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arxiv: 2510.21697 · v2 · submitted 2025-10-24 · 💻 cs.CV · cs.LG

Visual Diffusion Models are Geometric Solvers

Nir Goren , Shai Yehezkel , Omer Dahary , Andrey Voynov , Or Patashnik , Daniel Cohen-Or This is my paper

Pith reviewed 2026-05-18 04:13 UTC · model grok-4.3

classification 💻 cs.CV cs.LG
keywords visual diffusion modelsgeometric solversinscribed square problemSteiner tree problemimage space reasoningdiffusion for geometryJordan curvegeometric problem solving
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The pith

Visual diffusion models can solve hard geometric problems by generating approximate solutions directly as images from noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that standard visual diffusion models can function as geometric solvers by operating entirely in pixel space on images of problem instances. Training such a model allows it to convert Gaussian noise into images that depict valid approximate solutions for challenges like the Inscribed Square Problem on Jordan curves, the Steiner Tree Problem, and the Simple Polygon Problem. This recasts geometric reasoning as an image generation task rather than requiring parametric optimization or custom architectures. A sympathetic reader would care because it proposes a general and accessible route to approximating notoriously difficult geometry tasks using widely available generative tools. The results indicate that image-space methods could extend to a broader set of computational geometry problems.

Core claim

The authors claim that a standard visual diffusion model trained on images of geometric problem instances transforms Gaussian noise into images representing valid approximate solutions that closely match exact configurations. This holds for the Inscribed Square Problem where four points form a square on a Jordan curve, the Steiner Tree Problem for minimal connecting structures, and the Simple Polygon Problem. The model learns to correct noisy geometric structures into proper ones, showing that geometric problem solving can be performed through generative modeling in pixel space without specialized adaptations.

What carries the argument

A standard visual diffusion model that takes images of problem instances as input and performs iterative denoising to output solution configurations in pixel space.

If this is right

  • Approximate solutions to the Inscribed Square Problem can be produced as images for arbitrary Jordan curves.
  • Minimal Steiner trees for given point sets can be approximated through generated images of connecting structures.
  • Simple polygons satisfying the required properties can be output directly as solution images.
  • Geometric reasoning tasks in general can be reframed as learning to generate correct visual configurations from noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same image-space approach might provide useful initial approximations for other hard visualization-based optimization problems such as point-set packing.
  • Pairing the generated images with traditional exact solvers could accelerate computation by supplying strong starting points.
  • Problems involving visual or perceptual constraints that resist clean mathematical parameterization could become more tractable in this framework.

Load-bearing premise

That training a standard visual diffusion model on images of problem instances is sufficient to produce configurations that closely match exact geometric solutions without requiring problem-specific architectures or post-processing.

What would settle it

Testing the trained model on held-out instances of Jordan curves and measuring whether the generated images contain four points forming a square within a small tolerance would falsify the claim if success rates remain low.

Figures

Figures reproduced from arXiv: 2510.21697 by Andrey Voynov, Daniel Cohen-Or, Nir Goren, Omer Dahary, Or Patashnik, Shai Yehezkel.

Figure 1
Figure 1. Figure 1: We introduce a visual diffusion approach to solving [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a curve (black) with three inscribed [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Inscribed square 𝑥0 predictions across denoising steps. Each row corresponds to a different seed (inscribed square). Columns show selected 𝑥0 predictions for decreasing timesteps 𝑡 from left to right (leftmost: 𝑡=𝑇 ; penultimate: 𝑡=0). For 𝑡 ≠ 0 we render only the filled mask; at 𝑡=0 we also draw square edges and the minimum-area bounding box. The rightmost column (“snapped”) shows the rigidly snapped vers… view at source ↗
Figure 4
Figure 4. Figure 4: Solutions produced by our model. Each Jordan curve (black) is accompanied by predicted inscribed squares (colored). [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of a Steiner Minimal Tree. Left: The input [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Optimal solutions (left) vs. our model’s solutions (middle) and the difference between them (right). Input points are [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative examples of maximum area polygons (left) vs. polygons produced by our model (middle) and the difference [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Maximum area polygon𝑥0 predictions across denois￾ing steps. Each row corresponds to a different seed. Columns show selected 𝑥0 predictions for decreasing timesteps 𝑡 from left to right (leftmost: 𝑡=𝑇 ; rightmost: 𝑡=0). Input points are overlayed in red. differences often balance out, with areas lost in one region largely compensated elsewhere (see instances 9, 10 in the figure). Owing to the non-local natu… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper claims that visual diffusion models can serve as effective geometric solvers: they can directly reason about geometric problems by working in pixel space. It demonstrates this on the Inscribed Square Problem and extends to the Steiner Tree Problem and Simple Polygon Problem by treating each instance as an image and training a standard visual diffusion model to transform noise into images of valid approximate solutions that closely match exact ones, without specialized architectures or post-processing.

