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Rescale to Zero Terminal SNR

Diffusion ModelsNoise Schedules

Medium

Rescale a noise schedule so that its last timestep holds no signal at all, then return the new betas, alpha_bars and signal to noise ratios.

\bar\alpha_t = \prod_{k=0}^{t} (1 - \beta_k), \qquad a_t = \sqrt{\bar\alpha_t}, \qquad \tilde a_t = \left(a_t - a_{T-1}\right) \frac{a_0}{a_0 - a_{T-1}}

\tilde{\bar\alpha}_t = \tilde a_t^{\,2}, \qquad \tilde\beta_t = 1 - \frac{\tilde{\bar\alpha}_t}{\tilde{\bar\alpha}_{t-1}} \ \text{with}\ \tilde{\bar\alpha}_{-1} = 1, \qquad \mathrm{snr}_t = \frac{\tilde{\bar\alpha}_t}{1 - \tilde{\bar\alpha}_t}

Here betas holds the original per-step noise variances $\beta_t$ for timesteps $t = 0, \dots, T - 1$ , and $\bar\alpha_t$ is their cumulative signal fraction. The values $a_t$ are shifted so that the last one becomes 0 and scaled so that the first one keeps its original value. The new betas $\tilde\beta_t$ come from ratios of neighbouring new alpha_bars, so the last new beta equals 1.

Return a dict with exactly the keys "betas", "alpha_bars" and "snr", where each value is a float64 NumPy array of shape (T,) holding $\tilde\beta_t$ , $\tilde{\bar\alpha}_t$ and $\mathrm{snr}_t$ .

Examples

Input: betas = [0.1,0.2,0.3,0.4]

Output: {"betas":[0.1,0.439235,0.712845,1.0],"alpha_bars":[0.9,0.504688,0.144924,0.0],"snr":[9.0,1.018931,0.169486,0.0]}

Explanation: The square roots of the alpha_bars are shifted so the last one becomes 0 and stretched so the first one stays at sqrt(0.9). Squaring gives new alpha_bars ending at 0, so the last beta becomes 1 and the last SNR becomes 0.

Input: betas = [0.5,0.5]

Output: {"betas":[0.5,1.0],"alpha_bars":[0.5,0.0],"snr":[1.0,0.0]}

Input: betas = [0.3,0.6,0.99]

Output: {"betas":[0.3,0.630773,1.0],"alpha_bars":[0.7,0.258459,0.0],"snr":[2.333333,0.348543,0.0]}

Hint 1

np.sqrt(np.cumprod(1.0 - betas)) gives the original signal factors a_t.

Hint 2

a[0] and a[-1] are the two ends that the shift and scale remap, the first onto itself and the last onto 0.

Hint 3

np.concatenate(([1.0], arr[:-1])) builds the previous value for every timestep, with 1 in front of timestep 0.

Requirements

Use NumPy array operations.
Use 0-based timesteps, and treat the cumulative value before timestep 0 as 1.
Compute snr from the new alpha_bars at every timestep, including the last one, where it is 0.
Leave the input array betas unchanged.

Constraints

betas is a 1-D float64 NumPy array of shape (T,) with T between 2 and 1000.
Every beta satisfies 0 < beta < 1, so the alpha_bars are strictly decreasing and positive.
betas may be a non-contiguous view of a larger array.

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