# MixDiff-TTS: Mixture Alignment and Diffusion Model for Text-to-Speech
## 2 Materials and Methods
### Background  
#### 2.1.1 Diffusion Models  
Owing to their exceptional capability for modeling data, diffusion models are widely employed to characterize complex data distributions. As a subclass of probabilistic generative models, they have achieved cutting-edge performance in diverse domains (e.g., image and audio synthesis) in recent years, thereby demonstrating their remarkable capacity to produce high-fidelity samples [20; 21; 22; 23; 24; 25]. Inspired by non-equilibrium thermodynamics in physics, diffusion models employ a bidirectional stochastic process framework grounded in Markov chains. During training, the diffusion process gradually injects Gaussian noise into data samples, thereby progressively degrading their structural coherence until they evolve into pure Gaussian noise. Subsequently, the models achieve sample reconstruction by learning a reverse process, thereby recovering high-fidelity original samples from this noise-perturbed state. As depicted in Figure 1, the bidirectional stochastic process of diffusion models is illustrated. Given the predefined noise schedule \(\beta\) and diffusion steps \(t\), the constants corresponding to the diffusion process and the reverse process are calculated, as shown in the following equation:  
\[\alpha_{t}=\prod_{i=1}^{t}\sqrt{1-\beta_{i}} \tag{1}\]  
\[\sigma_{t}=\sqrt{1-\alpha_{t}^{2}} \tag{2}\]  
where \(\beta\) represents the noise schedule that governs the magnitude of noise injected during the diffusion process. \(\alpha_{t}\) denotes the cumulative retention rate at each diffusion step \(t\), measuring the signal attenuation from the original data to the current diffusion step. \(\sigma_{t}\) is the total noise standard deviation at diffusion step \(t\), indicating the proportion of noise in the current state. The diffusion process is typically defined mathematically by a memoryless Markov chain [26]. This Markov chain is characterized by two key properties: its parameters remain constant across all diffusion steps, and the current state depends solely on the previous state. The properties of the Markov chain significantly reduce the computational complexity  
Figure 1: Schematic diagram of the diffusion process and reverse process. of the diffusion process. By gradually adding Gaussian noise to the initial data \(x_{0}\), the diffusion process can transform \(x_{0}\) into a latent variable \(x_{T}\) over \(T\) steps, as shown below:  
\[q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}|x_{t-1}) \tag{3}\]  
where \(q(x_{1:T}|x_{0})\) represents the joint probability distribution of all intermediate variables \(x_{1}\) to \(x_{T}\) in the diffusion process, given that \(x_{0}\). \(q(x_{t}|x_{t-1})\) denotes the conditional probability of a single diffusion step, describing how \(x_{t}\) is generated from \(x_{t-1}\) in the diffusion process. According to the predefined noise schedule \(\beta\), the diffusion process can obtain the latent variable \(x_{T}\) by adding small Gaussian noise to \(x_{t-1}\). Therefore, \(q(x_{t}|x_{t-1})\) is given as follows:  
\[q(x_{t}|x_{t-1})=N(x_{t},\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I) \tag{4}\]  
where \(I\) denotes the identity matrix, ensuring that the noise is isotropic (i.e., independent across dimensions). This equation indicates that each step \(x_{t}\) is sampled from a Gaussian distribution with a mean of \(\sqrt{1-\beta_{t}}x_{t-1}\) and a variance of \(\beta_{t}I\). The reverse process is also modeled by a Markov chain. Unlike the diffusion process with fixed parameters, the parameters \(\theta\) of this Markov chain are learned through training.