Geometric Framework Unifies Differentiable Embedding Diagnostics

This paper presents a comprehensive geometric framework designed to help analysts determine the trustworthiness of nonlinear dimensionality reduction embeddings. Existing diagnostic methods, such as projection glyphs, map-continuity scores, and transport-based analyses, have previously appeared disparate. This new framework unifies these approaches by showing they all derive from a single geometric object inherent to any differentiable embedding, whether implicitly or explicitly defined. The framework offers two complementary perspectives: a differential view that explains local behavior, with its first-order term recovering projection glyphs and its second-order curvature quantifying the reliability of linear approximations. The integral view, on the other hand, tracks the same geometry along high-dimensional paths, revealing whether an embedding's output depends solely on the current state or also on the path taken to reach it. The research proves that map-continuity is a prerequisite for other analyses and demonstrates that the integral view captures path-dependent inconsistencies that local measurements cannot.