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

arXiv: https://arxiv.org/abs/2603.12166 · HTML: https://mlsanity.com/paper/2603.12166/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 40 of its 51 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)
- Multimodal LLMs / vision-language models / [Embodied and 3D vision-language models](https://mlsanity.com/mllm/embodied-and-3d-vision-language-models/)

## Builds on (most cited first, 30 of 40)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [Improved Baselines with Visual Instruction Tuning](https://mlsanity.com/paper/2310.03744/) (2023)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Qwen3-VL Technical Report](https://mlsanity.com/paper/2511.21631/) (2025)
- [InternVL: Scaling up Vision Foundation Models and Aligning for Generic Visual-Linguistic Tasks](https://mlsanity.com/paper/2312.14238/) (2023)
- [Qwen-VL: A Versatile Vision-Language Model for Understanding, Localization, Text Reading, and Beyond](https://mlsanity.com/paper/2308.12966/) (2023)
- [Expanding Performance Boundaries of Open-Source Multimodal Models with Model, Data, and Test-Time Scaling](https://mlsanity.com/paper/2412.05271/) (2024)
- [InternVL3.5: Advancing Open-Source Multimodal Models in Versatility, Reasoning, and Efficiency](https://mlsanity.com/paper/2508.18265/) (2025)
- [mPLUG-Owl: Modularization Empowers Large Language Models with Multimodality](https://mlsanity.com/paper/2304.14178/) (2023)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [DeepSeek-VL: Towards Real-World Vision-Language Understanding](https://mlsanity.com/paper/2403.05525/) (2024)
- [MathVerse: Does Your Multi-modal LLM Truly See the Diagrams in Visual Math Problems?](https://mlsanity.com/paper/2403.14624/) (2024)
- [R1-Onevision: Advancing Generalized Multimodal Reasoning through Cross-Modal Formalization](https://mlsanity.com/paper/2503.10615/) (2025)
- [SPHINX: The Joint Mixing of Weights, Tasks, and Visual Embeddings for Multi-modal Large Language Models](https://mlsanity.com/paper/2311.07575/) (2023)
- [Visual Sketchpad: Sketching as a Visual Chain of Thought for Multimodal Language Models](https://mlsanity.com/paper/2406.09403/) (2024)
- [GDPO: Group reward-Decoupled Normalization Policy Optimization for Multi-reward RL Optimization](https://mlsanity.com/paper/2601.05242/) (2026)
- [Math-LLaVA: Bootstrapping Mathematical Reasoning for Multimodal Large Language Models](https://mlsanity.com/paper/2406.17294/) (2024)
- [Machine Mental Imagery: Empower Multimodal Reasoning with Latent Visual Tokens](https://mlsanity.com/paper/2506.17218/) (2025)
- [MAVIS: Mathematical Visual Instruction Tuning with an Automatic Data Engine](https://mlsanity.com/paper/2407.08739/) (2024)
- [GeomVerse: A Systematic Evaluation of Large Models for Geometric Reasoning](https://mlsanity.com/paper/2312.12241/) (2023)
- [Latent Visual Reasoning](https://mlsanity.com/paper/2509.24251/) (2025)
- [Monet: Reasoning in Latent Visual Space Beyond Images and Language](https://mlsanity.com/paper/2511.21395/) (2025)
- [Chain-of-Visual-Thought: Teaching VLMs to See and Think Better with Continuous Visual Tokens](https://mlsanity.com/paper/2511.19418/) (2025)
- [MultiMath: Bridging Visual and Mathematical Reasoning for Large Language Models](https://mlsanity.com/paper/2409.00147/) (2024)
- [Math-PUMA: Progressive Upward Multimodal Alignment to Enhance Mathematical Reasoning](https://mlsanity.com/paper/2408.08640/) (2024)
- [Latent Sketchpad: Sketching Visual Thoughts to Elicit Multimodal Reasoning in MLLMs](https://mlsanity.com/paper/2510.24514/) (2025)
- [MMR1: Enhancing Multimodal Reasoning with Variance-Aware Sampling and Open Resources](https://mlsanity.com/paper/2509.21268/) (2025)
- [OpenMMReasoner: Pushing the Frontiers for Multimodal Reasoning with an Open and General Recipe](https://mlsanity.com/paper/2511.16334/) (2025)

## Abstract (arXiv)

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.