Polymath: A Challenging Multi-modal Mathematical Reasoning Benchmark
Pith reviewed 2026-05-23 20:01 UTC · model grok-4.3
The pith
Multi-modal models gain only about 4 percent when diagrams are replaced by text descriptions on a new math reasoning benchmark.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The central claim is that MLLMs do not truly comprehend visual diagrams and the spatial information they contain. This is shown by the low overall scores on PolyMATH and by the ablation result that textual descriptions produce only marginal gains over the actual images, leaving models prone to logical errors on tasks requiring drawn-out high-level reasoning.
What carries the argument
The PolyMATH benchmark itself, built from 5,000 high-quality images across ten distinct categories of textual and visual cognitive challenges.
If this is right
- Current MLLMs remain limited in handling spatial relations within diagrams even when prompted with Chain-of-Thought or Step-Back strategies.
- Models are prone to logical errors once tasks require integrating visual details over multiple steps.
- The benchmark can serve as a diagnostic tool to measure progress in visual abstraction beyond current training regimes.
- OpenAI o1 models matching human baseline performance indicates that scaling alone has not closed the gap on these visual reasoning tasks.
Where Pith is reading between the lines
- Training corpora may contain too few examples of complex diagram-based spatial relations, limiting what models can internalize from image-text pairs.
- Targeted pretraining on diagram parsing or synthetic spatial puzzles could be tested as a direct remedy for the observed weaknesses.
- Extending the benchmark with dynamic or interactive diagrams would reveal whether the current static-image limitation is fundamental.
Load-bearing premise
The 5,000 manually collected images form a fair, unbiased, and appropriately difficult sample of the visual and cognitive challenges that matter for mathematical reasoning.
What would settle it
A new model achieving above 70 percent accuracy on the full PolyMATH set or showing more than 15 percent improvement when given textual descriptions instead of the original images would undermine the claim of weak visual diagram comprehension.
Figures
read the original abstract
Multi-modal Large Language Models (MLLMs) exhibit impressive problem-solving abilities in various domains, but their visual comprehension and abstract reasoning skills remain under-evaluated. To this end, we present PolyMATH, a challenging benchmark aimed at evaluating the general cognitive reasoning abilities of MLLMs. PolyMATH comprises 5,000 manually collected high-quality images of cognitive textual and visual challenges across 10 distinct categories, including pattern recognition, spatial reasoning, and relative reasoning. We conducted a comprehensive, and quantitative evaluation of 15 MLLMs using four diverse prompting strategies, including Chain-of-Thought and Step-Back. The best scores achieved on PolyMATH are ~41%, ~36%, and ~27%, obtained by Claude-3.5 Sonnet, GPT-4o and Gemini-1.5 Pro respectively - highlighting the logical and visual complexity of these questions. A further fine-grained error analysis reveals that these models struggle to understand spatial relations and perform drawn-out, high-level reasoning. This is further strengthened by our ablation study estimating MLLM performance when given textual descriptions in place of diagrams. As evidenced by ~4% improvement over textual descriptions as opposed to actual images, we discover that models do not truly comprehend visual diagrams and the spatial information therein, and are thus prone to logical errors. Finally, we evaluate the OpenAI o1 models and find that their performance only matches the human baseline, highlighting the difficulty of the benchmark. The results on PolyMATH highlight the room for improvement in multi-modal reasoning and provide unique insights to guide the development of future MLLMs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces PolyMATH, a benchmark consisting of 5,000 manually collected images across 10 categories of cognitive challenges (pattern recognition, spatial reasoning, relative reasoning, etc.) to evaluate MLLMs on multi-modal mathematical reasoning. It reports evaluations of 15 MLLMs under four prompting strategies, with top scores of ~41% (Claude-3.5 Sonnet), ~36% (GPT-4o), and ~27% (Gemini-1.5 Pro); an ablation replacing images with textual descriptions yields only ~4% improvement, supporting the claim that models fail to comprehend visual diagrams and spatial information; o1 models match but do not exceed a human baseline.
Significance. If the 5,000-image sample is representative and the ablation comparison is well-controlled, the results would establish a valuable, high-difficulty benchmark exposing clear gaps in current MLLMs' spatial and multi-step visual reasoning, with the error analysis and human/o1 comparisons providing concrete guidance for future work. The scale and category coverage are strengths.
major comments (3)
- [Benchmark construction] Benchmark construction (abstract and § on data collection): The images are described only as 'manually collected' across 10 categories with no sourcing protocol, inclusion/exclusion criteria, diversity metrics, or verification steps against selection bias. This is load-bearing for the central claim, because both the headline ~41% ceiling and the ~4% image-vs-text gap could arise from non-representative sampling of unusually difficult cases rather than a fundamental deficit in diagram comprehension.
