Revert RePaint scheduler 'fix'

[GiusCat]

What does this PR do?

Reverts 'fix' introduced by pull request #10185; the PR adjusted the code accordingly to Algorithm 1 Line 5 of the RePaint paper (https://arxiv.org/pdf/2201.09865.pdf), however that line is reported incorrectly in the paper itself.

Algorithm 1 Line 5 refers to formula 8a of the same paper

$$x_{t-1}^{\text{known}} \sim \mathcal{N}(\sqrt{\bar{\alpha}} x_0, (1 - \bar{\alpha}) \mathbf{I})$$

which tells to sample from a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$ with $\mu = \sqrt{\bar{\alpha}} x_0$ and $\sigma^2 = (1 - \bar{\alpha})$. In order to obtain this result, the standard Gaussian distribution $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ should be stretched by a factor $\sigma$ and translated by a factor $\mu$ (source: Wikipedia). The correct operation should therefore be

$$x_{t-1}^{\text{known}} = \sqrt{\bar{\alpha}} x_0 + \sqrt{(1 - \bar{\alpha})} \epsilon$$

as it was indeed calculated before the change; Algorithm 1 Line 5 wrongly reports a stretching factor of $\sigma^2$. This PR restores the previous (correct) computation and adds some comments on the subject.

Fixes degraded performance using RePaint. Below are examples obtained using the example code in the documentation.

Diffusers v0.31.0 (before the 'fix')

Diffusers v0.32.1 (after the 'fix')

Who can review?

Anyone in the community is free to review the PR once the tests have passed. Feel free to tag
members/contributors who may be interested in your PR.

• Schedulers: @yiyixuxu

[yiyixuxu]

thanks @GiusCat

[yiyixuxu]

cc @hlky here too

[HuggingFaceDocBuilderDev]

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