Diffusion forward process

Background

Diffusion models generate data by learning to reverse a gradual noising process. The forward process is fixed: it slowly corrupts a clean sample $x_0$ into pure Gaussian noise over $T$ steps, following a variance schedule $\beta_1,\dots,\beta_T$. A key property is the closed form — you can jump directly to any timestep $t$ without iterating, which is what makes training efficient (you sample a random $t$ each step).

Problem statement

Implement forward_diffusion(x0, t, betas, noise) using the closed-form marginal:

$$\alpha_s = 1-\beta_s, \quad \bar\alpha_t = \prod_{s=0}^{t}\alpha_s, \quad x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon$$

where $\epsilon$ is the provided noise.

Input

• x0 — clean sample, np.ndarray.
• t — int, target timestep (0-indexed).
• betas — np.ndarray (T,), the variance schedule.
• noise — np.ndarray, Gaussian noise $\epsilon$ with the same shape as x0.

Output

An np.ndarray (same shape as x0): the noised sample $x_t$.

Examples

Example 1

Input:  x0 = [1, 1], betas = [0.1, 0.2], t = 1, noise = [0, 0]
Output: [0.8485, 0.8485]

Explanation: $\alpha=[0.9,0.8]$, so $\bar\alpha_1 = 0.9\cdot0.8 = 0.72$. With zero noise, $x_1 = \sqrt{0.72}\cdot[1,1] = [0.8485, 0.8485]$.

Constraints

• $\bar\alpha_t$ is the cumulative product of $(1-\beta)$ up to and including index t.
• Scale the signal by $\sqrt{\bar\alpha_t}$ and the noise by $\sqrt{1-\bar\alpha_t}$.
• Do not iterate step by step — use the closed-form marginal.

Notes

• As $t$ grows, $\bar\alpha_t \to 0$, so the signal fades and $x_t$ approaches pure noise — exactly the endpoint the reverse model learns to undo.
• The two coefficients satisfy $\bar\alpha_t + (1-\bar\alpha_t) = 1$, preserving unit variance when $x_0$ and $\epsilon$ are unit-variance.