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SNR from Continuous Log-SNR

MediumDiffusion Models 📚 Diffusion Models 1

Problem Statement

Continuous-time diffusion often works directly with lambda = log SNR. Given a log-SNR value, recover alpha_bar, SNR, and the noise standard deviation for the variance-preserving (VP) process.

Background

For the VP process, alpha_bar = sigmoid(lambda) because lambda = log(alpha_bar/(1-alpha_bar)) is exactly the logit of alpha_bar. From there:

$\bar{\alpha} = \sigma(\lambda), \qquad \text{SNR} = e^{\lambda}, \qquad \sigma_{\text{noise}} = \sqrt{1 - \bar{\alpha}}$

Your Task

Implement:

def from_log_snr(lam):

Return a dict with "alpha_bar", "snr", "sigma", each rounded to 6 decimals.

Input Format

lam (float): the log-SNR value.

Output Format

A dict of three floats.

Sample

print(from_log_snr(0.0))

Output:

{'alpha_bar': 0.5, 'snr': 1.0, 'sigma': 0.707107}

Example:

Input:

print(from_log_snr(0.0))

Output:

{'alpha_bar': 0.5, 'snr': 1.0, 'sigma': 0.707107}

Reasoning:

Compute $\bar{\alpha}$ using the sigmoid function, which maps the log-SNR to the signal retention coefficient: $\bar{\alpha} = \sigma(0) = \frac{1}{1 + e^0} = \frac{1}{2} = 0.5$ .
Determine the Signal-to-Noise Ratio (SNR) by exponentiating the log-SNR: $\text{SNR} = e^0 = 1.0$ .
Calculate the noise standard deviation $\sigma$ based on the variance-preserving property, where the noise variance is $1 - \bar{\alpha}$ : $\sigma = \sqrt{1 - 0.5} = \sqrt{0.5} \approx 0.70710678$ .
Round each result to 6 decimal places to match the required output format: $\bar{\alpha} \to 0.5$ , $\text{SNR} \to 1.0$ , and $\sigma \to 0.707107$ .
The final output is {'alpha_bar': 0.5, 'snr': 1.0, 'sigma': 0.707107}

Constraints:

lam is any real number.
alpha_bar = sigmoid(lam), snr = exp(lam), sigma = sqrt(1 - alpha_bar).
Round each to 6 decimals.

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1. Background Knowledge

In variance-preserving (VP) diffusion models, the forward process gradually adds Gaussian noise to a data point $x_0$ while keeping the overall variance constant. The noisy sample at continuous time is modeled as $x_t = \bar{\alpha}(t) x_0 + \sigma(t) \epsilon$ , where $\epsilon \sim \mathcal{N}(0, I)$ . The quantity $\bar{\alpha}$ controls how much of the original signal remains, and $\sigma$ controls the magnitude of the injected noise. Because the process is variance-preserving, the signal and noise variances must sum to one: $\bar{\alpha} + \sigma^2 = 1$ .

The signal-to-noise ratio (SNR) is defined as the ratio of signal variance to noise variance: $\text{SNR} = \bar{\alpha} / \sigma^2$ . Working directly with the log-SNR, denoted $\lambda = \log(\text{SNR})$ , is numerically stable and common in continuous-time formulations. The key insight is that $\lambda$ is exactly the logit (inverse sigmoid) of $\bar{\alpha}$ . This means $\bar{\alpha} = \sigma(\lambda) = \frac{1}{1 + e^{-\lambda}}$ , where $\sigma$ here is the sigmoid function, not the noise standard deviation. This relationship arises because $\text{SNR} = \bar{\alpha}/(1-\bar{\alpha})$ , so taking the log gives $\lambda = \log(\bar{\alpha}) - \log(1-\bar{\alpha})$ , which is the definition of the logit.

Once you have $\bar{\alpha}$ , the other quantities follow directly. The SNR is simply $e^{\lambda}$ , and the noise standard deviation is $\sigma_{\text{noise}} = \sqrt{1 - \bar{\alpha}}$ . Note the potential for confusion: the symbol $\sigma$ is used both for the sigmoid function and for the noise standard deviation. In this problem, the output key "sigma" refers to the noise standard deviation, not the sigmoid.

2. Algorithm Approach

This is a direct formula evaluation problem. There is no iterative algorithm, search, or optimization involved. The approach is:

Compute $\bar{\alpha}$ from $\lambda$ using the sigmoid function.
Compute $\text{SNR}$ as the exponential of $\lambda$ .
Compute $\sigma_{\text{noise}}$ as the square root of $1 - \bar{\alpha}$ .
Round each value to 6 decimal places and return them in a dictionary.

The entire computation is a sequence of three elementary mathematical operations applied to the input scalar.

3. Step-by-Step Strategy

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solution.py

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