Diffusions: Denoising Diffusion Probabilistic Model
Pipeline $$ x = x_0 \to x_1 \to x_2 \to \cdots \to x_{T-1} x_T = z $$which follows the workflow structure: $$ x_t = \alpha_t x_{t-1} + \beta_t \epsilon_t, \quad \epsilon_t \in \mathcal{N}(0, \mathbf{I}) $$where $\alpha_t, \beta_t > 0$ and $\alpha^2 + \beta^2 = 1$. We do this repeatedly, and we will get: $$ \begin{align*} x_t &= \alpha_t x_{t-1} + \beta_t \epsilon_t\\ &= \alpha_t(\alpha_{t-1} x_{t-2} + \beta_{t-1} \epsilon_{t-1}) + \beta_t \epsilon_t\\ &= \cdots\\ &= (\alpha_t \cdots \alpha_1)x_0 + (\alpha_t \cdots \alpha_2)\beta_1\epsilon_1 + (\alpha_t \cdots \alpha_3)\beta_2\epsilon_2 + \cdots + \alpha_t\beta_{t-1}\epsilon{t-1} + \beta_t\epsilon_t \end{align*} $${% note info %} ⚠ BUG: Usage for latex align failed! {% endnote %}