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A diffusion model is an image generator that learns by destroying pictures with noise and then learning to undo the damage. During training, a clean image is mixed with Gaussian noise (random values drawn from a bell curve) according to a fixed schedule, and a neural network is taught to predict the noise that was added at each level. Generation runs the process the other way: start from pure noise and remove a little at a time. A sampler is the recipe for that backward walk. DDPM and DDIM are the two foundational samplers, and both work with the same trained network.

DDPM stands for Denoising Diffusion Probabilistic Models, the paper by Jonathan Ho, Ajay Jain and Pieter Abbeel posted to arXiv on 19 June 2020 and published at NeurIPS 2020. It made a 2015 idea from Jascha Sohl-Dickstein and colleagues practical. The paper used T = 1,000 noise levels with a noise variance, called beta, rising linearly from 0.0001 to 0.02. Its network was a U-Net, an encoder-decoder convolutional network, with the timestep fed in as a sinusoidal embedding. The key simplification was to train the network to predict the added noise with a plain mean-squared-error loss, which the authors showed is closely related to denoising score matching. On CIFAR-10 the paper reported an Inception score of 9.46 and an FID of 3.17, where FID (Frechet Inception Distance) compares generated and real image statistics and lower is better.

DDPM sampling works like this. Draw x_T, an image-shaped array of pure noise. At each step t, the network predicts the noise in x_t. The sampler uses that prediction to compute the mean of a slightly cleaner image, then adds a fresh dose of random noise scaled to that step, giving x_(t-1). Repeat until t reaches zero. Because every step injects new randomness, the process is a Markov chain: each state depends only on the one before it. That randomness is mathematically required for the derivation, and it is also why DDPM needs nearly all 1,000 steps. The cost is real: the DDIM paper reports that sampling 50,000 32 by 32 images from a DDPM took around 20 hours on an Nvidia 2080 Ti, against under a minute for a GAN, and nearly 1,000 hours for 256 by 256 images.

DDIM, Denoising Diffusion Implicit Models, by Jiaming Song, Chenlin Meng and Stefano Ermon, was posted on 6 October 2020 and published at ICLR 2021. Its insight is that DDPM's training objective does not pin down a single backward process. A whole family of non-Markovian processes, where each step may depend on the starting point x_0 and not only the previous state, shares the same training loss. So a network trained as a DDPM can be sampled with any member of that family without retraining. DDIM picks members that allow skipping: instead of visiting all 1,000 levels, it can jump along a subsequence of, say, 50 levels.

A DDIM step has three parts. First, use the predicted noise to estimate the clean image x_0. Second, point back toward the current noise direction. Third, combine the two at the noise level of the next chosen timestep, optionally adding fresh randomness. A parameter called eta controls that randomness: eta = 1 recovers DDPM-like stochastic sampling, and eta = 0 gives fully deterministic DDIM, where the same starting noise always yields the same image. In the paper's Table 1, deterministic DDIM on CIFAR-10 reached an FID of 13.36 with 10 steps, 6.84 with 20, 4.67 with 50 and 4.16 with 100, while DDPM-style sampling (eta = 1) scored 41.07 at 10 steps, and the larger-variance variant Ho and colleagues used scored 367.43, essentially unusable. The authors report that DDIM matches 1,000-step quality within 20 to 100 steps, a 10 to 50 times wall-clock speedup, and that on CelebA a 100-step DDPM scores about the same as a 20-step DDIM.

Determinism has side benefits. Because noise maps to image along a fixed path, the path can be run in reverse to encode a real image into its starting noise, called DDIM inversion. The DDIM paper reports CIFAR-10 reconstruction error falling from 0.014 at 10 steps to 0.0001 at 1,000 steps. Interpolating between two starting noises gives smooth, meaningful blends of the two outputs, which DDPM cannot do because the same starting noise leads to different images. Null-text Inversion, posted in November 2022, builds real-photo editing on top of DDIM inversion, and notes that plain inversion breaks down when classifier-free guidance amplifies its small per-step errors.

Neither sampler is the end of the story. DDIM is a first-order method, meaning each step uses only the current slope. Later solvers treat sampling as solving an ordinary differential equation with higher-order steps: DPM-Solver, published at NeurIPS 2022, reports a CIFAR-10 FID of 4.70 in 10 steps and 2.87 in 20. Libraries such as diffusers now offer many schedulers, and DDIM remains a reference baseline. Known failure modes include blurry or washed-out images at very low step counts, accumulated error in inversion under strong guidance, and a training-inference mismatch: the diffusers documentation points to a 2023 paper showing that common schedules never reach pure noise, which limits very bright or dark outputs unless the betas are rescaled and timesteps are spaced to start from the last one.

In practice, both samplers cost one network evaluation per step, so total compute scales linearly with step count, as the DDIM paper's timing plots confirm. The DDPM paper's CIFAR-10 model trained for 800,000 steps in 10.6 hours on a TPU v3-8, roughly equal to eight V100 GPUs, and sampled 256 images in 17 seconds. Switching samplers needs no retraining, so choosing between them is purely an inference-time decision.