Forward Diffusion Process

$q(𝐱_{1:T}|𝐱_0)=∏_{t=1}^Tq(𝐱_t|𝐱_{t-1})$
$q(𝐱_t|𝐱_{t-1})=𝒩(𝐱_t;\sqrt{1-\beta_t}𝐱_{t-1},\beta_t𝐈)$

• We can sample at any arbitary time step $t$ by using reparameterization trick
  • Let $\alpha_t =1-\beta_t,$ and $\bar{\alpha_t}=∏_{i=1}^t\alpha_i$

$\frac{𝐱-\mu}{\sigma}\sim𝒩{(0,1)}, 𝐱=\mu+\sigma\cdot\epsilon$

$𝐱_t=\sqrt{\alpha_t}𝐱_{t-1}+\sqrt{1-\alpha_{t-1}}\epsilon_{t-1}$

$=\sqrt{\alpha_t}(\sqrt{\alpha_t-1}𝐱_{t-2}+\sqrt{1-\alpha_{t-1}}\epsilon_{t-2})+\sqrt{1-\alpha_t}\epsilon_{t-1}$
$=\sqrt{\alpha_t\alpha_{t-1}}𝐱_{t-2}+\sqrt{1-\alpha_t\alpha_{t-1}}\bar\epsilon_{t-2}$
$=...$
$=\sqrt{\bar\alpha_t}𝐱_0+\sqrt{1-\bar\alpha_t}\epsilon$
;where $\epsilon_{t-1},\epsilon_{t-2},...\sim𝒩{(0,1)}$