Positional diffusion: Graph-based diffusion models for set ordering

Positional reasoning is the process of ordering an unsorted set of parts into a consistent structure. To address this problem, we present Positional Diffusion, a plug-and-play graph formulation with Diffusion Probabilistic Models. Using a diffusion process, we add Gaussian noise to the set elements' position and map them to a random position in a continuous space. Positional Diffusion learns to reverse the noising process and recover the original positions through an Attention-based Graph Neural Network. To evaluate our method, we conduct extensive experiments on three different tasks and seven datasets, comparing our approach against the stateof-the-art methods for visual puzzle-solving, sentence ordering, and room arrangement, demonstrating that our method outperforms long-lasting research on puzzle solving with up to +17% compared to the secondbest deep learning method, and performs on par against the state-of-the-art methods on sentence ordering and room rearrangement. Our work highlights the suitability of diffusion models for ordering problems and proposes a novel formulation and method for solving various ordering tasks. We release our code at https://github.com/IIT-PAVIS/Positional_Diffusion.