The ELF model's SDE (Stochastic Differential Equation) sampler, specifically Algorithm 6, and its mathematical consistency with the paper's interpolation convention.

Hi, thanks for the great work and for releasing the code/paper.

I have a question about the SDE sampler in Algorithm 6. The paper uses the interpolation convention

```text
z_t = t * x + (1 - t) * eps
```

so `t` is the data coefficient and `1 - t` is the noise coefficient. In Algorithm 6, the sampler defines

```text
alpha = 1 - gamma * dt
t_back = alpha * t
z_back = alpha * z + (1 - alpha) * e
```

If we substitute `z = t * x + (1 - t) * eps`, then

```text
z_back = alpha * t * x + alpha * (1 - t) * eps + (1 - alpha) * e
```

The clean-data coefficient is indeed `alpha * t = t_back`. However, the noise part is a mixture of the previous noise and newly injected independent noise:

```text
alpha * (1 - t) * eps + (1 - alpha) * e
```

If `eps` and `e` are independent (this is naturally true), the total noise standard deviation is

```text
sqrt(alpha^2 * (1 - t)^2 + (1 - alpha)^2)
```

whereas a sample truly at timestep `t_back = alpha * t` under the paper's interpolation would require noise coefficient

```text
1 - t_back = 1 - alpha * t
```

These are generally not equal! Therefore, it seems that `z_back` matches the target clean-data coefficient, but not the target total noise level / marginal distribution at `t_back`.

This related to the coefficient-preserving sampling discussed in our paper, which emphasizes that stochastic flow samplers sho...