Aotian Chen

Shan Zhang, Aotian Chen, Kai Zou et al.

Symbolic computer vision represents diagrams through explicit logical rules and structured representations, enabling interpretable understanding in machine vision. This requires fundamentally different learning paradigms from pixel-based visual models. Symbolic visual learners parse diagrams into geometric primitives-points, lines, and shapes-whereas pixel-based learners operate on textures and colors. We propose a novel self-supervised symbolic auto-encoder that encodes diagrams into structured primitives and their interrelationships within the latent space, and decodes them through our executable engine to reconstruct the input diagrams. Central to this architecture is Symbolic Hierarchical Process Reward Modeling, which applies hierarchical step-level parsing rewards to enforce point-on-line, line-on-shape, and shape-on-relation consistency. Since vanilla reinforcement learning exhibits poor exploration in the policy space during diagram reconstruction; we thus introduce stabilization mechanisms to balance exploration and exploitation. We fine-tune our symbolic encoder on downstream tasks, developing a neuro-symbolic system that integrates the reasoning capabilities of neural networks with the interpretability of symbolic models through reasoning-grounded visual rewards. Evaluations across reconstruction, perception, and reasoning tasks demonstrate the effectiveness of our approach: achieving a 98.2% reduction in MSE for geometric diagram reconstruction, surpassing GPT-4o by 0.6% with a 7B model on chart reconstruction, and improving by +13% on the MathGlance perception benchmark, and by +3% on MathVerse and GeoQA reasoning benchmarks.

Yanpeng Sun, Shan Zhang, Wei Tang et al.

Diagrams serve as a fundamental form of visual language, representing complex concepts and their inter-relationships through structured symbols, shapes, and spatial arrangements. Unlike natural images, their inherently symbolic and abstract nature poses significant challenges for Multimodal Large Language Models (MLLMs). However, current benchmarks conflate perceptual and reasoning tasks, making it difficult to assess whether MLLMs genuinely understand mathematical diagrams beyond superficial pattern recognition. To address this gap, we introduce MATHGLANCE, a benchmark specifically designed to isolate and evaluate mathematical perception in MLLMs. MATHGLANCE comprises 1.2K images and 1.6K carefully curated questions spanning four perception tasks: shape classification, object counting, relationship identification, and object grounding, covering diverse domains including plane geometry, solid geometry, and graphical representations. Our evaluation of MLLMs reveals that their ability to understand diagrams is notably limited, particularly in fine-grained grounding tasks. In response, we construct GeoPeP, a perception-oriented dataset of 200K structured geometry image-text pairs explicitly annotated with geometric primitives and precise spatial relationships. Training MLLM on GeoPeP leads to significant gains in perceptual accuracy, which in turn substantially improves mathematical reasoning. Our benchmark and dataset establish critical standards for evaluating and advancing multimodal mathematical understanding, providing valuable resources and insights to foster future MLLM research.