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Ancestral Sampling Trajectory Norm

HardDiffusion Models 📚 Diffusion Models 1

Problem Statement

Run a short deterministic DDIM trajectory end-to-end over a given alpha-bar subsequence, using a constant noise prediction at every step (a toy oracle), and report the L2 norm of the final sample. This checks that you chain the per-step updates correctly.

Background

Given a descending list of alpha-bar values abs = [ab_0, ab_1, ..., ab_{K}] (from noisiest to a final 1.0) and a fixed predicted noise eps, run deterministic DDIM (eta=0) starting from x_start:

for each consecutive pair (ab_t, ab_prev):

x0_hat = (x - sqrt(1-ab_t)*eps)/sqrt(ab_t)
x      = sqrt(ab_prev)*x0_hat + sqrt(1-ab_prev)*eps

After the loop, return round(||x||_2, 4) where ||.||_2 is the Euclidean norm over all dimensions.

Your Task

Implement:

def trajectory_norm(x_start, eps, alpha_bars):

alpha_bars: list of K+1 alpha-bar values (descending, last is the target, e.g. 1.0).

Return the final sample's L2 norm rounded to 4 decimals.

Input Format

x_start, eps: lists of length D.
alpha_bars: list of >= 2 values in (0, 1].

Output Format

A float rounded to 4 decimals.

Sample

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

Output:

1.4142

Example:

Input:

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

Output:

1.4142

Reasoning:

Initialize the state vector $x$ with the input $[1.4142, 0.0]$ and the noise vector $\epsilon$ with $[1.0, -1.0]$ . The algorithm iterates through the alpha-bar pairs; here, there is only one step from $ab_t = 0.5$ to $ab_{prev} = 1.0$ .
Compute the estimated clean sample $x_0$ $x_{0}$ using the current noise level $ab_t = 0.5$ $a b_{t} = 0.5$ . The term $\sqrt{1 - 0.5} = \sqrt{0.5} \approx 0.7071$ $1 - 0.5 = 0.5 \approx 0.7071$ scales the noise, and dividing by $\sqrt{0.5} \approx 0.7071$ $0.5 \approx 0.7071$ normalizes the signal.
- Dimension 0: $x_0[0] = (1.4142 - 0.7071 \cdot 1.0) / 0.7071 = 0.7071 / 0.7071 = 1.0$
- Dimension 1: $x_0[1] = (0.0 - 0.7071 \cdot (-1.0)) / 0.7071 = 0.7071 / 0.7071 = 1.0$
- Resulting $x_0 = [1.0, 1.0]$ .
Update $x$ $x$ to the next step using $ab_{prev} = 1.0$ $a b_{p re v} = 1.0$ . Since $\sqrt{1.0} = 1.0$ $1.0 = 1.0$ and $\sqrt{1 - 1.0} = 0$ $1 - 1.0 = 0$ , the noise term vanishes, and $x$ $x$ becomes exactly $x_0$ $x_{0}$ .
- $x[0] = 1.0 \cdot 1.0 + 0 \cdot 1.0 = 1.0$
- $x[1] = 1.0 \cdot 1.0 + 0 \cdot (-1.0) = 1.0$
- Resulting $x = [1.0, 1.0]$ .
Calculate the L2 norm of the final vector $x$ : $\sqrt{1.0^2 + 1.0^2} = \sqrt{2} \approx 1.41421356$ .
The final output is 1.4142

Constraints:

len(x_start) == len(eps); len(alpha_bars) >= 2, values in (0, 1].
Chain deterministic DDIM steps over consecutive alpha-bar pairs.
Return the Euclidean norm of the final sample, rounded to 4 decimals.

More from Diffusion Models 1

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1. Background Knowledge

Diffusion models learn to generate data by gradually adding Gaussian noise to a sample over many timesteps, then learning to reverse that process. The forward process is parameterized by a schedule of noise levels, often expressed through $\bar{\alpha}_t$ , the cumulative product of per-step retention factors. A larger $\bar{\alpha}_t$ means less noise has been added; $\bar{\alpha}_T \approx 0$ corresponds to pure noise, while $\bar{\alpha}_0 = 1$ corresponds to the clean data.

DDIM (Denoising Diffusion Implicit Models) provides a deterministic sampling scheme that, given a predicted noise $\epsilon_\theta(x_t, t)$ , can jump directly between any two noise levels without simulating every intermediate step. The key identity is that the predicted clean sample $\hat{x}_0$ can be recovered from the noisy sample $x_t$ and the predicted noise $\epsilon$ via:

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

Once $\hat{x}_0$ is known, the next (less noisy) sample at level $\bar{\alpha}_{t-1}$ is reconstructed as:

x_{t-1} = \sqrt{\bar{\alpha}_{t-1}}\,\hat{x}_0 + \sqrt{1 - \bar{\alpha}_{t-1}}\,\epsilon

This is the deterministic update ( $\eta = 0$ ); no stochastic term is added. In this problem, $\epsilon$ is a fixed constant vector (a "toy oracle"), so the trajectory is fully determined by the input and the $\bar{\alpha}$ schedule.

The L2 norm $\|x\|_2 = \sqrt{\sum_i x_i^2}$ measures the overall magnitude of the final sample across all dimensions.

2. Algorithm Approach

This is a straightforward iterative chain of vector operations. You process consecutive pairs of $\bar{\alpha}$ values from the noisiest end toward the cleanest end. For each pair $(\bar{\alpha}_t, \bar{\alpha}_{t-1})$ :

Compute $\hat{x}_0$ from the current $x$ and the constant $\epsilon$ .
Reconstruct the next $x$ at the previous $\bar{\alpha}$ level.
Update $x$ in place.

After all pairs are processed, compute the Euclidean norm of the final $x$ and round to 4 decimal places. There is no branching, no optimization, and no state beyond the current vector $x$ .

3. Step-by-Step Strategy

Initialize: Set x to a copy of x_start (avoid mutating the input).
Loop over indices i from 0 to len(alpha_bars) - 2:
Let ab_t = alpha_bars[i] and ab_prev = alpha_bars[i+1].
Compute sqrt_ab_t = sqrt(ab_t) and sqrt_1_minus_ab_t = sqrt(1 - ab_t).
Compute x0_hat element-wise: (x - sqrt_1_minus_ab_t * eps) / sqrt_ab_t.
Compute sqrt_ab_prev and sqrt_1_minus_ab_prev similarly.
Update x element-wise: sqrt_ab_prev * x0_hat + sqrt_1_minus_ab_prev * eps.
Final norm: Compute sqrt(sum(x_i^2 for all i)).
Return round(norm, 4).

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