Log-SNR of a Schedule

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:

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