Fubini Study geometry of representation drift in high dimensional data

TL;DR

This paper introduces a geometric framework based on the Fubini Study metric to measure intrinsic representation drift in high-dimensional data, effectively separating genuine structural changes from parametrization artifacts.

Contribution

It proposes a projective geometric approach to quantify representation drift, invariant under gauge transformations, and demonstrates its advantages over traditional Euclidean and cosine metrics.

Findings

Fubini Study metric isolates intrinsic evolution of representations.

Conventional metrics overestimate change due to gauge ambiguities.

The approach provides a diagnostic to distinguish meaningful structural evolution from artifacts.

Abstract

High dimensional representation drift is commonly quantified using Euclidean or cosine distances, which presuppose fixed coordinates when comparing representations across time, training or preprocessing stages. While effective in many settings, these measures entangle intrinsic changes in the data with variations induced by arbitrary parametrizations. We introduce a projective geometric view of representation drift grounded in the Fubini Study metric, which identifies representations that differ only by gauge transformations such as global rescalings or sign flips. Applying this framework to empirical high dimensional datasets, we explicitly construct representation trajectories and track their evolution through cumulative geometric drift. Comparing Euclidean, cosine and Fubini Study distances along these trajectories reveals that conventional metrics systematically overestimate change whenever representations carry genuine projective ambiguity. By contrast, the Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations. We further show that the difference between cosine and Fubini Study drift defines a computable, monotone quantity that directly captures representation churn attributable to gauge freedom. This separation provides a diagnostic for distinguishing meaningful structural evolution from parametrization artifacts, without introducing model-specific assumptions. Overall, we establish a geometric criterion for assessing representation stability in high-dimensional systems and clarify the limits of angular distances. Embedding representation dynamics in projective space connects data analysis with established geometric programs and yields observables that are directly testable in empirical workflows.