Visual Diffusion Models are Geometric Solvers

TL;DR

This paper demonstrates that standard visual diffusion models can be used as geometric solvers by transforming geometric problems into images and generating solutions directly in pixel space.

Contribution

It introduces a novel approach of applying standard visual diffusion models to solve complex geometric problems without specialized architectures.

Findings

Successfully applied to the Inscribed Square Problem, Steiner Tree Problem, and Simple Polygon Problem.

The diffusion model generates approximate solutions that closely match exact geometric configurations.

Highlights the potential of image-based generative models for solving hard geometric problems.

Abstract

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.