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arxiv: 2605.14068 · v2 · pith:U2GPRN34new · submitted 2026-05-13 · 💻 cs.CV · cs.LG

CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

Amirreza Mohseni , Mona Mohammadi , Morteza Saghafian , Naser Talebizadeh Sardari This is my paper

Pith reviewed 2026-05-20 20:34 UTC · model grok-4.3

classification 💻 cs.CV cs.LG
keywords Jordan curvescontainment hierarchytopological reasoningvision language modelsbenchmark datasetstructured predictionnested structuresvisual topology
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The pith

Vision models achieve only 71 percent accuracy recovering containment trees from nested curve images.

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

The paper presents CurveBench as a new benchmark for testing how well AI systems can understand hierarchical containment from pictures of simple closed curves. It supplies 756 images in different styles, each with a tree that describes which curve encloses which region. Testing shows that even the best model gets 71.1 percent right on easy pictures and just 19.1 percent on hard ones. After fine-tuning, an open model improves but still lags behind on tougher cases. This matters because accurate topological understanding is needed for tasks like interpreting diagrams or analyzing spatial arrangements where mistakes in nesting can lead to wrong conclusions.

Core claim

CurveBench formulates exact topological reasoning as the recovery of a rooted tree that encodes the containment relations among regions defined by non-intersecting Jordan curves in an image, and demonstrates through model evaluations that current vision-language models do not yet possess reliable capability for this structured prediction task.

What carries the argument

The rooted containment tree induced by the curves, which represents the full hierarchy of nesting and serves as the exact target output for the visual reasoning task.

If this is right

  • Targeted training on containment tree prediction improves model performance on simpler instances of the task.
  • Accuracy declines sharply when moving from easy polygonal configurations to dense or maze-like ones.
  • The benchmark provides a quantitative measure for tracking progress in topology-aware visual reasoning.
  • Remaining performance gaps indicate the need for advances in models' ability to handle exact nesting relations.

Where Pith is reading between the lines

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

  • Similar benchmarks could be created for other topological properties such as connectivity or genus to broaden evaluation of spatial reasoning.
  • Models might be improved by incorporating explicit geometric priors for curve nesting rather than relying solely on learned patterns.
  • Applications in fields like map interpretation or circuit design could benefit from better performance on this type of reasoning.
  • Failure modes on hard cases may reveal specific weaknesses in handling high numbers of nested elements.

Load-bearing premise

The supplied ground-truth trees correctly capture the containment relations without errors, and measuring tree generation accuracy truly reflects understanding of topology instead of reliance on superficial image features.

What would settle it

A vision model that produces the correct containment tree for a large fraction of CurveBench-Hard images while failing on images where the topology is altered but low-level visual statistics are preserved would indicate that the models are learning genuine topological reasoning rather than shortcuts.

Figures

Figures reproduced from arXiv: 2605.14068 by Amirreza Mohseni, Mona Mohammadi, Morteza Saghafian, Naser Talebizadeh Sardari.

Figure 1
Figure 1. Figure 1: Representative examples from each category within the CurveBench dataset [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Examples of images consisting of disjoint Jordan curves. Understanding the nesting [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tree-reward learning dynamics for trained models. Left: training set Right: eval set environments for CurveBench-Easy and CurveBench-Hard, ensuring that all models are evaluated under identical conditions. The released environments are listed in Appendix B.3. 6 Results [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative examples from each category within the CurveBench dataset [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: CurveBench Dataset: Hierarchical Distribution CurveBench is released as a collection of benchmark datasets for evaluating visual topological reasoning. The released resources include CurveBench-Easy and the main CurveBench benchmark used for the harder evaluation set￾ting. Review mode. This submission is intended for the single￾blind review option in the NeurIPS 2026 Evaluations & Datasets Track. CurveBenc… view at source ↗
Figure 3
Figure 3. Figure 3: Tree-reward learning dynamics for trained models. Left: training set Right: eval set Parameter-Efficient Fine-Tuning. We employ Low-Rank Adaptation (LoRA) Hu et al. [2022] for parameter-efficient RL fine-tuning. Only LoRA adapter parameters are updated, while the base model weights remain frozen. We use the all-linear target-module configuration in TRL, which applies adapters to linear layers throughout th… view at source ↗
Figure 4
Figure 4. Figure 4: Per-category success-rates. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: CurveBench Dataset: Hierarchical Distribution CurveBench is released as a collection of benchmark datasets for evaluating visual topological reasoning. The released resources include CurveBench-Easy and the main CurveBench benchmark used for the harder evaluation set￾ting. Review mode. This submission is intended for the single￾blind review option in the NeurIPS 2026 Evaluations & Datasets Track. CurveBenc… view at source ↗
Figure 5
Figure 5. Figure 5: Stacked reward greakdown for CurveBench-Hard. Darkest, medium, and Lightest color [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Per-category success-rates. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Stacked reward greakdown for CurveBench-Hard. Darkest, medium, and Lightest color [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of \textbf{756 images} of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only \textbf{71.1\%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1\%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over \texttt{Qwen-3-VL-8B-Thinking} from \textbf{2.8\%} to \textbf{33.3\%} tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.

