A single math framework predicts the best way to generate images and text

One lattice to rule them all

Diffusion models and autoregressive (AR) models have long been treated as separate families of generative AI. Diffusion models, which gradually strip noise from an entire image or sequence at once, became the standard for continuous data like pictures and videos. Autoregressive models, which generate one token at a time conditioned on what came before, dominated discrete domains like language ¹. Recent hybrid approaches have tried to blend the two, but each one locks in its own decoding schedule by design.

A new paper argues that this division is unnecessary. The researchers, T. Y. Tsui, Jiatao Gu, and Lingjie Liu from the University of Pennsylvania, present a single theoretical framework called a corruption lattice that places diffusion, AR, and every intermediate schedule on the same mathematical structure. Every possible decoder becomes a path through this lattice ¹.

Cost is discarded dependence

The key insight is how they define a schedule's quality. They measure the cost of any decoding schedule as the total correlation — the dependence — that its parallel steps throw away. This cost equals the schedule's divergence from exact sampling, meaning it quantifies exactly how far a given approach strays from perfect generation ¹.

From this definition emerges a clean prediction: the minimum number of steps a zero-cost schedule needs is determined entirely by the geometry of the data itself. For data that follows a Markov structure on a graph, that minimum equals the graph's treedepth — a well-known measure from graph theory. Treedepth grows logarithmically with sequence length and linearly with the side length of a grid ¹. This is the same rule for both discrete tokens and continuous fields.

Predicting before decoding

When a schedule uses fewer steps than the treedepth, it must pay a positive cost — some dependence is always lost. The paper shows that you can still rank those suboptimal schedules before any decoding begins, using a pairwise-dependence kernel estimated directly from pretrained model weights ¹. This means you do not need to run the full generation to know which schedule will perform best at a fixed step count.

The authors test these predictions across text generation, image generation, and video generation, verifying most of the ranking predictions under different metrics and benchmarks ¹. The work gives model designers a concrete principle for choosing and constructing decoding schedules, whether they are building future AR models, diffusion models, or hybrids somewhere in between.

Code for the project has been released publicly.

Why it matters

For years, picking a decoding schedule has been largely an engineering guess — tune it empirically for each model family. This framework turns that process into something predictable. If you know the dependence structure of your data and have pretrained weights, you can rank schedules ahead of time rather than running dozens of trial generations ¹. That shift from trial-and-error to theory-driven design could save significant compute across the next generation of generative models.

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Key facts

• Diffusion, AR, and hybrid models can all be represented as paths on a single corruption lattice
• The minimum zero-cost steps for Markov data equal the graph's treedepth — logarithmic in sequence length
• Schedule rankings can be predicted before decoding using a pairwise-dependence kernel from pretrained weights
• Predictions were verified across text, image, and video generation benchmarks
• Code is available at https://github.com/TSUITUENYUE/The-Lattice-of-Transition-Laws

Sources

1. Paper page - The Lattice of Transition Laws
2. 2610.11216 The Lattice of Transition Laws - arXiv.org
3. The Lattice of Transition Laws - arXiv.org
4. The Lattice of Transition Laws | Arxiv - DeepPaper — arxiv.deeppaper.ai
5. GitHub - TSUITUENYUE/The-Lattice-of-Transition-Laws