17.9  Further Reading

The chapter taught one idea on problems small enough to check: a network that predicts the clean point from a noisy one at every noise level, trained by regression and turned into a sampler. Each entry states what a later work changes; nothing in the book depends on it. Distillation by distribution matching, adversarial post-training and the pitfalls of FID are on Section 16.8; the design space in continuous time, Schrödinger bridges, optimal couplings and sliced score matching on Section 43.7.

17.9.1 Energy-Based Models and Score Matching

Training energy-based models. Song and Kingma (2021) survey the ways to fit a density known up to its normalizer. The likelihood gradient needs samples from the model, and the methods differ in where their chains start: contrastive divergence restarts them at the data (Hinton 2002) and so cannot correct mode weights, persistent chains keep their states (Tieleman 2008), and Du and Mordatch (2019) scaled Langevin chains with a replay buffer to images. Short chains restarted from noise give a good sampler but a poor density (Nijkamp et al. 2019). Grathwohl et al. (2020) read a classifier’s logits as the energy of inputs and labels, so one network classifies and generates.

Noise-contrastive estimation. Gutmann and Hyvärinen (2012) fit an unnormalized model by logistic regression of data against samples from a known noise distribution, the classifier of Section 16.1 with fixed noise in place of a generator.

What score matching cannot see. Wenliang and Kanagawa (2020) show that the score is blind to the weights of well-separated components, as Section 17.1 measures, and Koehler et al. (2022) tie the efficiency of score matching to isoperimetric constants, which are poor when modes are far apart.

Correcting and reusing Langevin steps. An accept-reject test removes the step-size bias of the Langevin update (Roberts and Tweedie 1996) but needs density ratios, which a score network does not give. Stochastic-gradient Langevin dynamics runs the update on the posterior of a network’s weights with minibatch gradients (Welling and Teh 2011).

Empirical Bayes. Tweedie’s formula comes from empirical Bayes, where a prior is estimated from the noisy observations it is applied to (Robbins 1956); Efron (2011) showed how the formula corrects selection bias, as in the largest of many noisy effect estimates.

17.9.2 Schedules, Weighting and Networks

Cosine schedule and learned variances. Nichol and Dhariwal (2021) destroy information more evenly across steps with a cosine schedule and learn each reverse step’s variance from the bound, which improves likelihoods at fewer steps.

Choosing the training levels. Esser et al. (2024) find that a logit-normal distribution of training times, concentrated at intermediate noise, works best for rectified flow at every model size; later text-to-image models adopt it.

Weighting as multi-task learning. Hang et al. (2023) clamp the weight of each noise level at the minimum of its signal-to-noise ratio and a constant, one point in the weighting space of Section 17.5, reported to converge several times faster.

Zero terminal SNR. Against the leftover signal of Section 17.5, Lin et al. (2024) rescale the schedule to end at zero signal, which needs a target other than the noise.

Resolution and averaging. At higher resolution the same noise level destroys less of an image, so Hoogeboom et al. (2023) shift the schedule toward more noise. Karras et al. (2024) keep the magnitudes of activations and weights constant and store snapshots of two moving averages, from which an average of any half-life, fixed before training in Section 17.3, is reconstructed afterwards.

The U-Net at ImageNet scale. Dhariwal and Nichol (2021) ablated the U-Net of Ho et al. (2020) and kept multi-resolution attention, residual resampling and adaptive group normalization, in which the time and class embedding set a scale as well as a shift. The result, ADM, is the usual baseline.

17.9.3 Samplers

Exponential integrators. Dividing by \(\alpha\) removes the part of the probability-flow ODE that is linear in \(\mathbf{x}\). DPM-Solver (Lu, Zhou, et al. 2022) approximates the remaining integral of the predicted noise by a polynomial in \(\log \bar\sigma\); its first-order version is DDIM (exercise 6). DPM-Solver++ (Lu et al. 2025) applies the idea to the clean-image prediction and stays stable at large guidance scales, and UniPC (Zhao et al. 2023) adds a corrector that gains an order of accuracy without an extra network call.

