DDPM keeps the reverse-process variance fixed and only learns the mean, simplifying the training target

The reverse Gaussian p_theta(x_{t-1}|x_t) has two parameters: mean and variance. The original DDPM paper sets the variance to a fixed schedule (either beta_t or the posterior variance beta_tilde_t) and trains the network only to learn the mean. Experimentally, both choices of fixed variance gave similar results in the original paper. A later paper (Improved DDPM, Nichol et al., 2021) showed that also learning the variance improves performance. The simplification to fixed variance greatly reduces the training complexity and was a key design decision in the original DDPM.

Examples

Code: Sigma_theta(x_t, t) = sigma_t^2 * I is not a learnable parameter — it is precomputed from the beta schedule.

Assessment

Explain why fixing the variance is a simplification and what the later ‘Improved DDPM’ paper changed. What two outcomes did the original paper find for the two fixed-variance choices?