refactor: unify Euler, Euler Ancestral and DDIM implementations#1474
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wbruna wants to merge 7 commits intoleejet:masterfrom
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refactor: unify Euler, Euler Ancestral and DDIM implementations#1474wbruna wants to merge 7 commits intoleejet:masterfrom
wbruna wants to merge 7 commits intoleejet:masterfrom
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The sigma_to == 0 simplification is: d = (x - denoised) / sigma x = x + d * (sigma_to - sigma) = x + (x - denoised) / sigma * (0 - sigma) = x + (x - denoised) * -1 = denoised For eta == 0, sigma_down = sigma_to, and sigma_up = 0. The non-flow case is straightforward: x = x + d * (sigma_down - sigma) = x + d * (sigma_to - sigma) The flow case: sigma_ratio = sigma_down / sigma = sigma_to / sigma x = sigma_ratio * x + (1 - sigma_ratio) * denoised = x * sigma_ratio + denoised * (1 - sigma_ratio) = x * sigma_to / sigma - denoised * (sigma_to / sigma + 1) = x + x * sigma_to / sigma - x - denoised * sigma_to / sigma + denoised = x + (x - denoised) * (sigma_to / sigma - 1) = x + (x - denoised) / sigma * (sigma_to - sigma) = x + d * (sigma_to - sigma)
Euler Ancestral does: d = (x - denoised) / sigma x = x + d * (sigma_down - sigma) = x + (x - denoised) / sigma * (sigma_down - sigma) = x + (x - denoised) * (sigma_down / sigma - 1) = x + (x - denoised) * (sigma_ratio - 1) = x + x * sigma_ratio - x - denoised * sigma_ratio + denoised = x * sigma_ratio + denoised * (1 - sigma_ratio) The ancestral noise is also identical, except for the alpha_scale. I've kept the explicit test just to avoid an unnecessary tensor multiplication. Also, use the same calculation for the deterministic Euler implementation: it has one less tensor operation, and slightly better numerical stability.
We have: model_output = (x - denoised) / sigma = d alpha_prod_t = 1 / (sigma² + 1) beta_prod_t = 1 - alpha_prod_t = sigma² / (sigma² + 1) Substitute alpha_prod_t: sqrt(1 / alpha_prod_t) = sqrt(sigma² + 1) sqrt(beta_prod_t) = sqrt(sigma² / (sigma² + 1)) = sigma / sqrt(sigma² + 1) Then: pred_original_sample = (x / sqrt(sigma² + 1) - sqrt(beta_prod_t) * d) * (1 / sqrt(alpha_prod_t)) = (x / sqrt(sigma² + 1) - (sigma / sqrt(sigma² + 1)) * d) * sqrt(sigma² + 1) = x - sigma * d = x - sigma * ((x - denoised) / sigma) = x - (x - denoised) = denoised
When eta = 0, std_dev_t = 0. The sqrt term becomes: sqrt((1 - alpha_prod_t_prev - std_dev_t^2) / alpha_prod_t_prev) = sqrt((1 - alpha_prod_t_prev) / alpha_prod_t_prev) = sqrt(beta_prod_t_prev / alpha_prod_t_prev) Given: alpha_prod_t = 1 / (sigma^2 + 1) beta_prod_t = sigma^2 / (sigma^2 + 1) alpha_prod_t_prev = 1 / (sigma_to^2 + 1) beta_prod_t_prev = sigma_to^2 / (sigma_to^2 + 1) sqrt(beta_prod_t_prev / alpha_prod_t_prev) = sqrt((sigma_to^2 / (sigma_to^2 + 1)) / (1 / (sigma_to^2 + 1))) = sqrt(sigma_to^2) = sigma_to So the deterministic step becomes: x = denoised + sigma_to * model_output = denoised + sigma_to * (x - denoised) / sigma = denoised + (x - denoised) * sigma_to / sigma = denoised + x * sigma_to / sigma - denoised * sigma_to / sigma = denoised * (1 - sigma_to / sigma) + x * sigma_to / sigma = x + denoised * (1 - sigma_to / sigma) + x * sigma_to / sigma - x = x + denoised * (1 - sigma_to / sigma) - x * (1 - sigma_to / sigma) = x + (denoised - x) * (1 - sigma_to / sigma) = x + (denoised - x) * (1 - sigma_to / sigma) = x + (x - denoised) * (sigma_to / sigma - 1) = x + (x - denoised) / sigma * (sigma_to - sigma) = x + d * (sigma_to - sigma);
From the DDIM definitions:
alpha_prod_t = 1 / (sigma² + 1)
beta_prod_t = 1 - alpha_prod_t = sigma² / (sigma² + 1)
d = (x - denoised) / sigma
We have the coefficient of d in the x update:
coeff² = (1 - alpha_prod_t_prev - std_dev_t²) / alpha_prod_t_prev
Where:
std_dev_t² = eta² * variance
variance = (beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)
= sigma_to² (sigma² - sigma_to²) / (sigma² (sigma_to² + 1))
Substituting variance:
coeff² = ( sigma_to² / (sigma_to² + 1) - eta² * sigma_to² (sigma² - sigma_to²) / (sigma² * (sigma_to² + 1)) ) * (sigma_to² + 1)
= sigma_to² - eta² * sigma_to² * (sigma² - sigma_to²) / sigma²
= sigma_to² * ( 1 - eta² * (sigma² - sigma_to²) / sigma² )
From get_ancestral_step:
sigma_down² = sigma_to² - sigma_up²
= sigma_to² - eta² * sigma_to² * (sigma² - sigma_to²) / sigma²
= coeff²
So coeff = sigma_down, and the x update becomes:
x = denoised + sigma_down * d
= denoised + sigma_down * (x - denoised) / sigma
= denoised + (x - denoised) * sigma_ratio
= x * sigma_ratio + denoised - denoised * sigma_ratio
= x * sigma_ratio + denoised * (1 - sigma_ratio)
And the noise coefficient:
noise_coeff = std_dev_t / sqrt(alpha_prod_t_prev)
= eta * sqrt( sigma_to² * (sigma² - sigma_to²) / (sigma² * (sigma_to² + 1)) ) * sqrt(sigma_to² + 1)
= eta * sigma_to * sqrt( (sigma² - sigma_to²) / sigma² )
= sigma_up
It is equivalent to Euler Ancestral with the Simple scheduler.
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This started as an attempt to simplify the DDIM sampler, and ended up removing it entirely 🙂 It turns out it is equivalent to Euler Ancestral. I've kept the algebraic demonstrations on each commit message.
I've also joined the original Euler Ancestral with the flow variant, with the helper function from #1436 . The same approach could probably be used for other ancestral implementations.
The Euler merge is a little less clear-cut, since it loses its original simplicity, but I believe the unified code path to be overall simpler to follow and maintain.