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Diffusion models can be viewed as latent variable generators: they introduce a sequence of hidden variables (noisy versions of the data) and learn a probabilistic process that maps between the data distribution and a simple prior. The forward diffusion process gradually corrupts a sample x0 by adding Gaussian noise over T steps, producing xt that eventually approaches a standard Gaussian N(0, I). The reverse (generative) process then uses a learned denoising model to iteratively remove noise, starting from a latent variable xT sampled from N(0, I), until it reconstructs a sample from the target data distribution q(x0).
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The forward diffusion process gradually transforms data sampled from the target distribution into nearly pure Gaussian noise. Starting with x_0, noise is added over T steps according to x_t = sqrt(1 - beta_t) x_{t-1} + sqrt(beta_t) z_t, where z_t is independently sampled from a standard normal distribution and beta_t controls the noise added at step t. The related quantity alpha_t = 1 - beta_t and its cumulative product alpha_bar_t = alpha_1 ... alpha_t determine how much of the original data remains. Because the process is Gaussian, the noisy sample at any time t can be generated directly without simulating every previous step: x_t | x_0 follows a normal distribution with mean sqrt(alpha_bar_t) x_0 and variance (1 - alpha_bar_t)I. Equivalently, x_t = sqrt(alpha_bar_t) x_0 + sqrt(1 - alpha_bar_t) z. As t increases, the signal contribution decreases and the noise contribution increases, causing x_t to converge toward the standard normal distribution N(0, I).
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