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DDIM Deterministic Step (eta = 0)

EasyDiffusion Models 📚 Diffusion Models 1

Problem Statement

The deterministic DDIM update (eta = 0) is the workhorse of fast sampling. Implement one step given x_t, the predicted noise, and the two alpha-bar values.

Background

With eta = 0 there is no injected noise, so the update is

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

The same predicted noise is reused as the "direction pointing to x_t."

Your Task

Implement:

def ddim_deterministic(x_t, eps, alpha_bar_t, alpha_bar_prev):

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

Input Format

x_t, eps: lists of length D.
alpha_bar_t, alpha_bar_prev (float) in (0, 1].

Output Format

A list of D floats rounded to 4 decimals.

Sample

print(ddim_deterministic([1.4142, 0.0], [1.0, -1.0], 0.5, 1.0))

Output:

[1.0, 1.0]

Example:

Input:

print(ddim_deterministic([1.4142, 0.0], [1.0, -1.0], 0.5, 1.0))

Output:

[1.0, 1.0]

Reasoning:

Compute the predicted clean sample $\hat{x}_0$ using the formula $\hat{x}_0 = \frac{x_t - \sqrt{1 - \bar{\alpha}_t}\, \hat{\varepsilon}}{\sqrt{\bar{\alpha}_t}}$ . With $\bar{\alpha}_t = 0.5$ , the scaling factors are $\sqrt{1 - 0.5} = \sqrt{0.5} \approx 0.7071$ and $\sqrt{0.5} \approx 0.7071$ .
For the first dimension, substitute $x_t = 1.4142$ and $\hat{\varepsilon} = 1.0$ : $\hat{x}_{0,1} = \frac{1.4142 - (0.7071 \cdot 1.0)}{0.7071} = \frac{0.7071}{0.7071} = 1.0$ .
For the second dimension, substitute $x_t = 0.0$ and $\hat{\varepsilon} = -1.0$ : $\hat{x}_{0,2} = \frac{0.0 - (0.7071 \cdot -1.0)}{0.7071} = \frac{0.7071}{0.7071} = 1.0$ . Thus, $\hat{x}_0 = [1.0, 1.0]$ .
Compute the next state $x_{t'}$ using $x_{t'} = \sqrt{\bar{\alpha}_{t'}}\, \hat{x}_0 + \sqrt{1 - \bar{\alpha}_{t'}}\, \hat{\varepsilon}$ . Since $\bar{\alpha}_{t'} = 1.0$ , the coefficients are $\sqrt{1.0} = 1.0$ and $\sqrt{1 - 1.0} = 0.0$ .
Apply the coefficients to the vectors: $x_{t'} = 1.0 \cdot [1.0, 1.0] + 0.0 \cdot [1.0, -1.0] = [1.0, 1.0]$ .
The final output is [1.0, 1.0]

Constraints:

len(x_t) == len(eps); alpha-bar values in (0, 1].
alpha_bar_prev = 1.0 yields exactly x0_hat (the direction term vanishes).
Round to 4 decimals; avoid -0.0.

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1. Background Knowledge

DDPM (Denoising Diffusion Probabilistic Models) and DDIM (Denoising Diffusion Implicit Models) are two frameworks for sampling from a learned noise distribution. In DDPM, each reverse step is stochastic: you predict the noise $\hat{\varepsilon}$ , estimate the clean signal $\hat{x}_0$ , and then add fresh Gaussian noise to move to $x_{t-1}$ . DDIM generalizes this by introducing a parameter $\eta \in [0, 1]$ that controls how much stochasticity remains. When $\eta = 0$ , the process becomes deterministic: the same predicted noise $\hat{\varepsilon}$ is reused as the "direction" toward the noisy sample, and no new random noise is injected.

The forward process defines $\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$ , where $\alpha_s = 1 - \beta_s$ . The noisy sample is $x_t = \sqrt{\bar{\alpha}_t}\, x_0 + \sqrt{1 - \bar{\alpha}_t}\, \varepsilon$ . The model predicts $\hat{\varepsilon}$ from $x_t$ . From this prediction you can recover an estimate of the clean signal:

$\hat{x}_0 = \frac{x_t - \sqrt{1 - \bar{\alpha}_t}\, \hat{\varepsilon}}{\sqrt{\bar{\alpha}_t}}$

For the deterministic DDIM step, you then reconstruct $x_{t'}$ (where $t' < t$ ) by blending $\hat{x}_0$ and $\hat{\varepsilon}$ using the previous alpha-bar value:

$x_{t'} = \sqrt{\bar{\alpha}_{t'}}\, \hat{x}_0 + \sqrt{1 - \bar{\alpha}_{t'}}\, \hat{\varepsilon}$

This is a linear combination, so the operation is element-wise over the feature dimension $D$ .

2. Algorithm Approach

The problem reduces to two vectorized (or element-wise) arithmetic operations:

Predict $\hat{x}_0$ : For each dimension $i$ , compute $\hat{x}_0[i] = \frac{x_t[i] - \sqrt{1 - \bar{\alpha}_t}\cdot \hat{\varepsilon}[i]}{\sqrt{\bar{\alpha}_t}}$ .
Reconstruct $x_{t'}$ : For each dimension $i$ , compute $x_{t'}[i] = \sqrt{\bar{\alpha}_{t'}}\cdot \hat{x}_0[i] + \sqrt{1 - \bar{\alpha}_{t'}}\cdot \hat{\varepsilon}[i]$ .

No loops over timesteps are needed—this is a single-step update. The key insight is that both $\sqrt{\bar{\alpha}_t}$ and $\sqrt{1-\bar{\alpha}_t}$ are scalars shared across all dimensions, so you compute them once and apply them element-wise.

3. Step-by-Step Strategy

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