视觉扩散模型是几何求解器
Visual Diffusion Models are Geometric Solvers
把几何/组合优化问题实例渲染成图像，用标准视觉扩散模型从高斯噪声直接生成近似解，为视觉研究者提供「以图像生成为通用求解器」的范式，可迁移到布局、路径规划等需要输出结构化几何配置的任务。
原文依据 trains a standard visual diffusion model that transforms Gaussian noise into an image representing a valid approximate solution
摘要译文
由模型翻译 · 术语统一 · 请对照原文核对关键结论本文表明，视觉扩散模型可以作为有效的几何求解器：它们能够在像素空间中工作，直接对几何问题进行推理。我们首先在内接正方形问题上展示了这一点，该问题是几何学中长期存在的问题，询问每条若尔当曲线是否都包含四个构成正方形的点。随后我们将该方法扩展到另外两个著名的困难几何问题：斯坦纳树问题和最大面积多边形问题。我们的方法将每个问题实例视为一幅图像，并训练一个标准的视觉扩散模型，将高斯噪声变换为表示有效近似解的图像，该近似解与精确解高度吻合。模型学会将带噪声的几何结构变换为正确的构型，从而有效地将几何推理重新表述为图像生成。与先前将扩散应用于参数化几何表示时需要专门架构和领域特定适配的工作不同，我们采用在问题的视觉表示上运行的标准视觉扩散模型。这种简洁性凸显了生成建模与几何问题求解之间令人惊讶的桥梁。除本文研究的具体问题之外，我们的结果指向一个更广泛的范式：在图像空间中运行，为近似求解那些众所周知的困难问题提供了一个通用且实用的框架，并为处理更广泛的一类具有挑战性的几何任务打开了大门。
英文原文摘要 1499 字符
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 Maximum Area 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.