## 1 Introduction
### Background
Therefore, Equation (4) can be written as:  
\[x_{t}= \sqrt{\alpha_{t}\alpha_{t-1}}x_{t-2}+\sqrt{1-\alpha_{t}\alpha_{t-1}}\ \epsilon \tag{5}\]  
By expanding Equation (5) further, we obtain the following equation:  
\[x_{t}= \sqrt{\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}}x_{0}+\sqrt{1-\alpha_{t} \alpha_{t-1}\ldots\alpha_{1}}\ \epsilon \tag{6}\]  
Defining \(\overline{\alpha}_{t}=\alpha_{t}\alpha_{t-1}\ldots\alpha_{1}\), Equation (6) can be written concisely as follows:  
\[x_{t}= \sqrt{\overline{\alpha}_{t}}x_{0}+\sqrt{1-\overline{\alpha}_{t}}\ \epsilon \tag{7}\]  
Given \(x_{0}\), Equation (7) allows us to generate \(x_{t}\) in a computationally efficient manner in just one step, rather than iteratively adding noise in \(t\) steps. This is achieved by sampling from the Gaussian distribution \(\mathcal{N}(x_{t};\mu_{t}=\sqrt{\overline{\alpha}_{t}}x_{0},\ \Sigma_{t}^{2}=(1-\overline{\alpha}_{t})I)\). \(\overline{\alpha}_{t}\) is defined using \(\alpha_{1},\ldots,\alpha_{t}\), where \(\alpha_{i}=1-\beta_{i}\ (1\leq i\leq t)\). \(\beta_{i}\) values are often defined using a linear [21] or cosine scheduler [21; 22], with the cosine schedule leading to superior results as it smoothly transitions the noise added to the data across the diffusion steps, which is essential for controlling the quality and characteristics of the generated data in diffusion models. A Cosine scheduler can be defined as follows:  
\[q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}\mid x_{t-1})\]0  
where \(\beta_{t}\) represents the noise level at step \(t\) (higher \(\beta_{t}\) values correspond to higher noise levels); \(t=1\) and \(t=T\) result in the minimum and maximum noise levels, respectively; \(T\) is the total number of steps in the diffusion process; and \(t\) is the current time step. All experiments in this study were conducted using \(\beta_{\min}=0.0001\) and \(\beta_{\max}=0.02\). As \(T\) approaches infinity, \(x_{T}\) converges to an isotropic Gaussian distribution, meaning that all components of \(x_{t}\) are normally distributed and decorrelated. Therefore, by having a procedure for learning the reverse distribution \(q(x_{t-1}\mid x_{t})\), we can start with sampling \(x_{T}\) from a normal distribution and follow the reverse distribution to arrive at \(x_{0}\). It should be noted that in the forward diffusion process, noise is added at each step; that is, \(x_{t}\) is derived from adding noise to \(x_{t-1}\) according to a distribution of \(q(x_{t}|x_{t-1})\). In the backward diffusion process, however, the aim is to reverse the forward process by removing the noise added to \(x_{t}\) to recover \(x_{t-1}\). This is achieved by a reverse transition using a distribution denoted as \(q(x_{t-1}|x_{t})\). In practical terms, the true reverse distribution \(q(x_{t-1}|x_{t})\) is unknown and impossible to compute directly, as accurately estimating it would require complex calculations involving the entire data distribution. Instead, we approximate \(q(x_{t-1}|x_{t})\) using a parameterized model, a deep neural network, denoted as \(p_{\theta}(x_{t-1}|x_{t})\). Given that \(q(x_{t-1}|x_{t})\) is also Gaussian for small enough values of \(\beta_{t}\) in the forward diffusion process, we can choose \(p_{\theta}(x_{t-1}|x_{t})\) to be a Gaussian distribution. In this case, the neural network is trained to parameterize the mean and variance of this Gaussian distribution.