{\beta_{t}\in(0,1)\}_{t=1}^{T}\) is a variance schedule that controls the step size of the diffusion process. Due to the diffusion process, the data sample \(x_{0}\) gradually loses its characteristics as \(t\) increases, and ultimately, as \(T\) goes to infinity, \(x_{T}\) converges to an isotropic Gaussian distribution. As mentioned in [13], by iteratively applying the reparameterization trick, it is possible to sample any step of noise conditioned directly on the input \(x_{0}\) through the diffusion process defined in Equation (6), which is defined as follows:  
\[q(x_{t}|x_{0})=\mathcal{N}\Big{(}x_{t};\sqrt{\overline{\alpha}_{t}}x_{0},(1- \overline{\alpha}_{t})\mathbf{I}\Big{)}, \tag{7}\]  
where \(\alpha_{t}=1-\beta_{t}\), and \(\overline{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}\). According to Equation (7), given \(x_{0}\), the sample \(x_{t}\) can be easily obtained by sampling a Gaussian vector \(\epsilon\ \sim\mathcal{N}(0,\mathbf{I})\):  
\[x_{t}=\sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{\alpha}_{t}}\epsilon. \tag{8}\]  
In order to generate true samples from Gaussian noise \(x_{T}\), the diffusion model first generates an unstructured noise vector from a prior distribution that is generally challenging to handle and then executes a learned reverse process to gradually remove the noise. Specifically, the reverse process is parameterized by the prior distribution \(p(x_{T})=\mathcal{N}(x_{T};0,\mathbf{I})\) and a learnable transition kernel \(p_{\theta}(x_{t-1}|x_{t})\). This learnable transition kernel takes the following form:  
\[p_{\theta}(x_{t-1}|x_{t})=\mathcal{N}\big{(}x_{t-1};\mu_{\theta}(x_{t},t), \tilde{\beta}_{t}\big{)}, \tag{9}\]  
\[\mu_{\theta}=\frac{1}{\sqrt{\alpha_{t}}}(x_{t}-\frac{1-\alpha_{t}}{\sqrt{1- \overline{\alpha}_{t}}}e_{\theta}(x_{t},t)), \tag{10}\]  
\[\tilde{\beta}_{t}=\frac{1-\overline{\alpha}_{t-1}}{1-\overline{\alpha}_{t}} \beta_{t}, \tag{11}\]  
where \(\theta\) represents the model parameter, the mean \(\mu_{\theta}\) being parameterized by a deep neural network such as U-Net, \(\tilde{\beta}_{t}\) is the variance of the reverse process, and \(e_{\theta}(x_{t},t)\) denotes the estimated Gaussian noise in \(x_{t}\). This reverse process allows us to generate the data sample \(\hat{x}_{0}\) by first sampling the noise vector \(x_{T}\sim p(x_{T})\). Subsequently, the learnable transition kernel \(p_{\theta}(x_{t-1}|x_{t})\) is repeatedly sampled until \(t=1\) to generate the data sample \(\hat{x}_{0}\).