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DDIM with Stochasticity (general eta)

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

Implement the general DDIM update with the interpolation parameter eta sweeping from deterministic (eta=0) to full DDPM (eta=1).

Background

$\sigma_t = \eta \sqrt{\frac{1-\bar{\alpha}_{t'}}{1-\bar{\alpha}_t}}\sqrt{1 - \frac{\bar{\alpha}_t}{\bar{\alpha}_{t'}}}$

$\hat{x}_0 = \frac{x_t - \sqrt{1-\bar{\alpha}_t}\,\hat{\varepsilon}}{\sqrt{\bar{\alpha}_t}}, \qquad x_{t'} = \sqrt{\bar{\alpha}_{t'}}\,\hat{x}_0 + \sqrt{1-\bar{\alpha}_{t'}-\sigma_t^2}\;\hat{\varepsilon} + \sigma_t z$

Clamp the direction-term square root argument at 0. Add sigma_t * z only when eta > 0.

Your Task

Implement:

def ddim_step(x_t, eps, alpha_bar_t, alpha_bar_prev, eta=0.0, z=None):

Return x_{t'} as a list rounded to 4 decimals.

Input Format

x_t, eps: lists of length D; z: list or None.
alpha_bar_t, alpha_bar_prev (float); eta (float).

Output Format

A list of D floats rounded to 4 decimals.

Sample

print(ddim_step([0.4, -0.6], [0.1, 0.2], 0.5, 0.8, 0.0))

Output:

[0.4612, -0.8484]

Example:

Input:

print(ddim_step([0.4, -0.6], [0.1, 0.2], 0.5, 0.8, 0.0))

Output:

[0.4612, -0.8484]

Reasoning:

Compute the noise scale $\sigma_t$ : Since $\eta = 0.0$ , the stochastic term vanishes, so $\sigma_t = 0$ .
Estimate the initial state $\hat{x}_0$ by removing the noise component from $x_t$ : Using $\sqrt{1 - \bar{\alpha}_t} = \sqrt{0.5} \approx 0.7071$ and $\sqrt{\bar{\alpha}_t} = \sqrt{0.5} \approx 0.7071$ , we get $\hat{x}_0 = \frac{x_t - 0.7071 \cdot \hat{\varepsilon}}{0.7071}$ . For the first element: $\frac{0.4 - 0.7071(0.1)}{0.7071} \approx 0.4596$ ; for the second: $\frac{-0.6 - 0.7071(0.2)}{0.7071} \approx -0.8485$ .
Calculate the deterministic direction term: The coefficient is $\sqrt{1 - \bar{\alpha}_{t'} - \sigma_t^2} = \sqrt{1 - 0.8 - 0} = \sqrt{0.2} \approx 0.4472$ . Multiplying by $\hat{\varepsilon}$ gives $[0.0447, 0.0894]$ .
Combine the scaled prediction and direction term to find $x_{t'}$ $x_{t^{'}}$ : Multiply $\hat{x}_0$ $\overset{x}{^}_{0}$ by $\sqrt{\bar{\alpha}_{t'}} = \sqrt{0.8} \approx 0.8944$ $\overset{α}{ˉ}_{t^{'}} = 0.8 \approx 0.8944$ to get $[0.4111, -0.7589]$ $[0.4111, - 0.7589]$ , then add the direction term: $[0.4111 + 0.0447, -0.7589 + 0.0894] = [0.4558, -0.6695]$ $[0.4111 + 0.0447, - 0.7589 + 0.0894] = [0.4558, - 0.6695]$ . Correction: Re-evaluating the exact algebraic simplification for $\eta=0$ $η = 0$ where $x_{t'} = \sqrt{\bar{\alpha}_{t'}}\hat{x}_0 + \sqrt{1-\bar{\alpha}_{t'}}\hat{\varepsilon}$ $x_{t^{'}} = \overset{α}{ˉ}_{t^{'}} \overset{x}{^}_{0} + 1 - \overset{α}{ˉ}_{t^{'}} \overset{ε}{^}$ :
- Element 1: $\sqrt{0.8}\left(\frac{0.4 - \sqrt{0.5}(0.1)}{\sqrt{0.5}}\right) + \sqrt{0.2}(0.1) \approx 0.4612$
- Element 2: $\sqrt{0.8}\left(\frac{-0.6 - \sqrt{0.5}(0.2)}{\sqrt{0.5}}\right) + \sqrt{0.2}(0.2) \approx -0.8484$
The final output is [0.4612, -0.8484]

Constraints:

len(x_t) == len(eps); alpha-bar values in (0, 1].
Clamp 1 - alpha_bar_prev - sigma^2 at 0 inside the square root.
Add sigma*z only when eta > 0; round to 4 decimals; avoid -0.0.

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1. Background Knowledge

DDPM (Denoising Diffusion Probabilistic Models) defines a forward noising process where Gaussian noise is gradually added to data, and a reverse denoising process where a neural network predicts the noise $\hat{\varepsilon}$ to recover the clean signal. The standard DDPM reverse step is stochastic and follows a specific Gaussian transition kernel.

DDIM (Denoising Diffusion Implicit Models) generalizes the reverse process by introducing a deterministic path. When the interpolation parameter $\eta = 0$ , the update becomes fully deterministic, meaning the same noise prediction $\hat{\varepsilon}$ always leads to the same $x_{t'}$ . This allows for faster sampling (fewer steps) and enables techniques like classifier guidance.

The general DDIM update bridges these two extremes. By varying $\eta$ from $0$ to $1$ , you interpolate between the deterministic DDIM path and the stochastic DDPM path. The key insight is that the "noise" added at each step is controlled by $\sigma_t$ , which depends on $\eta$ and the ratio of cumulative noise schedules ( $\bar{\alpha}$ ). When $\eta=1$ , $\sigma_t$ recovers the standard DDPM noise level.

2. Algorithm Approach

The implementation follows a direct formulaic approach:

Compute $\sigma_t$ : Use the provided formula involving $\eta$ , $\bar{\alpha}_t$ , and $\bar{\alpha}_{t'}$ (which is alpha_bar_prev in the function signature).
Predict $\hat{x}_0$ : Use the current noisy sample $x_t$ and the predicted noise $\hat{\varepsilon}$ to estimate the clean data.
Reconstruct $x_{t'}$ : Combine the predicted clean data $\hat{x}_0$ , the predicted noise $\hat{\varepsilon}$ , and the stochastic term $\sigma_t z$ using the specific coefficients derived from the DDIM derivation.
Handle Edge Cases: Ensure numerical stability by clamping square root arguments and conditionally adding the stochastic term.

3. Step-by-Step Strategy

Calculate $\sigma_t$ :

Compute the term inside the first square root: $\frac{1-\bar{\alpha}_{t'}}{1-\bar{\alpha}_t}$ .
Compute the term inside the second square root: $1 - \frac{\bar{\alpha}_t}{\bar{\alpha}_{t'}}$ .
Multiply these by $\eta^2$ (since $\sigma_t = \eta \sqrt{\dots}$ , so $\sigma_t^2 = \eta^2 \dots$ ).
Take the square root to get $\sigma_t$ .

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