Symbol in context

$x_0$

This entry links every published equation using this exact notation. Its meaning may change between equations.

Meanings in context

Article-grounded definition

a coordinate in the stated process.

Article-grounded definition

the input.

x_0

a coordinate in the stated process.

x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1-\bar{\alpha}_t}\, \epsilon, \qquad \hat{x}_0 = f_\theta(x_t, t),

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

ds^2 = -U(Z)^2\,dx_0^2 + dZ^2 + dx^2 + dy^2, \qquad x_0 := ct, \qquad U(Z) := 1 + \frac{gZ}{c^2}.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

x_0

the input.

\forall\, x \in B_\epsilon(x_0): \quad f(x) \in S,

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

x_0

the input.

\mathcal{L}_{\text{diffusion}} = \mathbb{E}_{x_0,\, \epsilon \sim \mathcal{N}(0, I),\, t} \Big[\, \big\| \epsilon - \epsilon_\theta(x_t, t, c) \big\|^2 \,\Big], \qquad x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1 - \bar{\alpha}_t}\, \epsilon

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.