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

2 major / 1 minor

Summary. The paper introduces CurveBench, a benchmark of 756 images depicting nested, non-intersecting Jordan curves in easy, polygonal, topographic, maze-like, and dense configurations. Each image is paired with a ground-truth rooted tree encoding containment relations among the induced planar regions. The central task is structured prediction: given an image, recover the exact containment tree. The authors report that Gemini 3.1 Pro reaches 71.1% tree-generation accuracy on the Easy subset and 19.1% on the Hard subset; they further show that RLVR-style fine-tuning lifts Qwen3-VL-8B from 2.8% to 33.3% on Easy, exceeding several closed models under their protocol. The work concludes that exact topology-aware visual reasoning remains far from solved.

Significance. If the supplied containment trees are verifiably error-free and the tree-generation metric isolates genuine topological understanding from low-level visual shortcuts or rendering artifacts, the benchmark would usefully quantify current VLM limitations on hierarchical spatial reasoning and supply a concrete target for future progress. The fine-tuning result demonstrates that modest gains are achievable with targeted training, lending the benchmark immediate utility for model development.

major comments (2)
  1. [Abstract and Dataset Construction] The headline performance numbers (71.1% Easy, 19.1% Hard) and the claim that exact topological reasoning remains unsolved rest on the assumption that the 756 ground-truth containment trees are free of annotation errors. The manuscript supplies no inter-annotator agreement statistics, no automated consistency checks (e.g., verification that containment is transitive and that every curve appears exactly once), and no release of raw SVG coordinates that would permit external re-derivation of the trees. Even a modest fraction of inverted parents or missing leaves on the Hard subset would render the 19.1% figure uninterpretable as a pure measure of model capability.
  2. [Experiments and Evaluation] The evaluation protocol does not report controls for visual shortcuts (e.g., curve thickness, fill color, or counting heuristics) that could allow models to achieve non-zero accuracy without recovering the true containment topology. Because the central claim concerns exact topological reasoning rather than pattern matching, the absence of such controls weakens the interpretation of the reported gaps.
minor comments (1)
  1. [Abstract] The abstract states that curves are 'pairwise non-intersecting' yet the Hard subset includes 'dense counting configurations'; a brief clarification of how non-intersection is enforced in the densest cases would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments identify important areas for strengthening claims about dataset reliability and isolating topological reasoning. We respond to each major comment below and will revise the manuscript to address them.

read point-by-point responses

  1. Referee: [Abstract and Dataset Construction] The headline performance numbers (71.1% Easy, 19.1% Hard) and the claim that exact topological reasoning remains unsolved rest on the assumption that the 756 ground-truth containment trees are free of annotation errors. The manuscript supplies no inter-annotator agreement statistics, no automated consistency checks (e.g., verification that containment is transitive and that every curve appears exactly once), and no release of raw SVG coordinates that would permit external re-derivation of the trees. Even a modest fraction of inverted parents or missing leaves on the Hard subset would render the 19.1% figure uninterpretable as a pure measure of model capability.

    Authors: CurveBench is generated via a fully procedural pipeline in which a target containment tree is first sampled and then realized as non-intersecting Jordan curves; the ground-truth tree is therefore exact by construction rather than the product of manual annotation. We will add an expanded Dataset Construction section that documents the generation algorithm, the automated verification steps (transitivity, uniqueness, and planarity), and the release of raw SVG files with the benchmark to enable external re-derivation. revision: yes

  2. Referee: [Experiments and Evaluation] The evaluation protocol does not report controls for visual shortcuts (e.g., curve thickness, fill color, or counting heuristics) that could allow models to achieve non-zero accuracy without recovering the true containment topology. Because the central claim concerns exact topological reasoning rather than pattern matching, the absence of such controls weakens the interpretation of the reported gaps.

    Authors: We agree that additional controls would strengthen the interpretation. In the revised manuscript we will include a new ablation subsection that reports model performance under randomized curve thickness and fill colors, as well as on specially constructed subsets that neutralize simple counting or boundary-length heuristics. These results will be used to argue that the observed gaps reflect limitations in hierarchical topological reasoning. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmark with no derivations or self-referential predictions

full rationale

This paper presents CurveBench as an empirical dataset and evaluation benchmark for visual topological reasoning, consisting of 756 images paired with manually annotated containment trees. No derivation chain, first-principles result, or prediction is claimed; performance numbers (e.g., 71.1% on Easy, 19.1% on Hard) are direct empirical measurements against the supplied ground-truth trees rather than outputs of any fitted model or self-referential equation. The fine-tuning experiment is likewise a standard RLVR-style training run whose results are reported as observed improvements, not as predictions derived from the benchmark itself. Because the work contains neither mathematical derivations nor load-bearing self-citations that reduce to the paper's own inputs, the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no free parameters, mathematical axioms, or invented entities; it relies on the standard topological definition of Jordan curves and containment relations.

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