A more efficient forward diffusion process

If we use the chained process to calculate a noisy image at t step, it first requires calculating the noisy image from 1 to t − 1 steps, which is not efficient. We can leverage a trick called reparameterization [10] to transform the original chained process into a one-step process. Here is what the trick looks like.

If we have a Gaussian distribution z with μ as the mean and σ 2 variance:

z ∼ 𝒩(μ, σ 2)

Then, we can rewrite the distribution as follows:

ϵ ∼ 𝒩(0,1)

z = μ+ σϵ

The benefit brought by this trick is that we can now calculate an image at any step with a one-step calculation, which will greatly boost the training performance:

x t = √ _ 1 − β t  x t−1 + √ _ β t  ϵ t−1

Now, say we define the following:

α...