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The forward process starts with a clean data sample x_0 and repeatedly adds Gaussian noise.
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).
The forward process starts with a clean data sample x_0 and repeatedly adds Gaussian noise.
The noise schedule beta_t determines the amount of noise introduced at each timestep.
The cumulative product alpha_bar_t controls the remaining signal and allows direct sampling of x_t from x_0.
After sufficiently many steps, the data becomes approximately indistinguishable from standard Gaussian noise.
The forward process is fixed and known; the learned model is trained to reverse it during generation.
A fixed sequence of stochastic transformations that gradually adds noise to data until it approaches a Gaussian distribution.
The sequence of values beta_t that specifies how much noise is added at each diffusion timestep.
The variance coefficient controlling the amount of fresh Gaussian noise added at timestep t.
The signal-retention coefficient defined as alpha_t = 1 - beta_t.
The cumulative product of the alpha values through timestep t, representing the total signal retained from the original data.
A reformulation that expresses a noisy sample directly as a scaled clean sample plus scaled standard Gaussian noise.
Random noise sampled from a normal distribution, typically N(0, I), used to progressively corrupt the data.
“Can you explain what "The forward process starts with a clean data sample x_0 and repeatedly adds Gaussian noise." means in simple terms?”