Sampling

Sampling is the procedure that turns a learned generative model into concrete outputs.

For an unconditional model:

$$x\sim p_{\theta }(x)$$

For a conditional model:

$$x\sim p_{\theta }(x\mid c)$$

The sampling algorithm determines quality, diversity, latency, and controllability.

Sampling Patterns

Pattern	Base Randomness	Main Control	Common Cost
autoregressive sampling	token-level categorical draws	temperature, top-k, top-p, beam, constraints	one model call per step or block
diffusion sampling	Gaussian noise and denoising path	noise schedule, sampler, guidance, NFE	many denoising evaluations
flow sampling	base noise and ODE/invertible transform	solver, step count, path, guidance	ODE or inverse transform cost
latent sampling	latent $z\sim p(z)$	prior, decoder, latent conditioning	decoder plus latent search/filtering
rejection or filtering	candidate pool	verifier, score, validity rule, threshold	many candidates per kept sample

Here NFE means number of function evaluations. It is often the right unit for diffusion, score, and flow samplers:

$${\mathrm{cost}}_{\mathrm{sample}}\approx \mathrm{NFE}\times \mathrm{cost}(f_{\theta })+\mathrm{cost}(\mathrm{filter})$$

Diversity and Quality

Sampling often trades diversity for fidelity. If the sampler collapses to high-probability modes, outputs may look good but cover only a small part of the data distribution.

Effective Sample Distribution

Filtering, reranking, guidance, and beam search change the distribution being evaluated:

$$x\sim p_{\theta ,\mathrm{sampler}}(x\mid c)\ne p_{\theta }(x\mid c)$$

For post-filtering with acceptance rule $A(x)=1$:

$$p_{\mathrm{kept}}(x\mid c)\propto p_{\theta ,\mathrm{sampler}}(x\mid c)1[A(x)=1]$$

Reported validity, novelty, and utility should state whether they are computed before filtering, after filtering, or per kept sample.

Reporting Fields

Field	Required Detail
sampler	decoding rule, ODE/SDE solver, denoising schedule, or latent prior
stochasticity	seed policy, temperature, noise scale, top-k/top-p, restart count
guidance	guidance type, strength, schedule, and guide source
filtering	validity checks, score thresholds, rejection rate, reranking rule
budget	candidates generated, candidates kept, NFE, wall-clock, hardware, memory
distribution	whether metrics are pre-filter, post-filter, best-of-n, or oracle-selected

Checks

• What distribution is sampled first: token, noise, latent, graph, or coordinates?
• How many sampling steps or decoding calls are required?
• Are temperature, guidance strength, beam size, or rejection filters documented?
• Are invalid samples counted or silently removed?
• Are diversity, novelty, and task utility evaluated separately?
• Are best-of-n and reranking reported as part of the method rather than hidden evaluation?
• Is the comparison matched by quality at equal NFE/candidate budget, or only by headline metric?
• Does the sampler preserve constraints such as graph validity, chirality, coordinate frame, or sequence length?

Related

• Autoregressive model
• Diffusion model
• Score-based model
• Flow matching
• Normalizing flow
• Latent variable model
• Guidance
• Generation evaluation
• Decoding