Geometric Stability: The Missing Axis of Representations

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.

💡 Research Summary

The paper introduces “geometric stability” as a complementary axis to the widely used similarity metrics (RSA, CKA, PWCCA) for analyzing learned representations. While similarity measures how closely a representation aligns with an external reference, it tells nothing about the robustness of the internal geometry under perturbations, resampling, or context shifts. To fill this gap the authors propose Shesha, a framework that quantifies internal consistency of a representation by comparing Representational Dissimilarity Matrices (RDMs) derived from disjoint feature subsets (Feature‑Split Shesha) or disjoint data subsets (Sample‑Split Shesha). When task labels are available, Supervised Shesha evaluates robustness to label‑guided perturbations.

The authors evaluate Shesha across 2,463 encoder configurations spanning seven domains: language, vision, audio, video, neuroscience, protein sequence, and single‑cell molecular data. For each configuration they compute Shesha (stability) and CKA (similarity) to a domain‑specific reference. The aggregate Spearman correlation is ρ ≈ −0.01 (95 % CI