Geometry-Guided Constraint Learning for LLM Safety Classification

Fumiaki Uehara; Koo Imai; Masato Tsutsumi; Keigo Kansa; Sora Usui; Yuki Kobiyama

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arXiv · 2607.19366

Geometry-Guided Constraint Learning for LLM Safety Classification

Fumiaki Uehara·Koo Imai·Masato Tsutsumi·Keigo Kansa·Sora Usui·Yuki Kobiyama

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Abstract

Safety as Polytope (SaP) learns linear half-space constraints in LLM hidden space but requires per-category tuning of the constraint count K. We show that sparse autoencoder (SAE) feature extraction resolves this: K=2 becomes optimal for 12/14 categories on Qwen3.5-9B, achieving 96-99% accuracy per category on our BeaverTails classification benchmark, largely eliminating the need for exhaustive sweeps (K=4-25 with random initialization). This convergence to two planes is consistent with the Linear Representation Hypothesis, providing suggestive evidence that safety boundaries in this setting admit a low-dimensional linear description in the SAE feature space. Building on this geometric perspective, we introduce a cone constraint whose learnable aperture adapts to each category's cluster concentration, stabilized by a three-phase training

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Fumiaki Uehara, Koo Imai, Masato Tsutsumi, Keigo Kansa, Sora Usui, Yuki Kobiyama. 2026-06-09. Geometry-Guided Constraint Learning for LLM Safety Classification. https://arxiv.org/abs/2607.19366

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