Noise inside a deterministic solver. Karras et al. (2022) raise the noise level slightly before each Heun step, adding a stochastic sampler’s correction to a second-order method; it helps on some datasets and hurts on others. Restart sampling (Xu et al. 2023) alternates deterministic steps with one large jump back up the interval by forward noise, which contracts accumulated error.

Parallel sampling. Shih et al. (2023) refine all steps of a guessed trajectory at once by Picard iteration, trading compute for latency.

17.9.4 Guidance, Editing and Inverse Problems

Better guidance. Against the saturated colors of large guidance scales, dynamic thresholding (Saharia et al. 2022) clips the predicted clean image, and Sadat et al. (2025) down-weight the part of the update parallel to the conditional prediction.

Inversion and editing. The round trip of ddim_invert is inexact, more so under guidance. EDICT (Wallace et al. 2023) keeps two coupled copies of the state that invert each other exactly, and null-text inversion (Mokady et al. 2023) optimizes the unconditional text embedding so that the guided sampler retraces the inversion, which allows text-driven editing of real photographs.

Restoration without retraining. RePaint (Lugmayr et al. 2022) inpaints by re-imposing the noised known pixels at every step; DDRM (Kawar et al. 2022) solves linear inverse problems through the SVD of the measurement operator; DPS (Chung et al. 2023) follows the gradient of the measurement likelihood at the predicted clean image. ControlNet (Zhang et al. 2023) instead trains a copy of the encoder on edges, depth or pose.

17.9.5 Latent Spaces and Backbones

Latent systems. SDXL (Podell et al. 2024) scales the latent U-Net of Rombach et al. (2022) with two text encoders; SD3 (Esser et al. 2024) and FLUX.1 (Black Forest Labs 2024) are joint-attention transformers trained with rectified flow.

Better latents. REPA (Yu et al. 2025) aligns the transformer’s hidden states with a pretrained image encoder and trains many times faster. DC-AE (Chen et al. 2025) compresses far more than eightfold with sharp reconstructions, and representation autoencoders (Zheng et al. 2026) use a frozen pretrained encoder such as DINO.

Transformers on pixels. Two lines of work drop the autoencoder that limits the latent models of Section 17.6. Hourglass diffusion transformers (Crowson et al. 2024) give the transformer the U-Net’s multi-resolution shape, so its cost grows linearly with the pixels. JiT (Li and He 2026) keeps pixel patches of \(16 \times 16\) or more and predicts the clean image, while the loss stays on the velocity.

17.9.6 Few-Step Generation

Distillation and consistency. Progressive distillation (Salimans and Ho 2022) repeatedly halves a teacher’s steps; it introduced \(\mathbf{v}\)-prediction because a few-step student spends most of its evaluations at low signal-to-noise ratio, where a noise predictor’s \(\hat{\mathbf{x}}_0\) divides by a small \(\alpha\). Improved consistency training (Song and Dhariwal 2024) needs no teacher. sCM (Lu and Song 2025) trains consistency models in continuous time at ImageNet scale, which needs more than the identity of Section 17.7: a path that rotates between data and noise, a normalized tangent term and a warm-up.

General flow maps. Consistency trajectory models (Kim et al. 2024) jump between any two times; shortcut models (Frans et al. 2025) require one step of twice the size to equal two steps; improved MeanFlow (Geng et al. 2026) regresses on the instantaneous velocity and takes the guidance scale as an input. Boffi et al. (2025) turn any distillation scheme into training from scratch.

17.9.7 Likelihood

Bounds and weightings. Variational diffusion models (Kingma et al. 2021) learn the schedule to reduce the variance of the bound. Song et al. (2021) find the weighting under which score matching bounds the likelihood, and Lu, Zheng, et al. (2022) and Zheng et al. (2023) train and evaluate the likelihood of the probability-flow ODE.

