High-Dimensional Geometric Robustness in Representation

Abstract Contribution Type: This paper provides theoretical characterization of a phenomenon implicit in ML practice but not rigorously established—the geometric fate of normalized signal-plus-noise vectors. We clarify a persistent confusion between two distinct concepts: the 50% Angle Limit (about absolute angles between random pairs) versus retrieval accuracy (about relative angles between structured vs. random pairs). Main Result: We prove the High-Dimensional Vector Normalization Statistical Invariance Theorem: normalized signal-plus-noise vectors converge to a uniform distribution on the unit sphere, independent of noise intensity and type. This establishes a fundamental limit—directional information becomes irrecoverable under extreme noise regardless of noise characteristics. Why this matters: Classical concentration-of-measure theory describes how random vectors distribute on the sphere, but does not characterize the noise-intensity invariance we prove. Standard concentration bounds depend on variance alpha^2; our theorem shows the normalized distribution is identical across different alpha. This identity is not derivable from existing frameworks. Boundary Condition: Detection remains possible via magnitude-dependent statistics if and only if distributional asymmetry exists between target and distractors. In symmetric settings, performance collapses to chance—a clean theoretical boundary. Validation: Experiments across dimensions (5–1024), noise types (Gaussian, Uniform, Laplace, Poisson), and acoustic channel transmission validate the theory. Code: https://github.com/Winamin/car-complete-demo/tree/noise-test