# Geometry-Aware Test-Time Learning for Quantitative Spatial Reasoning

arXiv: https://arxiv.org/abs/2609.06004 · HTML: https://mlsanity.com/paper/2609.06004/

## 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 6 of its 54 references that are in the map.
- First posted on arXiv 5 Sep 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, 6 of 6)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Cosmos-Reason1: From Physical Common Sense To Embodied Reasoning](https://mlsanity.com/paper/2503.15558/) (2025)
- [ViewSpatial-Bench: Evaluating Multi-perspective Spatial Localization in Vision-Language Models](https://mlsanity.com/paper/2505.21500/) (2025)
- [Self-Improving LLM Agents at Test-Time](https://mlsanity.com/paper/2510.07841/) (2025)
- [Guided Trajectory Optimization with Sparse Scaling for Test-Time Diffusion](https://mlsanity.com/paper/2605.21907/) (2026)

## Abstract (arXiv)

Quantitative spatial reasoning in visual-language models (VLMs) aims to infer spatial distances and directional relationships among objects in 3D space from a 2D image and a natural language query. Despite recent progress, VLM spatial reasoning remains brittle under distribution shifts, largely due to the high cost of 3D supervision. As a result, models often produce inconsistent or contradictory predictions when faced with novel object configurations or rephrased spatial queries, revealing a misalignment between learned representations and underlying geometry. To address this, we propose TTL-SR, a geometry-aware Test-Time Learning framework for quantitative Spatial Reasoning that leverages geometric consistency constraints and unlabeled test data to adapt models to target domains. Specifically, TTL-SR augments the input query with geometrically coupled auxiliary queries, filters unreliable predictions via adaptive geometric triggering to construct structured token-level pseudo-labels, and updates model parameters under a geometry-aware multi-objective loss using only test data. Experimental results demonstrate that TTL-SR significantly boosts spatial reasoning performance, yielding 6.47% and 9.41% accuracy gains for Qwen3-VL-4B-Instruct and SpatialRGPT-VILA-1.5-8B on Q-Spatial-ScanNet dataset, respectively.