Reactivating Test-Time Scaling for Plane Geometry Problem Solving

2026-08-30

AI Summary

该研究针对平面几何问题求解中符号程序多样性和视觉基础不足的问题，提出了多轨迹合成和感知增强训练方法，有效提升了模型在不同规模下的解题性能。

Abstract

Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at https://github.com/Jason8Kang/ReTTS-PGPS.

Problem

• Test-Time Scaling
• Plane Geometry Problem
• Symbolic Program
• Visual Grounding
• Reasoning Diversity

Innovation

• Multi-Trace Synthesis (MTS)
• Perception-Augmented (PA) Training
• Consensus-Guided Multi-Trace Ensemble (CG-MTE)