Forward Diffusion Process in DDPM

The forward process (also called the diffusion process) systematically adds Gaussian noise to clean data until it eventually becomes nearly pure random noise.

Core Idea:
Start with a clean image (or data point), add a tiny amount of noise repeatedly over thousands of steps, and eventually obtain pure Gaussian noise.

Equation 1: Step-by-Step Noise Addition

q(x_t|x_t-1) = N(x_t; √(1-β_t)x_t-1, β_tI)

This equation describes how the noisy sample at timestep t is generated from the sample at timestep t−1.

Term	Meaning
q(xt|xt-1)	Probability of transitioning from step t−1 to step t
N(·)	Gaussian (Normal) distribution
xt	New noisy sample generated at timestep t
√(1−βt)xt−1	Mean of the Gaussian distribution
βtI	Variance of the Gaussian distribution

Why scale the previous image?
Without the factor √(1−β_t), variance would continuously grow and eventually explode. Scaling keeps the process mathematically stable.

Equation 2: Full Diffusion Trajectory

q(x_1:T|x₀) = ∏_t=1^T q(x_t|x_t−1)

This equation represents the probability of the entire diffusion trajectory from the original clean sample x₀ to the final noisy sample x_T.

Term	Meaning
q(x1:T|x0)	Joint probability of the complete noisy trajectory
∏	Product operator multiplying probabilities of every step
Markov Property	Each state depends only on its immediate predecessor

Important Observation:
The diffusion process forms a Markov Chain. The current state remembers only the previous state and ignores everything earlier.

Deriving the Closed-Form Sampling Formula

Instead of repeatedly executing thousands of diffusion steps, DDPM derives a direct mathematical shortcut that allows sampling x_t directly from x₀.

Step 1: Define New Variables

α_t = 1 − β_t

ᾱ_t = ∏_i=1^t α_i

Here:

• α_t = amount of original signal retained during one step
• ᾱ_t = cumulative signal retained after many diffusion steps

Reparameterization Form

x_t = √α_tx_t−1 + √(1−α_t)ε_t−1

where ε_t−1 ~ N(0,I)

This formulation explicitly separates:

• The preserved signal component
• The newly injected Gaussian noise

Interpretation:
Every diffusion step keeps part of the original image while injecting a small amount of fresh random noise.