NCSN

NCSN — Noise-Conditional Score Network — is the score-based counterpart of DDPM. Rather than predicting noise inside a discrete Markov chain, it directly learns the (Stein) score function $s_\theta(x, \sigma) \approx \nabla_x \log p_\sigma(x)$, where $p_\sigma$ is the data distribution convolved with Gaussian noise of standard deviation $\sigma$.

Training uses denoising score matching (Vincent 2011): for a noised sample $\tilde{x} = x + \sigma \epsilon,\; \epsilon \sim \mathcal{N}(0, \mathbf{I})$, the conditional score is available in closed form, $\nabla_{\tilde{x}} \log p_\sigma(\tilde{x} \mid x) = -(\tilde{x} - x)/\sigma^2$, giving the loss

with the standard weighting $\lambda(\sigma) = \sigma^2$ so every noise level contributes comparable magnitude. The architecture reuses the same U-Net as DDPM but conditions on $\sigma$ (continuous) rather than a discrete timestep $t$.

The key innovation is noise annealing: a geometric schedule $\sigma_1 > \sigma_2 > \cdots > \sigma_L$ (defaults $\sigma_1 = 50$, $\sigma_L = 0.01$, $L = 10$) where the largest $\sigma_1$ smooths the data distribution enough that low-density regions become traversable, then sampling refines toward $\sigma_L \approx 0$ where the score approximates that of clean data. Sampling uses annealed Langevin dynamics — for each level $i$ in order, run $T_i$ steps of

with the per-level step size $\alpha_i = \epsilon \cdot \sigma_i^2 / \sigma_L^2$ (Song 2019, §4.3) — large early on, decaying as noise shrinks. The score-based view unifies with diffusion through the continuous SDE framework (Song et al., 2021), where DDPM and NCSN become different discretisations of the same reverse-time process.