Significance. If the results hold with rigorous validation, this would establish a simple, general paradigm for approximating notoriously hard geometric problems via off-the-shelf generative image models. The approach recasts geometric reasoning as image generation and highlights an unexpected bridge between visual diffusion and geometry, potentially applicable to a wider class of tasks. The avoidance of problem-specific adaptations is a notable strength if empirically supported.

major comments (3)
  1. Abstract: The assertion that the trained model produces 'a valid approximate solution that closely matches the exact one' lacks any quantitative performance metrics, error analysis, comparisons to exact solvers, or baseline methods. This absence makes it impossible to assess whether the generated configurations are sufficiently accurate for the central claim of effective geometric solving.
  2. Abstract (method description): The claim that a 'standard visual diffusion model' operating in pixel space suffices without 'problem-specific architectures or post-processing' is undermined by the inherent discretization of the image grid. Continuous geometric properties (e.g., exact 90° angles for inscribed squares or 120° Steiner points) cannot be guaranteed at the pixel level, and extracting usable continuous coordinates from raster outputs would itself require post-processing, contradicting the stated simplicity.
  3. Abstract: While the Inscribed Square Problem is used as the primary demonstration, the extension to Steiner Tree and Simple Polygon problems provides no details on how image-based representations enforce problem constraints, validate solution quality, or handle varying instance complexities.
minor comments (1)
  1. Abstract: The description of the Inscribed Square Problem as 'a long-standing problem in geometry' would benefit from a brief citation to foundational references for context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We address each of the major comments below and have incorporated revisions to improve the clarity and rigor of our presentation. We believe these changes will better support the claims made in the paper.

read point-by-point responses

  1. Referee: Abstract: The assertion that the trained model produces 'a valid approximate solution that closely matches the exact one' lacks any quantitative performance metrics, error analysis, comparisons to exact solvers, or baseline methods. This absence makes it impossible to assess whether the generated configurations are sufficiently accurate for the central claim of effective geometric solving.

    Authors: We agree that quantitative metrics are essential for rigorously evaluating the performance. Although the full manuscript provides extensive qualitative visualizations and some error measurements in the experimental sections, we acknowledge that the abstract does not highlight these. In the revised version, we will include specific quantitative results, such as mean squared error against exact solutions, success rates for valid configurations, and comparisons to traditional geometric solvers and baseline generative models, directly in the abstract or as a summary in the introduction. revision: yes

  2. Referee: Abstract (method description): The claim that a 'standard visual diffusion model' operating in pixel space suffices without 'problem-specific architectures or post-processing' is undermined by the inherent discretization of the image grid. Continuous geometric properties (e.g., exact 90° angles for inscribed squares or 120° Steiner points) cannot be guaranteed at the pixel level, and extracting usable continuous coordinates from raster outputs would itself require post-processing, contradicting the stated simplicity.

    Authors: This is a valid point regarding the nature of pixel-based representations. Our method indeed operates in discrete pixel space, and the generated images approximate the continuous geometric properties. However, the diffusion model itself requires no problem-specific architecture or post-processing during the generation process; it directly outputs the solution image from noise. Any subsequent extraction of coordinates for precise comparison or application is an evaluation step, not part of the core solving mechanism. We will revise the manuscript to explicitly clarify this distinction and discuss the approximation quality due to discretization, including examples of how close the pixel-level solutions are to continuous optima. revision: partial

  3. Referee: Abstract: While the Inscribed Square Problem is used as the primary demonstration, the extension to Steiner Tree and Simple Polygon problems provides no details on how image-based representations enforce problem constraints, validate solution quality, or handle varying instance complexities.

    Authors: We appreciate the need for more details on the extensions. In the full manuscript, we describe the specific image encodings used for each problem (e.g., rendering the curve and points for inscribed square, graph edges for Steiner tree) and how the model is trained on datasets of valid solutions to implicitly learn the constraints. Validation is performed by checking geometric properties on the output images. To make this more accessible, we will expand the abstract slightly and add a dedicated paragraph in the introduction summarizing the approach for each problem, including how complexities are handled through diverse training instances. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical training on image instances is self-contained

full rationale

The paper's central claim rests on training a standard visual diffusion model to map noise to images of approximate geometric solutions for problems like inscribed squares and Steiner trees. This is a conventional supervised generative modeling procedure with no mathematical derivation chain, no self-definitional equations, and no load-bearing predictions that reduce to fitted inputs by construction. The method explicitly contrasts itself with prior specialized architectures, relying instead on pixel-space training data and standard diffusion objectives. No self-citation is invoked to justify uniqueness or to smuggle in an ansatz; results are presented as empirical outcomes evaluated against geometric ground truth. The approach is therefore independent of its own outputs and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not specify any free parameters, axioms, or invented entities; the method relies on standard diffusion training assumptions that are not detailed here.

pith-pipeline@v0.9.0 · 5753 in / 1034 out tokens · 22411 ms · 2026-05-18T04:13:05.998803+00:00 · methodology

discussion (0)

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    Christopher Wewer, Bart Pogodzinski, Bernt Schiele, and Jan Eric Lenssen. Spatial reasoning with denoising models, 2025. URL https://arxiv.org/abs/2502.21075. Goren and Yehezkel et al. Appendix A Regression Model Ablation For certain geometric problems, each input instance admits a unique optimal solution. In such cases, one may wonder whether training a ...

    work page arXiv 2025