Forward Diffusion

The problem, from first principles

Adding noise one step at a time would make training painfully slow, since producing a noisy image for step 500 would need 500 sequential operations. But Gaussian noise has a convenient property: the sum of Gaussians is Gaussian. Composing all the steps gives a closed form that jumps from the clean image x0 straight to step t: x_t = sqrt(alpha_bar_t) * x0 + sqrt(1 - alpha_bar_t) * eps with eps ~ N(0, I). This is a weighted blend of signal and noise whose weights satisfy a^2 + b^2 = 1, so if x0 has unit variance then so does x_t. The signal-to-noise ratio alpha_bar / (1 - alpha_bar) summarizes how corrupted the image is at each step.

From theory to code

Implement q_sample and snr.

Constraints

• q_sample(x0, t, alpha_bars, noise) returns sqrt(alpha_bars[t]) * x0 + sqrt(1 - alpha_bars[t]) * noise. t is a 0-based integer index into alpha_bars; x0 and noise have the same shape.
• snr(alpha_bars) returns alpha_bars / (1 - alpha_bars) as an array (assume alpha_bars < 1).
• Do not modify the inputs.

Hints