17.9.8 Evaluation, Memorization and Generalization

Text–image agreement. CLIPScore (Hessel et al. 2021) compares image and caption in a joint embedding; GenEval (Ghosh et al. 2023) checks counts, colors and positions with an object detector.

Memorization. Carlini et al. (2023) extract near-copies of training images from text-to-image models, and Somepalli et al. (2023) find replicated content by retrieval. Biroli et al. (2024) find two times in the reverse process: speciation, when a sample commits to a class, and collapse, when it commits to a training point.

Why networks generalize. The exact optimum of the regression on a finite set reproduces training points, and Bertrand et al. (2025) show that the noisy target is not what prevents this. Denoisers trained on disjoint halves of a large set learn nearly the same score (Kadkhodaie et al. 2024), and locality and equivariance alone predict their outputs (Kamb and Ganguli 2025). Massias and Bertrand (2026) survey the question.

17.9.9 Video

Video diffusion. Ho et al. (2022) factorize attention over space and time; Stable Video Diffusion (Blattmann et al. 2023) shows that curating video data matters as much as the architecture; Movie Gen (Polyak et al. 2024) trains one transformer on joint image and video latents.

17.9.10 Discrete Diffusion and Diffusion Language Models

Continuous time and ratios. In continuous time the masked chain of Section 17.8 becomes a continuous-time Markov chain (Campbell et al. 2022), and the score becomes ratios \(p_t(\mathbf{y})/p_t(\mathbf{x})\) between sequences that differ in one position. Concrete score matching (Meng et al. 2022) and SEDD (Lou et al. 2024) learn these ratios; for the absorbing kernel a ratio is a time factor times the clean conditional, what the masked denoiser learns. Discrete flow matching (Gat et al. 2024; Campbell et al. 2024) builds the same models from probability paths, including joint discrete and continuous data such as protein sequences and structures.

The time input. Ou et al. (2025) show that the continuous-time bound equals the loss of an order-agnostic autoregressive model. Zheng et al. (2025) show that sampling categories in low precision lowers the effective temperature and flatters reported generative perplexity.

Uniform noise and revision. Under a uniform kernel every token can change at every step: Schiff et al. (2025) add guidance to such models, and Sahoo et al. (2025) relate them to a Gaussian process and distill them to few steps. ReMDM (Wang et al. 2025) adds remasking to the masked sampler so that early mistakes can be revised. On puzzles such as Sudoku, a reveal order chosen by the model helps substantially (Kim et al. 2025), the other side of the mode-seeking measured in Section 17.8.

Scale. Masked diffusion follows autoregressive scaling laws with a constant-factor compute gap (Nie, Zhu, Du, et al. 2025) and overtakes autoregression under repeated data (Prabhudesai et al. 2025). LLaDA (Nie, Zhu, You, et al. 2025) trains an 8-billion-parameter model from scratch and decodes in blocks; Dream (Ye et al. 2025) and DiffuLLaMA (Gong et al. 2025) adapt autoregressive models instead. Block diffusion (Arriola et al. 2025) makes the blocks part of the model, interpolating between a GPT and a masked model, and Fast-dLLM (Wu et al. 2026) adds caching and parallel decoding without retraining.

Images and the debate. MaskGIT (Chang et al. 2022) unmasks the codes of an image tokenizer in eight to twelve confidence-ordered steps, and Muse (Chang et al. 2023) conditions it on text. Dieleman (2026) traces why masked diffusion displaced continuous diffusion for language and why continuous models are returning.

17.9.11 Post-Training by Reward and Preference

Policy gradients. After pretraining, a reward can replace the regression target, with the reverse steps as actions. DDPO (Black et al. 2024) applies the clipped policy-gradient update of Section 15.8 to the Gaussian reverse steps, and Flow-GRPO (Liu et al. 2025) brings the group-relative update to flow models through a stochastic sampler with the same marginals.

Preferences. Diffusion-DPO (Wallace et al. 2024) adapts direct preference optimization, built on the Bradley–Terry model of Section 15.3, by replacing the intractable log-likelihood with the variational bound of Section 17.5.

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