Geometric Structure Predicts Per-Example Neighborhood Loss Under PCA Truncation

Sriram Loganathan, Gokul Anand, Ragavan Arivazhagan

Dimensionality reduction can preserve the average performance of text embeddings even when a substantial fraction of their dimensions is removed. Aggregate metrics, however, do not reveal whether degradation is distributed uniformly across examples. We study per-example neighborhood stability under PCA truncation across static word embeddings and modern sentence encoders. Per-item neighbor retention varies widely and is predictable from geometric properties of the representation and its alignment with the retained PCA subspace. The fraction of centered-vector energy retained by the truncated subspace is strongest for CBOW and RI, with neighborhood contrast leading on SkipGram and the modern/Matryoshka settings (GTE approximately tied), while local density is associated with retention in marginal analysis but adds little unique information once energy and contrast are controlled (R-squared increments 0.001 to 0.010). Local intrinsic dimensionality accounts for more variation than density on modern sentence encoders in pairwise comparison. On SciFact, query density predicts retrieval degradation under aggressive compression; on NFCorpus the primary 32d association does not…