# Path-Dependent Denoising: A Non-Conservative Field Perspective on Order Collapse in Diffusion Language Models

arXiv: https://arxiv.org/abs/2605.09303 · HTML: https://mlsanity.com/paper/2605.09303/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 12 of its 19 references that are in the map.
- First posted on arXiv 10 May 2026.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)

## Builds on (most cited first, 12 of 12)

- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Train for the Worst, Plan for the Best: Understanding Token Ordering in Masked Diffusions](https://mlsanity.com/paper/2502.06768/) (2025)
- [Likelihood-Based Diffusion Language Models](https://mlsanity.com/paper/2305.18619/) (2023)
- [A Deep and Tractable Density Estimator](https://mlsanity.com/paper/1310.1757/) (2013)
- [KLASS: KL-Guided Fast Inference in Masked Diffusion Models](https://mlsanity.com/paper/2511.05664/) (2025)
- [Why Diffusion Language Models Struggle with Truly Parallel (Non-Autoregressive) Decoding?](https://mlsanity.com/paper/2602.23225/) (2026)
- [Masks Can Be Distracting: On Context Comprehension in Diffusion Language Models](https://mlsanity.com/paper/2511.21338/) (2025)
- [CONTESTS: a Framework for Consistency Testing of Span Probabilities in Language Models](https://mlsanity.com/paper/2409.19984/) (2024)

## Abstract (arXiv)

Diffusion language models (DLMs) offer a structural alternative to autoregressive generation: denoising can update tokens in arbitrary orders or in parallel rather than along a fixed left-to-right chain. In practice, fast DLM decoding remains strongly order-sensitive and often drifts toward autoregressive-like trajectories. We trace this tension to compatibility. At each reverse-time step, a DLM provides local denoising conditionals over the unresolved tokens. Arbitrary-order denoising becomes well defined when these local conditionals compose into order-invariant pseudo-joints. We formalize this view by defining order-induced pseudo-joints and a local denoising circulation: the log-ratio between the two pseudo-joints obtained by swapping a pair of unresolved positions. This circulation is zero under compatible conditionals, and global order gaps decompose into sums of local circulations along adjacent swaps. We further separate incompatibility-driven path dependence from conditional-dependence error in parallel updates and from order-specific estimation error. The resulting framework provides inference-only diagnostics for testing when DLM decoding is genuinely order-free.