# Positional Diffusion: Ordering Unordered Sets with Diffusion Probabilistic Models

arXiv: https://arxiv.org/abs/2303.11120 · HTML: https://mlsanity.com/paper/2303.11120/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 27 of its 49 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Placed in

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

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension](https://mlsanity.com/paper/1910.13461/) (2019)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [MaskGAN: Towards Diverse and Interactive Facial Image Manipulation](https://mlsanity.com/paper/1907.11922/) (2019)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Skip-Thought Vectors](https://mlsanity.com/paper/1506.06726/) (2015)
- [MERLOT: Multimodal Neural Script Knowledge Models](https://mlsanity.com/paper/2106.02636/) (2021)
- [Visual Storytelling](https://mlsanity.com/paper/1604.03968/) (2016)
- [Deep Attentive Ranking Networks for Learning to Order Sentences](https://mlsanity.com/paper/2001.00056/) (2019)
- [Topological Sort for Sentence Ordering](https://mlsanity.com/paper/2005.00432/) (2020)
- [Graph-based Neural Sentence Ordering](https://mlsanity.com/paper/1912.07225/) (2019)
- [Neural Sentence Ordering](https://mlsanity.com/paper/1607.06952/) (2016)
- [End-to-End Neural Sentence Ordering Using Pointer Network](https://mlsanity.com/paper/1611.04953/) (2016)
- [Learning Representations of Sets through Optimized Permutations](https://mlsanity.com/paper/1812.03928/) (2018)
- [Is Everything in Order? A Simple Way to Order Sentences](https://mlsanity.com/paper/2104.07064/) (2021)
- [Sort Story: Sorting Jumbled Images and Captions into Stories](https://mlsanity.com/paper/1606.07493/) (2016)
- [GANzzle: Reframing jigsaw puzzle solving as a retrieval task using a generative mental image](https://mlsanity.com/paper/2207.05634/) (2022)
- [Neural Sentence Ordering Based on Constraint Graphs](https://mlsanity.com/paper/2101.11178/) (2021)

## Built on by (most cited first, 5 of 5)

- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Scalable Diffusion for Materials Generation](https://mlsanity.com/paper/2311.09235/) (2023)
- [DiffAssemble: A Unified Graph-Diffusion Model for 2D and 3D Reassembly](https://mlsanity.com/paper/2402.19302/) (2024)
- [Nash Meets Wertheimer: Using Good Continuation in Jigsaw Puzzles](https://mlsanity.com/paper/2410.16857/) (2024)
- [ReassembleNet: Learnable Keypoints and Diffusion for 2D Fresco Reconstruction](https://mlsanity.com/paper/2505.21117/) (2025)

## Abstract (arXiv)

Positional reasoning is the process of ordering unsorted parts contained in a set into a consistent structure. We present Positional Diffusion, a plug-and-play graph formulation with Diffusion Probabilistic Models to address positional reasoning. We use the forward process to map elements' positions in a set to random positions in a continuous space. Positional Diffusion learns to reverse the noising process and recover the original positions through an Attention-based Graph Neural Network. We conduct extensive experiments with benchmark datasets including two puzzle datasets, three sentence ordering datasets, and one visual storytelling dataset, demonstrating that our method outperforms long-lasting research on puzzle solving with up to +18% compared to the second-best deep learning method, and performs on par against the state-of-the-art methods on sentence ordering and visual storytelling. Our work highlights the suitability of diffusion models for ordering problems and proposes a novel formulation and method for solving various ordering tasks. Project website at https://iit-pavis.github.io/Positional_Diffusion/