Problems
Loading...
Step 1 of 5
Forward Diffusion
Medium

Implement the closed-form forward diffusion step used by a Denoising Diffusion Probabilistic Model (DDPM). The cumulative signal retention through timestep t is

αˉt=∏s=1t(1−βs)\bar{\alpha}_t = \prod_{s=1}^{t}(1-\beta_s)

The noisy sample at timestep t is

xt=αˉt x0+1−αˉt ϵx_t = \sqrt{\bar{\alpha}_t}\,x_0 + \sqrt{1-\bar{\alpha}_t}\,\epsilon

Here, x_0 is the clean sample, beta_s is the noise variance at step s, t is a one-based timestep, and epsilon is the supplied Gaussian-noise sample. Use NumPy float64 calculations. Return the complete alpha-bar schedule as a list of floats from get_alpha_bar, and return x_t as a nested list from forward_diffusion.

Loading visualization...

Examples

Input: x_0=[[1,2],[3,4]], t=1, betas=[0.0001,0.005075,0.01005,0.015025,0.02], epsilon=[[0.5,-0.3],[0.1,-0.8]]

Output: [[1.0049,1.9969],[3.0008,3.9918]]

Explanation: At t=1, the first cumulative alpha value scales the clean sample while the supplied epsilon contributes a small amount of noise.

Input: x_0=[[1,2],[3,4]], t=3, betas=[0.0001,0.005075,0.01005,0.015025,0.02], epsilon=[[-0.2,0.7],[0.4,-0.5]]

Output: [[0.9677,2.071],[3.0264,3.908]]

Input: betas=[0.0001,0.005075,0.01005,0.015025,0.02]

Output: [0.9999,0.994826,0.984828,0.97003,0.95063]

Hint 1

Use np.cumprod on one minus the float64 beta array.

Hint 2

Select alpha_bar[t - 1] because t is one-based.

Hint 3

Convert x_0 and epsilon with np.asarray before combining them.

Requirements

  • Compute the cumulative product of one minus each beta value.
  • Use the supplied epsilon when computing x_t; do not sample new noise.
  • Treat t as one-based when selecting the cumulative alpha value.
  • Round alpha-bar values to six decimals and x_t values to four decimals.
  • Return Python lists whose shapes match the corresponding schedules or samples.

Constraints

  • betas is a nonempty list of float values strictly between 0 and 1.
  • t is an integer from 1 through the length of betas.
  • x_0 and epsilon are equally shaped, nonempty nested lists of real numbers.
  • Numerical calculations use NumPy float64.
Prev1 / 5Next