How can we predict the start image from the noise at x_t?

[vedantroy]

This is mostly a question for @lucidrains, if they have a moment to answer.
In the code-base, there's this line:

imagen-pytorch/imagen_pytorch/imagen_pytorch.py

Line 294 in 2535012

def predict_start_from_noise(self, x_t, t, noise):

which seems to predict the starting image given the noise predicted by the model at timestep t.

I've also seen this in the implementation of https://arxiv.org/pdf/2102.09672.pdf
https://github.com/openai/improved-diffusion/blob/783b6740edb79fdb7d063250db2c51cc9545dcd1/improved_diffusion/gaussian_diffusion.py#L328.

On the surface, this equation seems kinda crazy. It's allowing the model to predict the original image, from the noise predicted at an arbitrary timestep, t, which seems like it's skipping over a large portion of the reverse diffusion process ??

The part I am confused about is, I can't find this equation anywhere in the paper. Do you know where it comes from / is there an explanation?

[vedantroy]

Solved.

[vedantroy]

(Re-arrange equation 9 in the paper https://arxiv.org/abs/2102.09672 to solve for x_0)