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<!DOCTYPE html>
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content="Geometrically-Constrained Agent (GCA): A training-free agentic paradigm that resolves the semantic-to-geometric gap in Spatial Reasoning.">
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<h1 class="title is-3 publication-title">Geometrically-Constrained Agent for Spatial Reasoning</h1>
<div class="is-size-5 publication-authors">
<span class="author-block">
<a href="https://github.com/Zx55">Zeren Chen</a><sup>1,2*</sup>,</span>
<span class="author-block">
<a href="https://github.com/ursulalujun">Xiaoya Lu</a><sup>2,3*</sup>,</span>
<span class="author-block">
<a href="#">Zhijie Zheng</a><sup>1,2</sup>,
</span>
<span class="author-block">
<a href="#">Pengrui Li</a><sup>1</sup>,
</span>
<span class="author-block">
<a href="#">Lehan He</a><sup>1,4</sup>,
</span>
<span class="author-block">
<a href="#">Yijin Zhou</a><sup>2,3,4</sup>,
</span>
<br>
<span class="author-block">
<a href="https://amandajshao.github.io">Jing Shao</a><sup>2</sup>,
</span>
<span class="author-block">
<a href="#">Bohan Zhuang</a><sup>5†</sup>,
</span>
<span class="author-block">
<a href="#">Lu Sheng</a><sup>1†</sup>
</span>
</div>
<div class="is-size-5 publication-authors">
<span class="author-block"><sup>1</sup>School of Software, Beihang University,</span>
<span class="author-block"><sup>2</sup>Shanghai AI Laboratory,</span>
<span class="author-block"><sup>3</sup>Shanghai Jiao Tong University,</span><br>
<span class="author-block"><sup>4</sup>Shanghai Innovation Institute,</span>
<span class="author-block"><sup>5</sup>ZIP Lab, Zhejiang University</span>
</div>
<div class="is-size-6 publication-authors">
<span class="author-block">*Equal Contribution, †Corresponding Author</span>
</div>
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<a href="https://arxiv.org/abs/2511.22659" target="_blank" class="button"><i class="ai ai-arxiv"></i> arXiv</a>
<a href="https://github.com/gca-spatial-reasoning/gca" target="_blank" class="button"><i class="fa-light fa-code"></i> Code</a>
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<img src="./static/images/teaser_page.jpg" alt="Semantic-Geometric Gap vs Geometrically-Constrained Spatial Reasoning" style="width:100%; height:auto;">
<h2 class="subtitle has-text-centered">
<strong>Geometrically-Constrained Agent (GCA)</strong> resolves the semantic-to-geometric gap by decoupling the reasoning process into
Task Formalization and Constrained Geometric Computation.
</h2>
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<h2 class="subtitle has-text-centered">
Case Study #1: Unique Object Counting across Multiple Views
</h2>
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<source src="./static/videos/case_2.mp4" type="video/mp4">
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<h2 class="subtitle has-text-centered">
Case Study #2: Spatial Relationship Reasoning under Direction-Based Reference Frame
</h2>
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<h2 class="title is-3">Abstract</h2>
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<p>
Vision Language Models (VLMs) exhibit a fundamental <strong>semantic-to-geometric gap</strong> in spatial reasoning: they excel
at qualitative semantic inference but their reasoning operates within a lossy semantic space, misaligned with high-fidelity geometry.
Current paradigms fail to bridge this gap. Training-based methods suffer from an "oracle paradox,"
learning flawed spatial logic from imperfect oracles. Tool-integrated methods constrain the final computation but critically
leave the VLM's planning process unconstrained, resulting in geometrically flawed plans.
In this work, we propose <strong>Geometrically-Constrained Agent (GCA)</strong>, a training-free agentic paradigm that resolves this gap by introducing
a formal task constraint. Specifically, we strategically decouple the VLM's role into two stages:
1) Acting as a semantic analyst, the VLM translates the user's ambiguous query into a formal, verifiable task constraint, which defines the reference frame and objective.
