These are my notes on diffusion models. I loosely mirror the canonical organization of this reference (arXiv:2510.21890). This post introduces discrete-time Markov chains in DDPMs as a complement to the encoders in (H)VAEs, whose ELBO decompositions are tractable under some forward posterior. Score-matching (and contrastive divergence) avoids the partition constant in EBMs but requires computing the Jacobian gradient, which is especially intensive in high-dimensional spaces. DSM avoids this and provides the motivation behind the perturbation kernel, NCSN, and the Bayes-optimal quantities (noise, target, score) via Tweedie’s. Continuous-time formulations then configure the closed-form Gaussians for VP, VE, and sub-VP SDEs for more stable generation. We provide the motivation behind the Fokker-Planck and continuity PDEs under Anderson’s time-reversal, and why quadratic variation induces second-order terms via Itô’s lemma, which standard calculus is not equipped to consider.

Latent-variable models

VAE ELBO

Let \(x\) be an observation and \(z\) a latent variable with a joint model \(p_\phi(x, z)\) and an approximate posterior \(q_\theta(z \mid x)\). The identity that everything rests on is a single application of Bayes’ rule inside an expectation:

Because the KL term is nonnegative, \(\mathcal{L}_{\mathrm{ELBO}}(x) \le \log p(x)\) — hence evidence lower bound. Splitting the joint the other way, using \(p_\phi(x,z) = p_\phi(x \mid z)\,p(z)\), gives the form we actually optimize:

Taking the data expectation,

From a hierarchical VAE to DDPM

A diffusion model is a hierarchical VAE with a fixed Gaussian forward chain \(x_0 \to x_1 \to \cdots \to x_L\) and no learned encoder. Write the variational bound on the negative log-likelihood, factor the reverse model as \(p_\phi(x_0, x_{1:L}) = p_{\mathrm{prior}}(x_L)\prod_{i=1}^{L} p_\phi(x_{i-1}\mid x_i)\), and expand the forward posterior with Bayes’ rule conditioned on \(x_0\). The full chain of steps:

The telescoping in the fourth line uses \(q(x_i \mid x_{i-1}) = q(x_i \mid x_{i-1}, x_0) = \dfrac{q(x_{i-1}\mid x_i, x_0)\,q(x_i \mid x_0)}{q(x_{i-1}\mid x_0)}\); the crossed-out term is the self-consistent \(q(x_0 \mid x_1, x_0)\), which is a point mass and contributes nothing. The intermediate \(q(x_i\mid x_0)/q(x_{i-1}\mid x_0)\) factors telescope across the sum, leaving only the boundary term \(q(x_L\mid x_0)\) (folded into \(\mathcal{L}_{\mathrm{prior}}\)) and the per-step posteriors.

The three pieces have clean interpretations: \(\mathcal{L}_{\mathrm{prior}}\) asks that the fully-noised data match the prior (no learnable parameters — it is essentially zero for a well-chosen schedule); \(\mathcal{L}_{\mathrm{recon}}\) is the final denoising step back to data; and \(\mathcal{L}_{\mathrm{diff}}\) is a sum of denoising matching terms, one per timestep, each asking the learned reverse kernel \(p_\phi(x_{i-1}\mid x_i)\) to match the true forward posterior \(q(x_{i-1}\mid x_i, x_0)\).

The Gaussian transition kernels

For DDPM the forward chain is Gaussian with a variance schedule \(\{\beta_i\}\); write \(\alpha_i = 1 - \beta_i\) and \(\bar\alpha_i = \prod_{j=1}^{i}\alpha_j\). The single-step forward kernel, the closed-form marginal (from composing Gaussians), and the \(x_0\)-conditioned reverse posterior (by conjugacy / completing the square) are

with posterior mean and variance

Every KL in \(\mathcal{L}_{\mathrm{diff}}\) is therefore between two Gaussians and has a closed form, which is what makes the objective trainable.

Score-based Modeling

Motivation behind the score

An energy-based model writes \(p_\phi(x) = \frac{e^{-E_\phi(x)}}{Z_\phi}\) with partition function \(Z_\phi = \int e^{-E_\phi(x)}\, dx\), which is intractable in high dimensions. The score sidesteps \(Z_\phi\) entirely since it’s constant in \(x\).

