# Visual Diffusion Models are Geometric Solvers

arXiv: https://arxiv.org/abs/2510.21697 · HTML: https://mlsanity.com/paper/2510.21697/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in 3D, point cloud and scene generation, one of 14 directions in Diffusion models.
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Placed in

- Diffusion models / [3D, point cloud and scene generation](https://mlsanity.com/diffusion/3d-point-cloud-and-scene-generation/)

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [Fast T2T: Optimization Consistency Speeds Up Diffusion-Based Training-to-Testing Solving for Combinatorial Optimization](https://mlsanity.com/paper/2502.02941/) (2025)
- [Spatial Reasoning with Denoising Models](https://mlsanity.com/paper/2502.21075/) (2025)
- [NN-Steiner: A Mixed Neural-algorithmic Approach for the Rectilinear Steiner Minimum Tree Problem](https://mlsanity.com/paper/2312.10589/) (2023)
- [Vulcan: Solving the Steiner Tree Problem with Graph Neural Networks and Deep Reinforcement Learning](https://mlsanity.com/paper/2111.10810/) (2021)
- [Computing Steiner Trees using Graph Neural Networks](https://mlsanity.com/paper/2108.08368/) (2021)
- [Deep-Steiner: Learning to Solve the Euclidean Steiner Tree Problem](https://mlsanity.com/paper/2209.09983/) (2022)
- [Greedy and Local Search Heuristics to Build Area-Optimal Polygons](https://mlsanity.com/paper/2106.14728/) (2021)
- [Computing Area-Optimal Simple Polygonizations](https://mlsanity.com/paper/2111.05386/) (2021)
- [Area-Optimal Simple Polygonalizations: The CG Challenge 2019](https://mlsanity.com/paper/2111.07304/) (2021)

## Built on by (most cited first, 2 of 2)

- [DiffThinker: Towards Generative Multimodal Reasoning with Diffusion Models](https://mlsanity.com/paper/2512.24165/) (2025)
- [DisPOSE: Projected Polystochastic Diffusion for Self-Supervised Multi-View 3D Human Pose Estimation](https://mlsanity.com/paper/2606.07419/) (2026)

## Abstract (arXiv)

In this paper we show that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. We first demonstrate this on the Inscribed Square Problem, a long-standing problem in geometry that asks whether every Jordan curve contains four points forming a square. We then extend the approach to two other well-known hard geometric problems: the Steiner Tree Problem and the Simple Polygon Problem. Our method treats each problem instance as an image and trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution that closely matches the exact one. The model learns to transform noisy geometric structures into correct configurations, effectively recasting geometric reasoning as image generation. Unlike prior work that necessitates specialized architectures and domain-specific adaptations when applying diffusion to parametric geometric representations, we employ a standard visual diffusion model that operates on the visual representation of the problem. This simplicity highlights a surprising bridge between generative modeling and geometric problem solving. Beyond the specific problems studied here, our results point toward a broader paradigm: operating in image space provides a general and practical framework for approximating notoriously hard problems, and opens the door to tackling a far wider class of challenging geometric tasks.