- [Ablation study] Ablation study (results section): The ~4% performance lift when substituting textual descriptions for images is used to conclude that 'models do not truly comprehend visual diagrams,' yet the manuscript supplies no information on how the textual descriptions were generated, their fidelity to the original diagrams, or any validation that they preserve all spatial relations. Without these controls the ablation cannot isolate visual comprehension.
- [Experimental setup] Evaluation protocol (experimental setup): No inter-annotator agreement, difficulty calibration procedure, or statistical tests (e.g., confidence intervals or significance of the 4% gap) are reported for the quantitative scores or human baseline. These omissions undermine the reliability of the performance numbers that underpin the main conclusions.
minor comments (1)
- [Abstract] The abstract lists 'four diverse prompting strategies' but only names Chain-of-Thought and Step-Back; the remaining two should be enumerated for reproducibility.
Simulated Author's Rebuttal
We thank the referee for the constructive feedback highlighting areas where additional methodological details would strengthen the paper. We address each major comment below and will incorporate the suggested clarifications and controls in a revised manuscript.
read point-by-point responses
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Referee: [Benchmark construction] The images are described only as 'manually collected' across 10 categories with no sourcing protocol, inclusion/exclusion criteria, diversity metrics, or verification steps against selection bias. This is load-bearing for the central claim, because both the headline ~41% ceiling and the ~4% image-vs-text gap could arise from non-representative sampling of unusually difficult cases rather than a fundamental deficit in diagram comprehension.
Authors: We agree that the current description of data collection is insufficiently detailed. In the revision we will add a dedicated subsection describing the sourcing (publicly available cognitive puzzle repositories and textbooks), explicit inclusion criteria (clear diagrams, unambiguous ground-truth answers, coverage of the 10 categories), diversity metrics (balanced distribution across categories and difficulty levels as measured by pilot human accuracy), and verification (two independent annotators reviewed each item for bias and clarity, with disagreements resolved by discussion). These additions will directly support the representativeness of the 5,000-image sample. revision: yes
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Referee: [Ablation study] The ~4% performance lift when substituting textual descriptions for images is used to conclude that 'models do not truly comprehend visual diagrams,' yet the manuscript supplies no information on how the textual descriptions were generated, their fidelity to the original diagrams, or any validation that they preserve all spatial relations. Without these controls the ablation cannot isolate visual comprehension.
Authors: We acknowledge the omission of generation and validation details for the textual descriptions. The revision will specify that descriptions were produced by human experts following a template that required explicit enumeration of all spatial relations, object positions, and visual elements while omitting the solution; a separate validation set of 200 items was rated by two additional annotators for completeness (mean fidelity score 4.7/5), with inter-annotator agreement reported. These controls will be added to the ablation section to better isolate the contribution of visual input. revision: yes
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Referee: [Experimental setup] No inter-annotator agreement, difficulty calibration procedure, or statistical tests (e.g., confidence intervals or significance of the 4% gap) are reported for the quantitative scores or human baseline. These omissions undermine the reliability of the performance numbers that underpin the main conclusions.
Authors: We agree these statistical and reliability details are missing. The revised experimental setup section will report (i) inter-annotator agreement (Cohen’s kappa) for the human baseline, (ii) the difficulty calibration procedure (pilot testing on 100 items with 10 participants to stratify by accuracy bands), and (iii) bootstrap confidence intervals plus paired t-tests for all model comparisons, including the 4% image-vs-text gap. These additions will quantify the reliability of the reported numbers. revision: yes
Circularity Check
No circularity: direct empirical evaluation on collected dataset
full rationale
The paper reports performance measurements and an ablation on a newly collected set of 5,000 images. No equations, fitted parameters, predictions derived from inputs, or self-citation chains appear in the derivation of the central claims. The ~4% text-vs-image gap and overall scores are direct empirical observations on the provided data, not quantities forced by construction or renamed from prior fits. The manual collection step is an input to the evaluation rather than a self-referential loop. This is a standard benchmark paper with no load-bearing reductions to its own outputs.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Manually collected images across 10 categories accurately capture distinct cognitive challenges without selection bias.
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URL https://openreview.net/forum?id=d4UiXAHN2W. Xiang Zhang, Senyu Li, Zijun Wu, and Ning Shi. Lost in translation: When gpt-4v (ision) can’t see eye to eye with text. a vision-language-consistency analysis of vllms and beyond. arXiv preprint arXiv:2310.12520, 2023f. Xiaoman Zhang, Chaoyi Wu, Ziheng Zhao, Weixiong Lin, Ya Zhang, Yanfeng Wang, and Weidi Xi...
