What is the mathematical representation of the forward diffusion process in diffusion models as described in the chapter?

The forward diffusion process in diffusion models is mathematically represented by adding Gaussian noise with variance βt to an image xt-1 to generate a new image xt. The update process is xt = √(1-βt)xt-1 + √βtε, where ε is a standard Gaussian noise. This ensures that the variance of xt remains constant over time, approximating a standard Gaussian distribution for large T.

In the forward diffusion process, an image x0 is gradually corrupted by adding Gaussian noise over a series of steps until it becomes indistinguishable from standard Gaussian noise. The process is defined by a function q that adds Gaussian noise with variance βt to an image xt-1 to produce a new image xt. Mathematically, this is expressed as xt = √(1-βt)xt-1 + √βtε, where ε is a standard Gaussian noise with zero mean and unit variance. This formulation ensures that if the original image x0 is normalized to have zero mean and unit variance, then xt will also have these properties for all t, including the final image xT. This allows for easy sampling of xT and facilitates the reverse diffusion process through the trained neural network model.