Reverse Process Parameterization

When working with diffusion models, you need to generate realistic data by reversing the gradual noise corruption applied in the forward process. While the forward process is straightforward—adding small amounts of noise at each step—the reverse process is not directly accessible. This is because the true reverse transitions, denoted as $p(x_{t-1} | x_t)$, are not analytically tractable for complex data distributions. Therefore, you must model this reverse process with a parameterized distribution, often written as $pθ(x_{t-1} | x_t)$, where $θ$ represents the learnable parameters of a neural network or similar function approximator.

The reverse process in diffusion models is typically defined as a Markov chain that gradually removes noise from a sample. Its mathematical form is:

Here, $p(x_T)$ is usually a simple prior, such as a standard Gaussian, and each reverse transition $pθ(x_{t-1} | x_t)$ is parameterized, commonly as a Gaussian with mean and variance predicted by a neural network. The parameterization choice for $pθ(x_{t-1} | x_t)$ can vary:

• Predict the mean and variance directly;
• Predict only the mean and use a fixed variance schedule;
• Predict a noise component, from which the mean is computed.

These choices affect both the model's flexibility and the complexity of training.

The conceptual sampling procedure for the reverse process in a diffusion model can be described as follows:

Given: a final noise sample $xₜ ∼ 𝒩(0, I)$ for $t = T$

Repeat for $t = T, T−1, …, 1$:

• Sample $x_{t-1} \sim p_θ(x_{t-1} | x_t)$ (the learned reverse diffusion distribution).

Return: $x₀$, which is the generated data sample.

This pseudocode highlights the iterative nature of the reverse process, where at each step, you use the parameterized distribution to move from a noisier to a less noisy sample, ultimately producing a realistic data point.