视觉扩散模型是几何求解器

摘要

本文首次证明视觉扩散模型可作为有效的几何求解器：其能直接在像素空间中对几何问题进行推理。我们首先以内接正方形问题为例验证这一观点——该几何学难题长期探讨是否所有若尔当曲线都包含可构成正方形的四个点。随后将方法拓展至另外两个著名几何难题：斯坦纳树问题与简单多边形问题。我们的方法将每个问题实例视为图像，并训练标准视觉扩散模型使其将高斯噪声转换为能紧密逼近精确解的有效近似解图像。该模型通过学习将含噪几何结构转换为正确配置，成功将几何推理重构为图像生成任务。与先前研究在应用扩散模型至参数化几何表征时需专门架构及领域适配不同，我们采用标准视觉扩散模型直接处理问题的视觉表征。这种简洁性凸显了生成建模与几何问题求解间令人惊喜的桥梁。除本文研究的特定问题外，我们的成果指向更广泛的范式：在图像空间中操作为逼近著名难题提供了通用实用框架，并为攻克更庞大类别的几何难题开启新途径。

English

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.

视觉扩散模型是几何求解器

Visual Diffusion Models are Geometric Solvers

摘要

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