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Aojun Zhou, Ke Wang, Zimu Lu, Weikang Shi, Sichun Luo, Zipeng Qin, Shaoqing Lu, Anya Jia, Linqi Song, Mingjie Zhan, et al. Solving challenging math word problems using gpt-4 code interpreter with code-based self-verification. arXiv preprint arXiv:2308.07921,
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MiniGPT-4: Enhancing Vision-Language Understanding with Advanced Large Language Models
Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mohamed Elhoseiny. MiniGPT-4: En- hancing vision-language understanding with advanced large language models. arXiv preprint arXiv:2304.10592, 2023a. Wanrong Zhu, Jack Hessel, Anas Awadalla, Samir Yitzhak Gadre, Jesse Dodge, Alex Fang, Youngjae Yu, Ludwig Schmidt, William Yang Wang, and Yejin Choi. Multimod...
work page internal anchor Pith review Pith/arXiv arXiv 2021
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word problems with limited scope. Subsequent efforts, including MATH (Hendrycks et al., 2021b), GSM8K (Cobbe et al., 2021), MMLU (Hendrycks et al., 2021a), and others (Zhou et al., 2023; Yue et al., 2023b; Wang et al., 2024a; Gao et al., 2023a; Luo et al., 2023), expanded the range and quality of textual mathematical problems, establishing robust benchmar...
work page 2021
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provide limited coverage of rigorous scientific domains crucial for general-purpose AI assistants. While these benchmarks assess text-only mathematical reasoning, the rapid progress of MLLMs necessitates high-quality benchmarks for evaluating visual mathematical problem-solving. Prior 20 Preprint attempts like GeoQA (Chen et al., 2021a), while MathVista (...
work page 2024
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and large vision models (Radford et al., 2021; Kirillov et al., 2023; Zhang et al., 2023d;c;e), have become increasingly prominent. They extend LLMs to diverse tasks and modalities, including 2D images (Li et al., 2022; Dai et al., 2023; Alayrac et al., 2022; Li et al., 2023a), 3D point clouds (Guo et al., 2023; Xu et al., 2023b; Hong et al., 2024), audio...
work page 2021
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However, their closed-source nature hinders broader application and development of MLLMs
exhibit exceptional visual reasoning capabilities, setting new benchmarks in multi-modal performance. However, their closed-source nature hinders broader application and development of MLLMs. Concur- rently, open-source MLLMs like LLaMA-Adapter (Zhang et al., 2024; Gao et al., 2023b), LLaV A (Liu et al., 2023b; 2024; 2023a), MiniGPT-4 (Zhu et al., 2023a; ...
work page 2024
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for image encoding and LLaMA (Touvron et al., 2023a) for multi-modal instruction tuning, advancing MLLMs’ visual understanding and generalization. Despite comprehensive benchmarks (Fu et al., 2023a; Liu et al., 2023d; Li et al., 2023b; Xu et al., 2023a) for general visual instruction-following scenarios, the specific potential of MLLMs for visual mathemat...
work page 2015
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evaluate LMMs’ general visual question answering abilities on open-ended image queries. Additionally, works have assessed LMMs’ specific skills beyond natural scenes, such as abstract shapes (Antol et al., 2015; Lu et al., 2021b; Ji et al., 2022), geometry diagrams (Seo et al., 2015; Lu et al., 2021a; Chen et al., 2022a; Cao & Xiao, 2022), charts (Methani...
work page 2015
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leverage paired (Schuhmann et al., 2022; Sharma et al., 2018; Lin et al.,
work page 2022
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and interleaved (Zhu et al., 2023b) image-text data. Additionally, specialized versions like LLaV AR (Zhang et al., 2023h; Ye et al., 2023a) emphasize document understanding and math comprehension. Recent works like Visit-Bench (Bitton et al., 2023), LVLM-eHub (Yu et al., 2023), MMBench (Liu et al., 2023d; Xu et al., 2023a; Shao et al.,
work page 2023
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assess these models’ instruction-following and reasoning capabilities. Large language models (LLMs) have demonstrated remarkable reasoning abilities, further enhanced by approaches like chain-of-thought (CoT) (Wei et al., 2022b), program-of-thought (PoT) (Chen et al., 2022b), and inductive reasoning (Wang et al., 2023a; Tan & Motani, 2023). The feasibilit...
work page 2023
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