Geometric Stability: The Missing Axis of Representations

• URL: http://arxiv.org/abs/2601.09173v2
• Date: Mon, 19 Jan 2026 18:16:24 GMT
• Title: Geometric Stability: The Missing Axis of Representations
• Authors: Prashant C. Raju,
• Abstract summary: We introduce $geometric$ $stability, a distinct dimension that quantifies how reliably representational geometry holds under perturbation.<n>Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated.<n>For safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2$times$ more sensitively than CKA.<n>For model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Analysis of learned representations has a blind spot: it focuses on $similarity$, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce $geometric$ $stability$, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present $Shesha$, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated ($ρ\approx 0.01$) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2$\times$ more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability ($ρ= 0.89$-$0.96$); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying $how$ $reliably$ systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

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