Diffusion Model

A diffusion model learns to reverse a gradual noising process: it is trained to denoise corrupted data and generates samples by iteratively denoising from noise.

A common forward noising form is:

$$x_t=\sqrt{{\bar{\alpha }}_t}{x}_0+\sqrt{1-{\bar{\alpha }}_t}ϵ,ϵ\sim \mathcal{N}(0,I)$$

The model is often trained to predict $ϵ$, $x_0$, or a related velocity target.

Forward Process

In discrete-time DDPM notation:

$$q(x_t\mid x_{t-1})=\mathcal{N}\left(\sqrt{{\alpha }_t}{x}_{t-1},(1-{\alpha }_t)I\right)$$

and the closed-form marginal is:

$$q(x_t\mid x_0)=\mathcal{N}\left(\sqrt{{\bar{\alpha }}_t}{x}_0,(1-{\bar{\alpha }}_t)I\right)$$

where:

$${\bar{\alpha }}_t=\prod _{s=1}^t{\alpha }_s$$

This marginal is what allows training with randomly sampled timesteps instead of simulating every forward step.

A common noise-prediction objective is:

$${\mathcal{L}}_{ϵ}={\mathbb{E}}_{t,x_0,ϵ}\left[{\Vert ϵ-{ϵ}_{\theta }(x_t,t)\Vert }_2^2\right]$$

The reverse process samples from noise by repeatedly applying a learned denoising transition:

$$p_{\theta }(x_{t-1}\mid x_t)=\mathcal{N}\left({\mu }_{\theta }(x_t,t),{\Sigma }_{\theta }(x_t,t)\right)$$

Alternative parameterizations predict $x_0$ or velocity $v$:

$$v={\alpha }_tϵ-{\sigma }_t{x}_0$$

The parameterization affects stability, guidance behavior, and solver design.

Parameterization Choices

Target	Model Predicts	Useful When	Caveat
noise $ϵ$	${ϵ}_{\theta }(x_t,t,c)$	classic DDPM-style training	sample quality depends on noise schedule and solver
clean data $x_0$	${\hat{x}}_{0,\theta }(x_t,t,c)$	interpretable reconstruction target	can be unstable at high noise
score	${\nabla }_{x_t}\log p_t(x_t)$	score-based SDE/ODE view	scale depends on noise convention
velocity $v$	$v_{\theta }(x_t,t,c)$	diffusion/flow bridge and stable interpolation	must define ${\alpha }_t,{\sigma }_t$ convention

For the common parameterization

$$x_t={\alpha }_t{x}_0+{\sigma }_tϵ,ϵ\sim \mathcal{N}(0,I),$$

a velocity target is often written:

$$v_t={\alpha }_tϵ-{\sigma }_t{x}_0.$$

Always record which target a paper uses before comparing losses or samplers.

The score associated with the same Gaussian convention is:

$$s_{\theta }(x_t,t)\approx -\frac{{ϵ}_{\theta }(x_t,t)}{{\sigma }_t}$$

with ${\sigma }_t^2=1-{\bar{\alpha }}_t$ in the simple variance-preserving notation. This connects diffusion noise prediction to score-based models.

Conditioning

Conditional diffusion adds context $c$:

$${ϵ}_{\theta }(x_t,t,c)$$

Guidance changes the effective score or denoising direction, which can improve fidelity but reduce diversity.

Classifier-free guidance often combines conditional and unconditional predictions:

$${ϵ}_{\mathrm{guided}}={ϵ}_{\theta }(x_t,t,\varnothing )+w\left[{ϵ}_{\theta }(x_t,t,c)-{ϵ}_{\theta }(x_t,t,\varnothing )\right]$$

where $w$ is the guidance scale. Larger $w$ can improve condition satisfaction while reducing diversity or validity.

Sampling Budget

Diffusion quality is tied to solver and step count:

$$x_T\sim \mathcal{N}(0,I),x_{t-1}={\mathrm{Step}}_{\theta }(x_t,t,c;\eta )$$

where $\eta$ represents sampler settings such as stochasticity, guidance scale, schedule, and step size. A benchmark should report these settings because a model can look better simply by spending more sampling steps or stronger guidance.

Sampler Families

Sampler	Shape	Claim Affected
ancestral DDPM	stochastic reverse chain	diversity and baseline comparability
DDIM-style	deterministic or partially stochastic trajectory	fewer steps, different diversity
SDE sampler	stochastic continuous-time reverse process	score quality and noise schedule
probability-flow ODE	deterministic score-derived ODE	likelihood route and solver budget
distilled/consistency sampler	one or few steps	speed at possible quality loss

NFE, wall time, memory, and invalid-sample denominator should be reported with sample quality.

Evaluation Boundary

Claim	Required Evidence
better likelihood or denoising	NLL/ELBO or denoising loss under matched schedule
better sample quality	task-specific sample metrics and human/domain checks
better conditional control	condition satisfaction measured separately from diversity
faster generation	quality at matched NFE, wall time, memory, and hardware
better molecular/protein samples	validity, novelty, diversity, constraints, and downstream utility

Why It Matters

• Strong sample quality and stable training across images, molecules, and proteins.
• Flexible conditioning enables guided and inverse-problem generation.
• Iterative sampling is the main cost, motivating faster samplers.

Failure Modes

• Too few sampling steps can produce invalid or low-detail samples.
• Strong guidance can collapse diversity or overfit the condition.
• The noise schedule may be poorly matched to data scale or geometry.
• Evaluation can confuse validity, novelty, diversity, and downstream utility.

Checks

• Does the noise schedule fit the data scale?
• Is conditioning/guidance trading diversity for fidelity?
• How many sampling steps are needed for acceptable quality?
• Are sampling steps and guidance scales fixed across comparisons?
• Is the prediction target $ϵ$, $x_0$, score, or velocity clearly stated?
• Are NFE, solver, stochasticity, and guidance scale matched across baselines?
• Are validity, diversity, novelty, and task utility reported separately?
• Is the noising convention variance-preserving, variance-exploding, or another schedule?
• Are filtered or repaired samples counted in the denominator?

Related

• Score-based model
• Score matching
• Probability flow ODE
• Sampling
• Guidance
• Flow matching
• Consistency model
• Conditional generation
• Generation evaluation