Where do the extra DDPM sampling steps come from?

[AlejandroBaron]

I see this line matching the original paper

diffusers/src/diffusers/schedulers/scheduling_ddpm.py

Line 449 in 1e8cf27

pred_original_sample = (sample - beta_prod_t ** (0.5) * model_output) / alpha_prod_t ** (0.5)

But some extra steps are in place

diffusers/src/diffusers/schedulers/scheduling_ddpm.py

Lines 468 to 476 in 1e8cf27

	# 4. Compute coefficients for pred_original_sample x_0 and current sample x_t
	# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
	pred_original_sample_coeff = (alpha_prod_t_prev ** (0.5) * current_beta_t) / beta_prod_t
	current_sample_coeff = current_alpha_t ** (0.5) * beta_prod_t_prev / beta_prod_t
	# 5. Compute predicted previous sample µ_t
	# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
	pred_prev_sample = pred_original_sample_coeff * pred_original_sample + current_sample_coeff * sample

Sampling Algorithm 2 does not perform this operation in the original paper

Where do these come from?

[bmaxdk]

These additional steps defines the calculation of the predicted sample during the reverse diffusion process which introducing coefficients that balance the contributions of the predicted original sample x_0 and the current noisy sample x_t. This will be improving both numerical stability and sample quality in the final output. That's true that those are not shown in the original paper but have likely been added based on practical considerations and optimization for real-world use cases

[jadevaibhav]

These formula comes from reverse diffusion process, where at each time step we sample from a Gaussian distribution (Step 4). The first term is basically predicted mean (computed using current sample and predicted noise). The implementation uses reformulation of the same.
For 3 different models of training , namely prediction of 'noise', 'sample' and 'velocity prediction'; its easier to use reformulation with predicted original sample(pred_x_0) and current sample (x_t). I think that's what the code does.