几何稳定性：表征理论中缺失的坐标轴

Geometric Stability: The Missing Axis of Representations

January 14, 2026

作者: Prashant C. Raju

cs.AI

摘要

学习表征分析存在一个盲区：当前方法聚焦于相似性度量，即衡量嵌入向量与外部参照的匹配程度，但相似性仅能揭示表征内容，无法判断结构是否稳健。我们提出几何稳定性这一全新维度，用于量化表征几何在扰动下的保持可靠性，并推出测量框架Shesha。通过对七大领域2463种配置的实验，我们发现稳定性与相似性在经验上无关（ρ≈0.01）且机制迥异：移除主成分后相似性度量会失效，而稳定性仍能敏感捕捉细粒度流形结构。这种差异具有实践价值：在安全监控方面，稳定性可作为功能性几何预警指标，其检测结构漂移的灵敏度比CKA提高近2倍，同时能过滤刚性距离度量中引发误报的非功能性噪声；在可控性方面，监督式稳定性可预测线性导向能力（ρ=0.89-0.96）；在模型选择方面，稳定性与可迁移性解耦，揭示了迁移优化所产生的几何代价。超越机器学习领域，稳定性还能预测CRISPR扰动一致性与神经-行为耦合度。通过量化系统维持结构的可靠性，几何稳定性为生物与计算系统的表征审计提供了相似性度量之外的必要补充维度。

English

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 (ρapprox 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 2times 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.