Score matching

Instead of a likelihood, fit the score \(s(x) = \nabla_x \log p_{\mathrm{data}}(x)\) directly with a model \(s_\phi\). The explicit objective is the Fisher divergence

This is not directly usable because the target \(\nabla_x \log p_{\mathrm{data}}\) is unknown. Expand the square into three terms:

The last term is irrelevant to optimization. We handle the cross-term by applying IbP:

The boundary term vanishes because \(p_{\mathrm{data}}(x)\to 0\) at infinity faster than \(s_\phi\) grows. In the weak formulation of a PDE, the derivative moves off the unknown density and onto the model, which we can differentiate. Our expanded objective becomes

The two terms have geometric interpretations: the magnitude term pushes the score toward zero in high-density regions, and the trace term contracts those stationary points inward for the induced vector field.

Denoising score matching and Tweedie’s formula

The trace term is expensive, and the clean score is ill-defined when data lies on a low-dimensional manifold. Both problems dissolve if we add noise. Let

so the conditional corruption density is \(p_\sigma(\tilde{x}\mid x) = \mathcal{N}(\tilde{x};\, \alpha x,\, \sigma^2 I)\) with closed-form conditional score

The object we actually want is the score of the noisy marginal \(p_\sigma(\tilde{x}) = \int p_\sigma(\tilde{x}\mid x)\, p_{\mathrm{data}}(x)\, dx\). Differentiating under the integral,

So the marginal score is the Rao–Blackwellized posterior expectation of the conditional score. Substituting the Gaussian conditional score and rearranging yields Tweedie’s formula:

Because a conditional expectation is the minimizer of the squared error, the optimal network trained on the conditional target equals the marginal score:

This is the denoising-score-matching (DSM) equivalence: for any fixed noise level, the tractable conditional objective and the intractable marginal objective share a minimizer.

One object, three parameterizations

Tweedie’s formula shows the score predictor, the noise predictor, and the \(x_0\)-predictor (denoiser) are algebraically interchangeable. Writing the DDPM optimum for noise prediction \(\epsilon^\star\) against the score optimum \(s^\star\),

Continuous time formulation

From discrete updates to SDEs

Both the variance-exploding (NCSN) and variance-preserving (DDPM) chains are Euler–Maruyama discretizations of an SDE. Take a general update

The conditional mean and covariance to first order in \(\Delta t\) are \(\mathbb{E}[\Delta x \mid x_t] = a\,\Delta t\) and \(\operatorname{Cov}(\Delta x \mid x_t) = bb^\top \Delta t\), which identify the limiting SDE

where \(dw_t \sim \mathcal{N}(0, dt \; I)\).

For DDPM, expand the drift coefficient with the scalar Taylor series \(\sqrt{1 - z} = 1 - \tfrac{z}{2} + O(z^2)\):

so the drift and diffusion are \(f(x,t) = -\tfrac12\beta(t)x\) and \(g(t) = \sqrt{\beta(t)}\); typically, the drift is non-positive in $x_t$ to corrupt the signal to the zero-centered mean. This is a (time-inhomogeneous) Ornstein–Uhlenbeck (OU) process. NCSN is non-affine in the drift, and only the variance tends to 1, \(\sqrt{\sigma^2_{t+\Delta t} - \sigma^2_t} \approx \sqrt{\tfrac{d\sigma_t^2}{dt}}\sqrt{\Delta t}\), giving zero drift.

Ordinary calculus is not enough, because Brownian increments scale as \(\Delta w = O(\sqrt{\Delta t})\), so the \((\Delta w)^2 = O(\Delta t)\) term remains at the same order as the drift. Concretely, in the second-order Taylor expansion of \(h(x_t, t)\) the products of differentials obey the Itô multiplication rules

which hold in mean-square: \(dt\cdot dt\) and \(dt\cdot dw_t\) are \(O(\Delta t^{3/2})\) or smaller and vanish, while \(\mathbb{E}[(dw_t)^2] = dt\) with vanishing variance survives. Keeping only the surviving second-order term gives the Itô correction:

The two canonical forward SDEs:

The VP process has closed-form marginals \(x_t \mid x_0 \sim \mathcal{N}(\alpha(t)x_0,\,(1-\alpha(t)^2)I)\) with \(\alpha(t) = \exp\left(-\tfrac12\int_0^t \beta(s)\,ds\right)\). This mean factor comes from the integrating factor of the (deterministic part of the) linear SDE: for \(\tfrac{d}{dt}\mathbb{E}[x_t] = -\tfrac12\beta(t)\,\mathbb{E}[x_t]\), multiplying by \(\exp\!\left(\tfrac12\int_0^t\beta\right)\) makes the left side an exact derivative, giving \(\mathbb{E}[x_t] = \alpha(t)x_0\). The variance then follows from the Itô isometry, and the exponentially-decaying mean with variance approaching \(I\) is preserves the variance, i.e. \(\alpha(t)^2 + (1-\alpha(t)^2) = 1\) at all times when \(x_0\) is standardized.

