# Smooth Operator: Smooth Verifiable Reward Activates Spatial Reasoning Ability of Vision-Language Model

arXiv: https://arxiv.org/abs/2601.07695 · HTML: https://mlsanity.com/paper/2601.07695/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 33 of its 52 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-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 33)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Qwen2.5-VL Technical Report](https://mlsanity.com/paper/2502.13923/) (2025)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Imagine while Reasoning in Space: Multimodal Visualization-of-Thought](https://mlsanity.com/paper/2501.07542/) (2025)
- [MMSI-Bench: A Benchmark for Multi-Image Spatial Intelligence](https://mlsanity.com/paper/2505.23764/) (2025)
- [MedVLM-R1: Incentivizing Medical Reasoning Capability of Vision-Language Models (VLMs) via Reinforcement Learning](https://mlsanity.com/paper/2502.19634/) (2025)
- [VLM-3R: Vision-Language Models Augmented with Instruction-Aligned 3D Reconstruction](https://mlsanity.com/paper/2505.20279/) (2025)
- [SpaceR: Reinforcing MLLMs in Video Spatial Reasoning](https://mlsanity.com/paper/2504.01805/) (2025)
- [LlamaV-o1: Rethinking Step-by-step Visual Reasoning in LLMs](https://mlsanity.com/paper/2501.06186/) (2025)
- [Cambrian-S: Towards Spatial Supersensing in Video](https://mlsanity.com/paper/2511.04670/) (2025)
- [Reinforcing Spatial Reasoning in Vision-Language Models with Interwoven Thinking and Visual Drawing](https://mlsanity.com/paper/2506.09965/) (2025)
- [SAT: Dynamic Spatial Aptitude Training for Multimodal Language Models](https://mlsanity.com/paper/2412.07755/) (2024)
- [Self-Rewarding Vision-Language Model via Reasoning Decomposition](https://mlsanity.com/paper/2508.19652/) (2025)
- [MindCube: Spatial Mental Modeling from Limited Views](https://mlsanity.com/paper/2506.21458/) (2025)
- [Semantic Abstraction: Open-World 3D Scene Understanding from 2D Vision-Language Models](https://mlsanity.com/paper/2207.11514/) (2022)
- [Visual Spatial Tuning](https://mlsanity.com/paper/2511.05491/) (2025)
- [SpatialReasoner: Towards Explicit and Generalizable 3D Spatial Reasoning](https://mlsanity.com/paper/2504.20024/) (2025)
- [MetaSpatial: Reinforcing 3D Spatial Reasoning in VLMs for the Metaverse](https://mlsanity.com/paper/2503.18470/) (2025)
- [3D-R1: Enhancing Reasoning in 3D VLMs for Unified Scene Understanding](https://mlsanity.com/paper/2507.23478/) (2025)
- [Perception-R1: Advancing Multimodal Reasoning Capabilities of MLLMs via Visual Perception Reward](https://mlsanity.com/paper/2506.07218/) (2025)
- [Spatial-SSRL: Enhancing Spatial Understanding via Self-Supervised Reinforcement Learning](https://mlsanity.com/paper/2510.27606/) (2025)
- [EDGE-GRPO: Entropy-Driven GRPO with Guided Error Correction for Advantage Diversity](https://mlsanity.com/paper/2507.21848/) (2025)
- [AgentThink: A Unified Framework for Tool-Augmented Chain-of-Thought Reasoning in Vision-Language Models for Autonomous Driving](https://mlsanity.com/paper/2505.15298/) (2025)
- [FASIONAD : FAst and Slow FusION Thinking Systems for Human-Like Autonomous Driving with Adaptive Feedback](https://mlsanity.com/paper/2411.18013/) (2024)
- [FASIONAD++ : Integrating High-Level Instruction and Information Bottleneck in FAt-Slow fusION Systems for Enhanced Safety in Autonomous Driving with Adaptive Feedback](https://mlsanity.com/paper/2503.08162/) (2025)
- [LEGO-Motion: Learning-Enhanced Grids with Occupancy Instance Modeling for Class-Agnostic Motion Prediction](https://mlsanity.com/paper/2503.07367/) (2025)

## Built on by (most cited first, 2 of 2)

- [WaferSAGE: Large Language Model-Powered Wafer Defect Analysis via Synthetic Data Generation and Rubric-Guided Reinforcement Learning](https://mlsanity.com/paper/2604.27629/) (2026)
- [GaugeVLM: Structuring Spatial Supervision with Measured Geometric Interventions](https://mlsanity.com/paper/2609.38285/) (2026)

## Abstract (arXiv)

Vision-Language Models (VLMs) face a critical bottleneck in achieving precise numerical prediction for 3D scene understanding. Traditional reinforcement learning (RL) approaches, primarily based on relative ranking, often suffer from severe reward sparsity and gradient instability, failing to effectively exploit the verifiable signals provided by 3D physical constraints. Notably, in standard GRPO frameworks, relative normalization causes "near-miss" samples (characterized by small but non-zero errors) to suffer from advantage collapse. This leads to a severe data utilization bottleneck where valuable boundary samples are discarded during optimization. To address this, we introduce the Smooth Numerical Reward Activation (SNRA) operator and the Absolute-Preserving GRPO (AP-GRPO) framework. SNRA employs a dynamically parameterized Sigmoid function to transform raw feedback into a dense, continuous reward continuum. Concurrently, AP-GRPO integrates absolute scalar gradients to mitigate the numerical information loss inherent in conventional relative-ranking mechanisms. By leveraging this approach, we constructed Numerical3D-50k, a dataset comprising 50,000 verifiable 3D subtasks. Empirical results indicate that AP-GRPO achieves performance parity with large-scale supervised methods while maintaining higher data efficiency, effectively activating latent 3D reasoning in VLMs without requiring architectural modifications.