# A Diffusion Modeling-Based System for Teaching Dance to Digital Human
## 3 Theoretical Framework of Digital Human Dance Teaching System
### Diffusion Model Training  
The foundation of this system is the diffusion model, specifically the Denoising Diffusion Probabilistic Model (DDPM). This model is employed by companies such as OpenAI, Google, and Stability AI to achieve advancements in text-to-image generation, thereby advancing the field of Artificial Intelligence Generated Content (AIGC). In this system, the diffusion model is utilized to perform noise prediction for dance sequences. The model  
Figure 4: Schematic of the FineDance dataset.  
operates in a Gaussian space to predict and match dance sequences, fulfilling the requirements for driving the dance generation process. Compared to other AI generation methods, the diffusion model offers significant advantages, including effective sequence prediction and enhanced diversity in the generated dance movements, providing robust technical support for the system.  
In general, the diffusion model comprises two main processes: the forward diffusion process and the backward sampling process, as illustrated in Figure 5[16]. The forward diffusion process involves adding Gaussian noise to transform the dance sequence into random noise. The backward sampling process, or denoising process, employs a series of Gaussian distributions parameterized by a neural network. The diffusion model utilizes these trained networks to constitute the final generative model. Specifically, noise is first added to the dance sequence, and then the denoising process is applied to predict and generate the dance sequence.  
The diffusion model employed in this system is based on the Denoising Diffusion Probabilistic Model (DDPM). It defines the diffusion process as a Markov process characterized by a forward noise process \(q(z_{t}|x)\). The backward process \(x\sim p(x)\) is derived from the data distribution, with the forward noise process described as follows:  
\[q(z_{t}|x)\sim\mathcal{N}(\sqrt{\pi_{t}}x,(1-\pi_{t})I) \tag{1}\]  
where \(\pi_{t}\in(0,1)\) is a constant that follows a monotonically decreasing time and approximates \(z_{T}\sim\mathcal{N}(0,I)\) as \(\pi_{t}\) approaches zero.  
Where the musical condition \(\pi_{t}\in(0,1)\) is set for diffusion restriction, and the forward propagation process is inverted by predicting the model parameters \(\theta\), and thus estimating \(\hat{x}_{\theta}(z_{t},t,c)\approx x\), in time step \(t\). We optimize the objective function using the following:  
\[\mathcal{L}_{\text{simple}}=\mathbb{E}_{x,t}\Big{[}\parallel x-\hat{x}_{ \theta}(z_{t},t,c)\parallel_{2}^{2}\Big{]} \tag{2}\]  
where \(\mathbb{E}\) denotes the mathematical expectation, \(z_{t}\) is the dance sequence, \(t\) is the time step, and \(c\) is the musical condition.  
This system utilizes the Transformer architecture for the forward prediction in diffusion models. The Transformer is known for its ability to \"vectorize\" any text or sentence, thereby capturing and maximizing the precise meaning of the input. It has demonstrated significant success across various domains in AI, including natural language processing, computer vision, and audio processing. In 2021, a paper co-authored by Stanford University referred to models based on the Transformer architecture as \"foundational models\", highlighting its role in driving a paradigm shift in AI [17]. In this system, Nx = 6 denotes that the encoder consists of 6 stacked encoder blocks. Each block, as depicted in Figure 6, comprises a Multi-Head Attention mechanism and a Fully Connected Feed-Forward Network.  
Figure 5: Diffusion modeling to predict dance sequences.  
The attention mechanism is the core of the Transformer, and using it can excellently accomplish the task of dance sequence prediction, the specific process is as follows:  
\[Attention(Q,K,V)=Softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V \tag{3}\]  
where \(Q\in R^{N\times d_{k}}\), \(K\in R^{N\times d_{k}}\), \(V\in R^{N\times d_{v}}\) are matrices consisting of query, key, and value vectors, respectively.