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A diffusion model learns to generate data by first learning to undo a deliberate corruption. During training, real examples are gradually buried in noise, and a neural network learns how to move a noisy input back toward the structure of the original data. To generate something new, the model starts from pure noise and applies that learned correction many times until a sample with the character of the training data appears.
The idea works because the network does not need to memorize one exact path from noise back to a specific image. It needs to estimate a direction, called a score, that tells a noisy input which way leads toward more probable data at that noise level. This article explains that idea in terms of probability distributions, contrasts the discrete DDPM formulation with the continuous-time score-SDE framework, and introduces DDIM as a faster sampling method. The benchmark figures quoted here come from 2020 papers and describe those papers’ experiments, not today’s leading systems.
Two directions, one training signal
Diffusion modeling has two linked halves. The forward half is fixed: it takes a data point and corrupts it step by step, usually by adding Gaussian noise according to a schedule. The reverse half is learned: it takes a noisy point and produces a slightly cleaner one. Generation runs the learned half, beginning from the simple noise distribution used at the end of the forward process.
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In the continuous-time formulation by Yang Song and coauthors (arXiv, 2020), the forward process is a stochastic differential equation (SDE) that does not depend on the data and has no trainable parameters. Its only job is to specify how the distribution of data is progressively smoothed into a tractable noise distribution. The learning problem sits entirely in the reverse direction.
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Why reversing noise is possible at all
Noise erases the details of any single image, so it is tempting to think the process cannot be reversed. The resolution is that the question changes. The model does not need to recover which clean image produced a given noisy input. At every noise level, the corrupted data still follows a well-defined probability distribution, and that distribution has a gradient.
That gradient is the score: the gradient of the log density of the data distribution after corruption to time t, written ∇x log pt(x). It points toward regions where noisy inputs are more likely at that noise level. If a network can estimate this score well for every noise level, it can move a noisy point toward realistic data.
This is also why “reverse” does not mean subtracting the exact noise that was added during corruption. A single noisy input is compatible with many different clean images, so the model learns an average direction that reflects the training distribution as a whole. Generation is a sequence of approximate, learned corrections, not an exact undoing of a known random draw.
DDPM: corruption as a Markov chain
Jonathan Ho, Ajay Jain, and Pieter Abbeel’s 2020 paper, “Denoising Diffusion Probabilistic Models,” presents the discrete version. The forward process is a Markov chain of T steps, each adding a small amount of Gaussian noise under a prescribed variance schedule. The reverse process is a chain of learned Gaussian transitions, and the network parameterizes them. The DDPM paper used T = 1,000 steps in its reported experiments.
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Training does not require running the forward chain one step at a time. Because the forward noise accumulates in a known way, a noisy version at any chosen step can be produced directly from the clean example. The paper’s simplified objective then trains a network to predict the noise that was added. The authors state that their objective is connected to denoising score matching, which is why this approach and the score-based view describe closely related mechanisms.
Training one example
- Draw a clean training example x0 from the dataset.
- Choose a random time step t from the schedule.
- Sample Gaussian noise ε and form the noisy input xt directly from x0, using the closed-form expression for the forward chain at step t.
- Update the network so that, given xt and t, it predicts ε. Repeat over the dataset.
Generating one sample
- Draw xT from the standard Gaussian prior.
- For t from T down to 1, use the network’s noise estimate to compute the mean of the learned reverse transition, then sample xt-1 from it. Fresh noise is added at each step except the last.
- Return x0 as the sample.
Generation therefore costs one network evaluation per step. With T = 1,000, a single sample requires 1,000 sequential passes through the network, which is the main practical cost that later sampling methods target.
Score-based SDEs: the continuous-time picture
Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole’s 2020 paper “Score-Based Generative Modeling through Stochastic Differential Equations” replaces the fixed sequence of noise levels with a continuum. Time runs over an interval, and a forward SDE describes how the data distribution spreads into noise. The authors summarize the asymmetry in one line: “Creating noise from data is easy; creating data from noise is generative modeling.”
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The forward SDE has a drift term and a diffusion term that are chosen in advance and do not depend on the data. Different choices recover different discrete schemes, which is part of the framework’s appeal: the design space is explicit rather than hidden in one particular chain.
The reverse-time SDE
Classical results on time reversal of SDEs say that the reverse process has the same marginal distributions as the forward process when run backward in time. Its drift adds a term involving the time-dependent score ∇x log pt(x). Since that score is unknown, a neural network is trained to estimate it at each time. With that estimate in place, a numerical SDE solver can generate samples from noise.
Predictor-corrector sampling
The paper also describes predictor-corrector samplers. A predictor takes a numerical step of the reverse-time SDE, and a corrector then applies a few Langevin dynamics steps at the current noise level to nudge the sample toward the correct distribution at that time. The two parts can be tuned separately, which gives practitioners control over the balance between speed and accuracy.
The probability-flow ODE
The same framework derives a probability-flow ordinary differential equation (ODE). It has the same marginal distributions as the reverse SDE but contains no injected randomness. Starting from the same noise, it therefore produces a deterministic mapping from noise to data. Because the mapping is deterministic, the same starting noise always yields the same sample, and the ODE also gives a route to computing exact likelihoods, which the paper uses in its experiments.