A third common choice interpolates between the two. The sub-VP SDE shares the VP drift but scales the diffusion so the marginal variance stays below the VP variance throughout:

Its variance is bounded by the VP variance at every \(t\), which empirically yields smoother likelihoods and lower sampling error while retaining the same limiting prior as VP.

Fokker–Planck and the probability current

The marginal density \(p_t\) of the forward SDE \(dX_t = f\,dt + G\,dW_t\) (with diffusion matrix \(a = GG^\top\)) evolves by the Fokker–Planck equation, which we write as a continuity equation:

with the probability current

Using the product rule, \(\dfrac{1}{p_t}\nabla\cdot(a p_t) = \nabla\cdot a + a\,\nabla\log p_t\), so

Anderson’s reverse-time SDE

Run the clock backward with \(s = T - t\), \(Y_s := X_{T-s}\), so \(Y_s\) has density \(q_s(x) = p_{T-s}(x)\). Since \(\partial_s q_s = \nabla\cdot J_{T-s}\) while the reverse process must satisfy its own continuity equation \(\partial_s q_s = -\nabla\cdot J_s^{\mathrm{R}}\), reversing time reverses the current: \(J_s^{\mathrm{R}} = -J_{T-s}\). Positing a reverse SDE with drift \(b^{\mathrm{R}}\) and the same local diffusion \(a\), matching currents gives

In the diffusion-model case \(G(x,t) = g(t)I\), we have \(a = g^2 I\) and \(\nabla\cdot a = 0\), so — written in the original time label integrated backward from \(T\) to \(0\) — this is the familiar reverse-time SDE:

For DDPM (\(f = -\tfrac12\beta x\), \(g^2 = \beta\)), the reverse dynamics are

The only unknown is the time-dependent score \(\nabla\log p_t\) — which is exactly what DSM trains the network to approximate.

A note on time labels. Writing the coefficients as \(f(\cdot, T-s)\) under a forward-running reverse clock \(s\) is not treating \(T-s\) as increasing time; it is the forward-time label of the slice currently being reversed. At \(s=0\) the reverse process starts from the noisy terminal \(p_T\), so it must use the schedule value \(\beta(T)\), i.e. \(\beta_{\mathrm{rev}}(s) = \beta(T-s)\). Using \(\beta(s)\) would incorrectly start from the data-end schedule value.

The probability-flow ODE and marginal consistency

The current can also be reproduced by a deterministic flow. Since \(-\tfrac12 g^2 \nabla p_t = -\tfrac12 g^2 p_t \nabla\log p_t\), the Fokker–Planck equation is equivalently a continuity equation for the velocity field \(v(x,t) = f(x,t) - \tfrac12 g^2(t)\nabla\log p_t(x)\):

This probability-flow ODE has the same marginals \(p_t\) as the SDE. It is the basis for deterministic samplers and exact likelihood computation.

Conceptual Summary

A diffusion model is a HVAE whose encoder is a fixed Gaussian noising process. Its ELBO decomposes into per-step denoising-matching KLs; each is minimized by learning the score of the noisy marginal, which DSM makes tractable via Tweedie’s formula. In continuous time the forward chain is an SDE whose marginals are governed by Fokker–Planck (the \(\mathcal{L}^2\) adjoint of the infintesimal generator); Anderson’s theorem reverses it using the score, and the PF-ODE reproduces the same marginals and admits time-reversable integration. Score matching, DDPMs, discrete diffusion, and SDEs are all views of the same underlying object: conditioning on initial states and corrupting the data yields tractable objectives. Studying the underlying density evolution allows us to formalize the theory without singularities; e.g., computing deterministic sampling trajectories.

These notes are my own derivations, written for my reference and anyone learning the material; corrections are welcome.