The forward diffusion process adds scheduled Gaussian noise over T steps until the signal becomes isotropic noise

Given a clean data point x_0, the forward process defines a sequence x_1, x_2, …, x_T where each x_t is x_{t-1} scaled by sqrt(1-beta_t) and mixed with Gaussian noise of variance beta_t. The beta_t schedule is a set of T small positive constants (the noise/variance schedule) that increase from near-0 to near-1. As T grows large and beta_t stays small, x_T approaches an isotropic Gaussian regardless of x_0 — the sample gradually loses its distinguishable features. Usually beta_1 < beta_2 < … < beta_T, since a larger update step is affordable once the sample is already noisy. The forward process has no learnable parameters; it is fully specified by the schedule and is only used to manufacture noisy training targets, never run at inference.

Examples

With T=1000 and a linear schedule (beta from 1e-4 to 0.02), the image is essentially pure noise well before t reaches T. The forward chain is the fixed corruption recipe the reverse network learns to undo.

Assessment

What is the role of the variance schedule beta_t in the forward process? Explain why increasing beta_t over the course of the chain is reasonable.