Predict Noise from x0 and xt

easyPROParameterizations and the Reverse PosteriorDiffusion ModelsDiffusion Models 1

Problem Statement

The three diffusion targets (eps, x0, v) are interconvertible given x_t and alpha_bar_t. Recover the noise eps from a known clean sample x0 and the noisy x_t.

Background

Since x_t = sqrt(alpha_bar) x0 + sqrt(1 - alpha_bar) eps, solving for the noise gives

$\varepsilon = \frac{x_t - \sqrt{\bar{\alpha}}\, x_0}{\sqrt{1 - \bar{\alpha}}}$

Your Task

Implement:

def eps_from_x0(x_t, x0, alpha_bar):

x_t, x0: NumPy-compatible nested lists (or scalars) of equal shape.

Return the noise as a list rounded to 4 decimals.

Input Format

x_t, x0: lists of equal length D.
alpha_bar (float) in (0, 1).

Output Format

A list of D floats rounded to 4 decimals.

Sample

print(eps_from_x0([1.4142, 0.0], [1.0, 1.0], 0.5))

Output:

[1.0, -1.0]

Example

Input

print(eps_from_x0([1.4142, 0.0], [1.0, 1.0], 0.5))

Output

[1.0, -1.0]

Reasoning

Compute the scaling factors from $\bar{\alpha} = 0.5$ : $\sqrt{\bar{\alpha}} = \sqrt{0.5} \approx 0.7071$ and $\sqrt{1 - \bar{\alpha}} = \sqrt{0.5} \approx 0.7071$ .
Calculate the numerator for the first dimension by subtracting the scaled clean sample from the noisy sample: $1.4142 - (0.7071 \times 1.0) \approx 0.7071$ .
Calculate the numerator for the second dimension: $0.0 - (0.7071 \times 1.0) = -0.7071$ .
Divide each numerator by the noise scaling factor to recover the noise: $\frac{0.7071}{0.7071} = 1.0$ and $\frac{-0.7071}{0.7071} = -1.0$ .
The final output is [1.0, -1.0]

Constraints

len(x_t) == len(x0), 0 < alpha_bar < 1.
eps = (x_t - sqrt(alpha_bar)*x0) / sqrt(1 - alpha_bar).
Round to 4 decimals; avoid -0.0.

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Concept primerPREVIEW

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 $\varepsilon \sim \mathcal{N}(0, I)$ :

x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1 - \bar{\alpha}_t}\, \varepsilon

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:

\varepsilon = \frac{x_t - \sqrt{\bar{\alpha}_t}\, x_0}{\sqrt{1 - \bar{\alpha}_t}}

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

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solution.py

Python 3.13

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