# Geometric Structure Predicts Per-Example Neighborhood Loss Under PCA Truncation

arXiv: https://arxiv.org/abs/2610.17742 · HTML: https://mlsanity.com/paper/2610.17742/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 34 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 17 of 17)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models](https://mlsanity.com/paper/2104.08663/) (2021)
- [MPNet: Masked and Permuted Pre-training for Language Understanding](https://mlsanity.com/paper/2004.09297/) (2020)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Towards General Text Embeddings with Multi-stage Contrastive Learning](https://mlsanity.com/paper/2308.03281/) (2023)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [All-but-the-Top: Simple and Effective Postprocessing for Word Representations](https://mlsanity.com/paper/1702.01417/) (2017)
- [All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality](https://mlsanity.com/paper/2109.04404/) (2021)
- [The geometry of hidden representations of large transformer models](https://mlsanity.com/paper/2302.00294/) (2023)
- [Simple and Effective Dimensionality Reduction for Word Embeddings](https://mlsanity.com/paper/1708.03629/) (2017)
- [Shared Global and Local Geometry of Language Model Embeddings](https://mlsanity.com/paper/2503.21073/) (2025)
- [Factors Influencing the Surprising Instability of Word Embeddings](https://mlsanity.com/paper/1804.09692/) (2018)
- [Redundancy, Isotropy, and Intrinsic Dimensionality of Prompt-based Text Embeddings](https://mlsanity.com/paper/2506.01435/) (2025)
- [Randomly Removing 50% of Dimensions in Text Embeddings has Minimal Impact on Retrieval and Classification Tasks](https://mlsanity.com/paper/2508.17744/) (2025)
- [Investigating the Frequency Distortion of Word Embeddings and Its Impact on Bias Metrics](https://mlsanity.com/paper/2211.08203/) (2022)

## Abstract (arXiv)

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 replicate, and a direct variance-restriction test leaves that null unresolved. Average compression performance conceals systematic per-example variation that geometry makes visible.