# Thinking with Images as Continuous Actions: Numerical Visual Chain-of-Thought

arXiv: https://arxiv.org/abs/2602.23959 · HTML: https://mlsanity.com/paper/2602.23959/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 19 of its 45 references that are in the map.
- First posted on arXiv 27 Feb 2026.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-reasoning/)
- Multimodal LLMs / vision-language models / [VLM benchmarks and visual reasoning](https://mlsanity.com/mllm/vlm-benchmarks-and-visual-reasoning/)

## Builds on (most cited first, 19 of 19)

- [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)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [InternVL3: Exploring Advanced Training and Test-Time Recipes for Open-Source Multimodal Models](https://mlsanity.com/paper/2504.10479/) (2025)
- [DeepEyes: Incentivizing "Thinking with Images" via Reinforcement Learning](https://mlsanity.com/paper/2505.14362/) (2025)
- [LLaVA-Mini: Efficient Image and Video Large Multimodal Models with One Vision Token](https://mlsanity.com/paper/2501.03895/) (2025)
- [OpenThinkIMG: Learning to Think with Images via Visual Tool Reinforcement Learning](https://mlsanity.com/paper/2505.08617/) (2025)
- [SoTA with Less: MCTS-Guided Sample Selection for Data-Efficient Visual Reasoning Self-Improvement](https://mlsanity.com/paper/2504.07934/) (2025)
- [Multimodal ArXiv: A Dataset for Improving Scientific Comprehension of Large Vision-Language Models](https://mlsanity.com/paper/2403.00231/) (2024)
- [Latent Visual Reasoning](https://mlsanity.com/paper/2509.24251/) (2025)
- [Adaptive Chain-of-Focus Reasoning via Dynamic Visual Search and Zooming for Efficient VLMs](https://mlsanity.com/paper/2505.15436/) (2025)
- [VGR: Visual Grounded Reasoning](https://mlsanity.com/paper/2506.11991/) (2025)
- [VisionReasoner: Unified Reasoning-Integrated Visual Perception via Reinforcement Learning](https://mlsanity.com/paper/2505.12081/) (2025)
- [VisualToolAgent (VisTA): A Reinforcement Learning Framework for Visual Tool Selection](https://mlsanity.com/paper/2505.20289/) (2025)
- [v1: Learning to Point Visual Tokens for Multimodal Grounded Reasoning](https://mlsanity.com/paper/2505.18842/) (2025)
- [CMMCoT: Enhancing Complex Multi-Image Comprehension via Multi-Modal Chain-of-Thought and Memory Augmentation](https://mlsanity.com/paper/2503.05255/) (2025)
- [InternVL-X: Advancing and Accelerating InternVL Series with Efficient Visual Token Compression](https://mlsanity.com/paper/2503.21307/) (2025)
- [Patch-as-Decodable-Token: Towards Unified Multi-Modal Vision Tasks in MLLMs](https://mlsanity.com/paper/2510.01954/) (2025)

## Built on by (most cited first, 4 of 4)

- [Metacognition as Reward: Reinforcing LLM Reasoning via Knowledge and Regulation Signals](https://mlsanity.com/paper/2605.23384/) (2026)
- [Efficient Reinforcement for Visual-Textual Thinking with Discrete Diffusion Model](https://mlsanity.com/paper/2606.14792/) (2026)
- [See2Think: Do Multimodal Models Really Use Intermediate Visual States?](https://mlsanity.com/paper/2607.26769/) (2026)
- [Dyad: Extending Large Language Models with Native Typed Decision-Making](https://mlsanity.com/paper/2609.36116/) (2026)

## Abstract (arXiv)

Recent MLLMs increasingly rely on visual chain-of-thought to perform region-grounded reasoning over images. However, existing approaches ground regions via either textified coordinates (causing modality mismatch and semantic fragmentation) or fixed-granularity patches that both limit precise region selection and often require non-trivial architectural changes. In this paper, we propose Numerical Visual Chain-of-Thought (NV-CoT), a framework that enables MLLMs to reason over images using continuous numerical coordinates. NV-CoT expands the MLLM action space from discrete vocabulary tokens to a continuous Euclidean space, allowing models to directly generate bounding-box coordinates as actions with only minimal architectural modification. The framework supports both supervised fine-tuning and reinforcement learning. In particular, we replace categorical token policies with a Gaussian (or Laplace) policy over coordinates and introduce stochasticity via reparameterized sampling, making NV-CoT fully compatible with GRPO-style policy optimization. Extensive experiments on three benchmarks against 11 representative visual reasoning baselines demonstrate that NV-CoT significantly improves localization precision and final answer accuracy, while also accelerating training convergence, validating the effectiveness of continuous-action visual reasoning in MLLMs. Project page: https://kesenzhao.github.io/NV-CoT/