The score function

The gradient of log density, and why anyone would learn it

For a density $p(x)$, the score is the vector field $\nabla_x \log p(x)$. It points in the direction in which the log density increases fastest, so following it climbs toward the modes. For a single Gaussian $\mathcal{N}(\mu, \sigma^2)$ there is nothing to compute — the score is $-(x-\mu)/\sigma^2$, a field of arrows all pointing straight at the mean with magnitude growing linearly with distance.

The reason score gets its own name is that it is often easier to estimate than the density itself. The density needs a normalising constant, which is an integral over everything and is generally intractable; the score does not, because the gradient of $\log Z$ is zero. That single observation is the foundation of score-based generative modelling: fit the gradient, never compute the constant, and never evaluate $p$ directly.

The difficulty is that in regions where the data are sparse, the score is badly estimated and the direction it gives is unreliable. The fix is to add noise: a noisy density has a smoother, better-behaved score, and if you anneal the noise down from large to small, the collection of scores at every level lets you walk from pure noise to data. That is exactly what the forward process was doing, and it is exactly what the network in part 3 was learning.

Drag the dot and watch it fall into a mode

If the score points uphill in probability, a point that repeatedly steps along it should find the modes. That is the idea of Langevin dynamics: take a gradient step, then add a little noise, and repeat. Written down, with a step size $\delta$,

$$ x_{k+1} = x_k + \delta\,\nabla_x\log p(x_k) + \sqrt{2\delta}\,z_k, \qquad z_k\sim\mathcal{N}(0,I) $$

The deterministic part climbs; the stochastic part stops the point from getting stuck at the top of a mode and lets it explore the whole distribution. The same $\sqrt{2\delta}$ balance that appears here reappears in the SDEs below, and it is not a liberty: with that coefficient, and only that coefficient, the stationary distribution of the walk is exactly $p$.

The demo draws the target density as blue shading and the score as a grid of arrows, coloured from grey (small) to blue (large). The pink dot is a starting point you can drag anywhere; the pink curve is 220 Langevin steps from that position. Drag it into a valley between two modes and watch it get pulled to one side by the field before the noise jitters it along.

One identity ties this part to the last three

Here is the bridge. The forward process gives a noisy density $p_t$ at every level, and for the Gaussian perturbation $q(x_t\mid x_0) = \mathcal{N}(\sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I)$ the score of the noisy density has a closed relation to the noise that was added:

$$ \nabla_{x_t}\log p_t(x_t) = -\,\frac{\mathbb{E}[\varepsilon \mid x_t]}{\sqrt{1-\bar\alpha_t}} = -\,\frac{\varepsilon_\theta(x_t,t)}{\sqrt{1-\bar\alpha_t}} $$

So a network trained to predict noise is a network trained to predict the score, up to one scalar that depends only on $t$. Nothing in part 3 needs to change; it was doing score matching the whole time under a different name. This equivalence is the SMLD-to-DDPM bridge: the older score-matching line of work learned $\nabla\log p$ with denoising autoencoders at a ladder of noise levels, and the diffusion line learned $\varepsilon$ at a ladder of timesteps, and the two are the same object rescaled.

The small plot below makes the rescaled factor visible. The curve is $-1/\sqrt{1-\bar\alpha_t}$, the multiplier that converts a unit noise prediction into a score, plotted against $\bar\alpha_t$. At high signal the factor is near one, so the two are almost the same number; near the end of the schedule it diverges, and that divergence is exactly the reason a sampler trained on noise predictions struggles to use them as scores at very low signal-to-noise ratio.

This identity also explains the sampler from the previous part. DDIM's deterministic update is a discretised ordinary differential equation whose drift is built from this score; DDPM's stochastic update adds the Langevin-style noise that keeps the walk at the right temperature. The two samplers are two ways of integrating the same field.

Two continuous-time limits of the same forward process

Take the number of forward steps to infinity and the discrete chain becomes a stochastic differential equation. Two parameterisations dominate the literature, and they differ in what they keep fixed as time runs.

• Variance-exploding (VE): the drift is zero and only noise is added, $dx = g(t)\,dw$ with $g(t) = \sqrt{2\sigma(t)\sigma'(t)}$. The marginal variance grows without bound, so the process ends as pure noise of enormous amplitude — the choice of the SMLD line.
• Variance-preserving (VP): the drift is $-\tfrac12\beta(t)x$ and the diffusion is $\sqrt{\beta(t)}$, so signal is removed exactly as fast as noise is added and the marginal variance stays at one. This is the continuous limit of DDPM's forward chain.

Both define a valid bridge from data to noise; both admit a reverse-time SDE driven by the score; and a sampler can be built from either. The demo plots the two quantities side by side, normalised so their shapes are comparable: either the marginal standard deviation as a function of time, or the diffusion coefficient $g(t)$ itself. Switch modes and move the slider to see how the choice of maximum noise scale reshapes the VE curve.

A third parameterisation, the variance-preserving with a cosine schedule, is what most image models actually use; and flow matching, which the later parts of this volume cover, replaces the SDE entirely with a deterministic interpolation between noise and data. All of them are members of one family, and this part is the reason they can be compared at all.

One field, many samplers

This is the conceptual end of the first arc of the guide. The remaining parts take the same objects into practice: conditioning the score on a prompt, compressing the space so sampling is affordable, replacing the U-Net with a transformer, and extending the whole apparatus to video and audio. Every one of them assumes the picture on this page — a vector field you learn, and a walk you take through it.

Further reading

The score-based line is older than diffusion and repays reading on its own terms. These references cover the field, the sampler, and the unification with diffusion, in that order.

If you read one, read the SDE paper: it is the paper in which all of the pieces in this volume finally became one piece.

• Yang Song and Stefano Ermon, "Generative Modeling by Estimating Gradients of the Data Distribution", 2019 — score matching with denoising and annealed Langevin sampling (SMLD).
• Yang Song, Jascha Sohl-Dickstein and coauthors, "Score-Based Generative Modeling through Stochastic Differential Equations", 2020 — the VE/VP SDEs and the equivalence to DDPM.
• Pascal Vincent, "Extracting and Composing Robust Features with Denoising Autoencoders", 2008 — the result that denoising learns the score.
• Aapo Hyvärinen, "Estimation of Non-Normalized Statistical Models by Score Matching", 2005 — the original score-matching objective.
• Gareth Roberts and Richard Tweedie, "Exponential Convergence of Langevin Distributions", 1996 — why the $\sqrt{2\delta}$ coefficient is the right one.

Cheat sheet