def p_sample(self, xt: torch.Tensor, t: torch.Tensor, eps_theta: torch.Tensor):
"""
#### Sample from $\textcolor{lightgreen}{p_\theta}(x_{t-1}|x_t)$
\begin{align}
\textcolor{lightgreen}{p_\theta}(x_{t-1} | x_t) &= \mathcal{N}\big(x_{t-1};
\textcolor{lightgreen}{\mu_\theta}(x_t, t), \sigma_t^2 \mathbf{I} \big) \\
\textcolor{lightgreen}{\mu_\theta}(x_t, t)
&= \frac{1}{\sqrt{\alpha_t}} \Big(x_t -
\frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\textcolor{lightgreen}{\epsilon_\theta}(x_t, t) \Big)
\end{align}
"""
# [gather](utils.html) $\bar\alpha_t$
alpha_bar = gather(self.alpha_bar, t)
# $\alpha_t$
alpha = gather(self.alpha, t)
# $\frac{\beta}{\sqrt{1-\bar\alpha_t}}$
eps_coef = (1 - alpha) / (1 - alpha_bar) ** .5
# $$\frac{1}{\sqrt{\alpha_t}} \Big(x_t -
# \frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\textcolor{lightgreen}{\epsilon_\theta}(x_t, t) \Big)$$
mean = 1 / (alpha ** 0.5) * (xt - eps_coef * eps_theta)
# $\sigma^2$
var = gather(self.sigma2, t)
# $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$
eps = torch.randn(xt.shape, device=xt.device)
# Sample
return mean + (var ** .5) * eps
def p_x0(self, xt: torch.Tensor, t: torch.Tensor, eps: torch.Tensor):
"""
#### Estimate $x_0$
$$x_0 \approx \hat{x}_0 = \frac{1}{\sqrt{\bar\alpha}}
\Big( x_t - \sqrt{1 - \bar\alpha_t} \textcolor{lightgreen}{\epsilon_\theta}(x_t, t) \Big)$$
"""
# [gather](utils.html) $\bar\alpha_t$
alpha_bar = gather(self.alpha_bar, t)
# $$x_0 \approx \hat{x}_0 = \frac{1}{\sqrt{\bar\alpha}}
# \Big( x_t - \sqrt{1 - \bar\alpha_t} \textcolor{lightgreen}{\epsilon_\theta}(x_t, t) \Big)$$
return (xt - (1 - alpha_bar) ** 0.5 * eps) / (alpha_bar ** 0.5)