Visual Diffusion Models are Geometric Solvers
Authors: name: Nir Goren, name: Shai Yehezkel, name: Omer Dahary, name: Andrey Voynov, name: Or Patashnik, name: Daniel Cohen-Or

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.