LatentGeo: Learnable Auxiliary Constructions in Latent Space for Multimodal Geometric Reasoning

2026-03-12

AI Summary

This work addresses the challenge that multimodal large language models struggle to effectively represent auxiliary geometric constructions—critical for geometric reasoning yet often absent from visual inputs. To this end, the authors propose LatentGeo, a novel framework that internalizes auxiliary constructions within a continuous latent space, enabling end-to-end differentiable geometric reasoning without relying on pixel-level rendering or external solvers. The approach integrates a three-stage curriculum learning strategy with a newly designed latent-aware Direct Preference Optimization (LaGDPO) algorithm to jointly refine latent representations and reasoning policies, further enhanced by multimodal alignment and geometric semantic supervision. Evaluated on the GeoAux and MathVerse benchmarks, LatentGeo substantially improves geometric reasoning performance, particularly on problems requiring auxiliary constructions, with ablation studies confirming the contribution of each component.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Despite recent advances in multimodal reasoning, representing auxiliary geometric constructions remains a fundamental challenge for multimodal large language models (MLLMs). Such constructions are absent from the original diagram and must be introduced before theorems apply. Existing approaches predominantly rely on explicit construction paradigms, including text-based geometric specification, visual-token interleaving during reasoning, and tool-augmented geometric execution. However, these methods either fail to faithfully represent complex spatial relationships, incur representation mismatch between discrete symbols and continuous geometric structures, or rely on external capabilities that hinder end-to-end optimization. To address these limitations, we propose LatentGeo, a framework that learns continuous latent visual representations to internalize auxiliary geometric constructions without pixel-level rendering or external executors. We design a three-stage curriculum that progressively aligns and internalizes these latent representations through auxiliary visual supervision, followed by LaGDPO, a latent-aware reinforcement learning procedure that stabilizes latent representations during policy optimization while improving end-task correctness. To systematically evaluate construction-centric representation quality, we introduce GeoAux, a new benchmark targeting visually dependent geometry problems, and conduct experiments on GeoAux and MathVerse. Results show that LatentGeo achieves substantial gains on geometric reasoning tasks, particularly those requiring auxiliary constructions. Extensive analyses and ablation studies further validate the effectiveness of each component in our framework.

Problem

• auxiliary geometric constructions
• multimodal reasoning
• geometric representation
• latent space
• multimodal large language models

Innovation

• latent visual representation
• auxiliary geometric constructions
• multimodal geometric reasoning
• LaGDPO
• curriculum learning