#reverse-sde

Question 1: Single choice

The forward process $d\mathbf{x} = g(t)\,d\mathbf{w}$ is simulated backwards in time with step $\Delta t$ , using the score $\mathbf{s}(\mathbf{x}, t) = \nabla_\mathbf{x}\log p_t(\mathbf{x})$ and fresh noise $\mathbf{z}_t \sim \mathcal{N}(\mathbf{0}, I)$ . Which update is one step of the reverse-time SDE?

$\mathbf{x}_{t-\Delta t} = \mathbf{x}_t + g(t)^2\,\mathbf{s}(\mathbf{x}_t, t)\,\Delta t + g(t)\sqrt{\Delta t}\,\mathbf{z}_t$ 1
$\mathbf{x}_{t-\Delta t} = \mathbf{x}_t - g(t)^2\,\mathbf{s}(\mathbf{x}_t, t)\,\Delta t + g(t)\sqrt{\Delta t}\,\mathbf{z}_t$ 2
$\mathbf{x}_{t-\Delta t} = \mathbf{x}_t + g(t)\,\mathbf{s}(\mathbf{x}_t, t)\,\Delta t + g(t)^2\sqrt{\Delta t}\,\mathbf{z}_t$ 3
$\mathbf{x}_{t-\Delta t} = \mathbf{x}_t - g(t)\sqrt{\Delta t}\,\mathbf{z}_t$ , with no score term at all4

Timed test