2) Acting as a task solver, the VLM generates and executes tool calls strictly within the deterministic bounds defined by the constraint.
This geometrically-constrained reasoning strategy successfully resolves the semantic-to-geometric gap, yielding a robust and verifiable reasoning pathway.
Comprehensive experiments demonstrate that GCA achieves SOTA performance on multiple spatial reasoning benchmarks, surpassing existing methods by <strong>~27%</strong>.
</p>
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<h2 class="title is-3">Overview of GCA</h2>
<div class="content has-text-justified">
<img src="./static/images/overall_paradigm_page.jpg" alt="Overall Paradigm of GCA" style="width:100%; height:auto;">
<p>
We introduce Geometrically-Constrained Agent (GCA), a training-free agentic paradigm for geometrically-constrained spatial reasoning.
This strategy leverages a formal task constraint, C<sub>task</sub>, to decouple the reasoning process into two stages:
</p>
<p>
1. <strong>Task Formalization.</strong> The VLM, acting as a semantic analyst, translates the ambiguous query and visual data into the formal, verifiable task constraint C<sub>task</sub>. This stage defines what to solve, establishing immutable sub-constraints: a reference frame constraint and an objective constraint. <br>
2. <strong>Constrained Geometric Computation.</strong> The VLM then, acting as a task solver, generates and executes tool calls to compute the final answer, operating strictly within the deterministic bounds defined by C<sub>task</sub>.
</p>
<img src="./static/images/ref_frame_page.jpg" alt="Overall Paradigm of GCA" style="display: block; margin: 0 auto; width: 75%; height: auto;">
<p>
Formal Task Constraint C<sub>task</sub> bridges the semantic-to-geometric gap, which contains two key sub-constraints: <br>
1. <strong>Reference Frame Constraint</strong> that defines the coordinate system for answering the query. <br>
2. <strong>Objective Constraint</strong> that specifies the objective to be measured within that frame.
</p>
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<h2 class="title is-3">Performance of GCA</h2>
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<p>
<strong>Performance across VLMs</strong> GCA proves to be a robust architectural solution, achieving an average of <strong>37%</strong> relative improvement across all tested models.
Notably, the magnitude of enhancement correlates with the base VLM's agentic proficiency, unlocking up to <strong>49%</strong> gain on stronger models like Gemini-2.5-Pro. <br>
<strong>Impact of Task Constraint</strong> Simply prompting the VLM or using unconstrained tools ("Tool (Prompt)") yields negligible improvements.
In contrast, our GCA delivers a substantial performance boost.
</p>
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<img src="./static/images/generalizability_ablation_page.jpg" alt="Right Image Description" style="width:100%; height:auto;">
</div>
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<img src="./static/images/formalize_ablation_page.jpg" alt="Left Image Description" style="width:90%; height:auto;">
</div>
</div>
<p>
<strong>Error Attribution and Failure Modes</strong> A key advantage of GCA is its interpretable reasoning pathway. We conduct a detailed error attribution analysis to identify bottlenecks.
Results indicate that <strong>30%</strong> of errors stem from the initial Task Formalization stage, while the remaining <strong>70%</strong> occur during the Geometric Computation stage.
</p>
<img src="./static/images/failure_case_page.jpg" alt="Overall Paradigm of GCA" style="width:100%; height:auto;">
</div>
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</section>
<section class="section" id="BibTeX">
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<h2 class="title">BibTeX</h2>
<pre><code>
@article{chen2025geometrically,
title={Geometrically-Constrained Agent for Spatial Reasoning},
author={Zeren, Chen and Xiaoya, Lu and Zhijie, Zheng and Pengrui, Li and Lehan, He and Yijin, Zhou and Jing, Shao and Bohan, Zhuang and Lu, Sheng},
journal={arXiv preprint arXiv:2511.22659},
year={2025}
}
</code></pre>
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