How DDPM and score-SDE relate
DDPM and score-based SDE models are not rival explanations of unrelated mechanisms. Song et al. state that the DDPM approach and score matching with Langevin dynamics can both be viewed as discretizations of different SDE choices. The discrete chain is one numerical view of a continuous process, and the score-based formulation is the continuous process itself.
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For most readers, the practical takeaway is this: the discrete Markov chain and the continuous SDE describe the same family of ideas at different levels of abstraction. The discrete view is easier to implement and reason about step by step. The continuous view makes it clearer which parts of the design are choices (the SDE, the solver, the schedule) and which are learned (the score or the noise estimate).
DDIM: same training, a different sampling path
The bottleneck of DDPM is sampling, not training. Jiaming Song, Chenlin Meng, and Stefano Ermon’s 2020 paper “Denoising Diffusion Implicit Models” starts from a sentence that describes the problem directly: DDPMs “require simulating a Markov chain for many steps to produce a sample.”
DDIM keeps DDPM’s training procedure and objective, but defines a family of non-Markovian forward processes with the same marginal distributions at each step. Different members of this family lead to different reverse samplers, and some of them allow the reverse process to skip steps rather than traversing every intermediate noise level. One setting of the family produces a deterministic sampler, so the same starting noise yields the same sample, similar in spirit to the probability-flow ODE.
The DDIM authors report generation that is 10× to 50× faster in wall-clock time than DDPM in their experiments. That figure depends on their datasets, architectures, step counts, and sampling settings, and the paper describes a trade-off: fewer steps reduce computation, and sample quality changes with the number of steps. Treat the range as a result from that paper’s setup, not a guarantee that any given model will speed up by the same factor.
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Comparing the sampling options
| Approach | Time representation | What the network learns | Sampling path | Main trade-off reported or implied |
|---|---|---|---|---|
| DDPM ancestral sampling | Discrete Markov steps (T = 1,000 in the paper) | Reverse transition mean, typically via noise prediction | Stochastic, one step at a time | Many sequential network evaluations per sample |
| Reverse-time SDE with predictor-corrector | Continuous time, solved numerically | Time-dependent score estimate | Stochastic; predictor steps plus Langevin corrector steps | Solver and corrector settings trade speed against accuracy; the paper reports results under its own configurations |
| Probability-flow ODE | Continuous time, solved numerically | Time-dependent score estimate | Deterministic: same noise gives same sample | Deterministic mapping and exact likelihood computation; sample quality depends on solver settings (not stated as a universal ranking) |
| DDIM sampling | Discrete steps, which can be skipped | Same trained model as DDPM | Non-Markovian family; can be deterministic or stochastic | Fewer steps and lower compute; 10× to 50× wall-clock speedup in the DDIM paper’s experiments, with quality depending on step count |
No single row is a universal winner. The source papers demonstrate particular trade-offs under their own experimental settings, and the choice of schedule, solver, and step count can change the outcome for a given model.
Reading the 2020 benchmark numbers
The papers report quantitative results that are useful for understanding what these methods could achieve in 2020, but they should not be read as current rankings. Each figure belongs to a specific dataset, model, and evaluation setup.
- DDPM, unconditional CIFAR-10 (Ho, Jain, and Abbeel, 2020): Inception score 9.46 and FID 3.17, as stated in the paper’s abstract.
- DDPM, 256×256 LSUN: the authors report sample quality similar to ProgressiveGAN, a 2017-era GAN, in their comparison. This is the authors’ own comparison, reported at that resolution and dataset.
- Score-based SDE, CIFAR-10 (Song et al., 2020): Inception score 9.89, FID 2.20, and likelihood 2.99 bits/dim under the paper’s described experiments.
- DDIM speedup (Song, Meng, and Ermon, 2020): 10× to 50× faster wall-clock sampling compared with DDPM in the paper’s experiments.
Inception score and FID both depend on a pretrained image classifier and on the sample set used to compute them, so they are sensitive to evaluation details. Comparing numbers across papers with different preprocessing, sample counts, or architectures is not a like-for-like comparison.
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What these papers do and do not establish
The three papers set out the conceptual foundations: a prescribed corruption process, a learned reverse process driven by a score or noise estimate, a discrete and a continuous view of the same family of methods, and a faster sampler that reuses the same training. They do not describe the latest implementations, the best currently available samplers, or the text-to-image systems that followed. Conditional tasks such as inpainting and colorization are demonstrated in the score-SDE paper, but how conditioning is implemented depends on the method, and the papers do not establish a single standard approach.
Quick Recap
Further reading
- Ho, Jain, and Abbeel, “Denoising Diffusion Probabilistic Models,” NeurIPS 2020 proceedings (abstract page): https://papers.neurips.cc/paper/2020/hash/4c5bcfec8584af0d967f1ab10179ca4b-Abstract.html
- Song et al., “Score-Based Generative Modeling through Stochastic Differential Equations,” arXiv:2011.13456 (2020): https://arxiv.org/abs/2011.13456
- Song, Meng, and Ermon, “Denoising Diffusion Implicit Models,” arXiv:2010.02502 (2020): https://arxiv.org/abs/2010.02502
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