Predict Noise from x0 and xt

1. Background Knowledge

In denoising diffusion probabilistic models (DDPMs), the forward process gradually adds Gaussian noise to a clean data sample $x_0$ over $T$ discrete timesteps. At each step $t$, the noisy sample $x_t$ is a linear combination of the original signal and the injected noise $\varepsilon \sim \mathcal{N}(0, I)$:

Here, $\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$ is the cumulative product of per-step signal retention factors, where $\alpha_s = 1 - \beta_s$ and $\beta_s$ is the noise schedule. As $t$ increases, $\bar{\alpha}_t$ decreases toward 0, meaning the signal contribution shrinks and the noise dominates.

The three common parameterizations of a diffusion model's output are:

• Epsilon ($\varepsilon$) prediction: The network predicts the noise added at step $t$.
• $x_0$ prediction: The network predicts the original clean sample.
• Velocity ($v$) prediction: A reparameterization that interpolates between the two.

These are algebraically interconvertible given $x_t$ and $\bar{\alpha}_t$. For instance, if you know $x_0$ and $x_t$, you can recover $\varepsilon$ by isolating it from the forward equation. This is useful in training pipelines where the model is trained with one parameterization but evaluated or analyzed with another.

2. Algorithm Approach

This is a direct algebraic inversion problem. Given the forward diffusion equation, solve for the unknown $\varepsilon$ by rearranging terms:

The approach is element-wise: for each dimension $d$, compute the numerator $x_t^{(d)} - \sqrt{\bar{\alpha}_t} \cdot x_0^{(d)}$, then divide by the scalar denominator $\sqrt{1 - \bar{\alpha}_t}$. No iterative methods or optimization are needed—this is a closed-form solution.

3. Step-by-Step Strategy