Log-SNR of a Schedule

easyPROSignal-to-Noise Ratio and Loss WeightingDiffusion ModelsDiffusion Models 1

Problem Statement

Continuous-time diffusion (VDM) parameterizes everything by lambda_t = log SNR_t. Convert an alpha_bar schedule to its log-SNR curve.

Background

The signal-to-noise ratio is SNR_t = alpha_bar_t / (1 - alpha_bar_t), and its log is

$\lambda_t = \log \text{SNR}_t = \log \bar{\alpha}_t - \log(1 - \bar{\alpha}_t)$

lambda runs from large positive (clean) to large negative (pure noise) and is the natural axis for noise schedules and loss weights.

Your Task

Implement:

def log_snr(alpha_bar):

Return the list of lambda_t values rounded to 4 decimals.

Input Format

alpha_bar: list of cumulative products, each strictly in (0, 1).

Output Format

A list of floats rounded to 4 decimals.

Sample

print(log_snr([0.5, 0.2]))

Output:

[0.0, -1.3863]

Example

Input

print(log_snr([0.5, 0.2]))

Output

[0.0, -1.3863]

Reasoning

Input: alpha_bar is a list of cumulative signal fractions $\bar{\alpha}_t$ , each strictly between 0 and 1. Here $\bar{\alpha}_1 = 0.5$ and $\bar{\alpha}_2 = 0.2$ : the first step keeps half the signal, the second keeps a fifth.
First element: $1 - \bar{\alpha}_1 = 0.5$ , so signal and noise are equal and $\text{SNR}_1 = 0.5 / 0.5 = 1$ . Then $\lambda_1 = \log 0.5 - \log 0.5 = 0$ , which prints as 0.0 (a positive zero, never a negative one).
Second element: $1 - \bar{\alpha}_2 = 0.8$ , so $\lambda_2 = \log 0.2 - \log 0.8 \approx -1.6094 - (-0.2231) = \log 0.25 \approx -1.386294$ , which rounds to 4 decimal places as $-1.3863$ .
Output: one natural-log value per schedule entry, same length and order as the input: [0.0, -1.3863].
Notice: $\lambda = 0$ marks the point where signal equals noise. Less signal gives a negative $\lambda$ , and the curve is antisymmetric: $\bar{\alpha} = 0.8$ would give $+1.3863$ , the exact mirror of $0.2$ .

Constraints

1 <= len(alpha_bar) <= 100000, values strictly in (0, 1).
lambda_t = log(alpha_bar_t) - log(1 - alpha_bar_t).
Round to 4 decimals; avoid -0.0.

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Concept primerPREVIEW

1. Background Knowledge

In Variational Diffusion Models (VDM), the forward noising process is parameterized by a single scalar $\lambda_t = \log \text{SNR}_t$ , where $\text{SNR}_t$ is the ratio of signal power to noise power at time step $t$ . This log-SNR axis is preferred over raw time $t$ or the cumulative product $\bar{\alpha}_t$ because it linearizes the relationship between noise level and model loss weighting. As $t \to 0$ , $\bar{\alpha}_t \to 1$ and $\lambda_t \to +\infty$ (clean signal). As $t \to T$ , $\bar{\alpha}_t \to 0$ and $\lambda_t \to -\infty$ (pure noise).

The quantity $\bar{\alpha}_t$ (often written as alpha_bar) is the cumulative product of per-step signal retention factors: $\bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s$ . It represents the fraction of original signal variance remaining after $t$ steps of the forward process. The corresponding noise variance fraction is $1 - \bar{\alpha}_t$ . Because both quantities are strictly positive for valid schedules, their logarithms are well-defined real numbers.

The conversion from $\bar{\alpha}_t$ to $\lambda_t$ is purely algebraic:

\lambda_t = \log \bar{\alpha}_t - \log(1 - \bar{\alpha}_t)

This identity follows directly from $\log(a/b) = \log a - \log b$ . Computing the two log terms separately (rather than forming the ratio first) is numerically more stable, especially when $\bar{\alpha}_t$ is very close to 0 or 1, where the ratio could underflow or overflow in floating-point arithmetic.

2. Algorithm Approach

This is a direct element-wise transformation problem. There is no iterative search, dynamic programming, or optimization involved. For each value in the input list, apply the closed-form formula and collect the results. The key design decision is to use the log-difference form rather than computing the SNR ratio first and then taking a single log. This avoids intermediate overflow/underflow and is the standard practice in diffusion model implementations.

3. Step-by-Step Strategy

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