논문 Preliminaries에 넣으면 좋을 듯

DDPM Reverse Process

$$ p(x_{t-1}|x_t)=\mathcal{N}(z_{t-1};\mu_{\theta}(x_t, t), \Sigma_{\theta}(x_t, t))\\

\mu_{\theta}(z_t, t)=\frac{1}{\sqrt{\alpha_t}}(z_t-\frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\epsilon_{\theta}(z_t, t)), \space \Sigma_{\theta}(z_t, t)=\sigma_t^2I $$

Sampling Formula - from DDIM paper

https://juniboy97.tistory.com/53

$$ \mathbf{x}{t-1} = \sqrt{\bar\alpha{t-1}} \underbrace{\left( \frac{\mathbf{x}t - \sqrt{1 - \bar\alpha_t} \epsilon\theta(\mathbf{x}t, t)}{\sqrt{\bar\alpha_t}} \right)}{\text{predicted } \\mathbf{x}_0 \\text{''}} + \\underbrace{\\sqrt{1 - \\bar\\alpha_{t-1} - \\sigma_t^2} \\cdot \\epsilon_\\theta(\\mathbf{x}_t, t)}_{\\text{direction pointing to } \mathbf{x}t \text{''}} + \underbrace{\sigma_t \epsilon_t}{\text{random noise}} $$

• where $\sigma_t=\eta_t\sqrt{(1-\bar\alpha_{t-1})/(1-\bar\alpha_t)}\sqrt{1-\bar\alpha_t/\bar\alpha_{t-1}}$
  • $\eta_t=1$: Forward Process - Markovian, Generative Process - DDPM
  • $\eta_t=0\space(\sigma_t=0)$: Forward Process - Deterministic ($\sigma_t \epsilon_t = 0$), Generative Process - DDIM

→ “Diffusion Model의 Sampling 과정(Denoising Process)이 Latent 뿐만 아니라 timestep $t$에 의존한다”

• noise prediction network $\epsilon_{\theta}(x_t, t)$는 timestep $t$를 명시적인 parameter로 받음
  • Latent Diffusion Model의 $\epsilon_{\theta}(x_t, t)$는 time-conditional UNet 사용
• $\sigma_t, \space \bar\alpha_t=\prod_{s=1}^{t}\alpha_s=\prod_{s=1}^{t}(1-\beta_s)$는 timestep $t$에 따라 다른 값을 가짐

(DDPM & DDIM 논문 인용하기!!)

Latent Diffusion Model Loss Function Formula

$$ L_{LDM}=\mathbb{E}{\mathcal{E}(x),\epsilon\sim\mathcal{N}(0,1),t}[\lVert \epsilon - \epsilon{\theta}(z_t, t) \rVert_2^2] $$

• Latent Diffusion Model의 Denoising Process가 예측하는 Noise는 Gaussian Noise를 따르도록 가정하고 학습