几何稳定性：表征中缺失的坐标轴

Geometric Stability: The Missing Axis of Representations

January 14, 2026

作者: Prashant C. Raju

cs.AI

摘要

對學習表徵的分析存在一個盲點：當前研究主要聚焦於相似性度量，即衡量嵌入向量與外部參考標準的對齊程度，但相似性僅能揭示表徵內容，無法反映表徵結構的穩健性。本文提出「幾何穩定性」這一全新維度，用於量化表徵幾何在干擾下的保持可靠性，並開發了測量框架Shesha。通過對七個領域2,463種配置的實驗表明：穩定性與相似性實證上無相關性（ρ≈0.01）且機制相異——移除頂部主成分後相似性度量會失效，而穩定性仍能敏感捕捉細粒度流形結構。這種區別具有實踐指導價值：在安全監控方面，穩定性可作為功能性幾何預警指標，其檢測結構漂移的靈敏度比CKA提升近2倍，同時能過濾觸發剛性距離度量誤報的非功能性噪聲；在可控性方面，監督式穩定性可預測線性可控性（ρ=0.89-0.96）；在模型選擇方面，穩定性與遷移性呈現解耦現象，揭示了遷移優化產生的幾何代價。超越機器學習領域，穩定性還能預測CRISPR干擾一致性和神經-行為耦合強度。通過量化系統維持結構的可靠性，幾何穩定性為審計生物與計算系統的表徵提供了相似性度量之外的必要補充維度。

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.