# Anisotropy Decides Cosine vs. Rank Metrics for Text Embeddings

arXiv: https://arxiv.org/abs/2606.29571 · HTML: https://mlsanity.com/paper/2606.29571/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 50 references that are in the map.
- First posted on arXiv 28 Jun 2026.

## Builds on (most cited first, 30 of 33)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [Text Embeddings by Weakly-Supervised Contrastive Pre-training](https://mlsanity.com/paper/2212.03533/) (2022)
- [SemEval-2017 Task 1: Semantic Textual Similarity - Multilingual and Cross-lingual Focused Evaluation](https://mlsanity.com/paper/1708.00055/) (2017)
- [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)
- [MTEB: Massive Text Embedding Benchmark](https://mlsanity.com/paper/2210.07316/) (2022)
- [C-Pack: Packed Resources For General Chinese Embeddings](https://mlsanity.com/paper/2309.07597/) (2023)
- [LLM2Vec: Large Language Models Are Secretly Powerful Text Encoders](https://mlsanity.com/paper/2404.05961/) (2024)
- [PAWS: Paraphrase Adversaries from Word Scrambling](https://mlsanity.com/paper/1904.01130/) (2019)
- [Improving Text Embeddings with Large Language Models](https://mlsanity.com/paper/2401.00368/) (2023)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [All-but-the-Top: Simple and Effective Postprocessing for Word Representations](https://mlsanity.com/paper/1702.01417/) (2017)
- [Whitening Sentence Representations for Better Semantics and Faster Retrieval](https://mlsanity.com/paper/2103.15316/) (2021)
- [All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality](https://mlsanity.com/paper/2109.04404/) (2021)
- [BERT Busters: Outlier Dimensions that Disrupt Transformers](https://mlsanity.com/paper/2105.06990/) (2021)
- [Is Cosine-Similarity of Embeddings Really About Similarity?](https://mlsanity.com/paper/2403.05440/) (2024)
- [WhiteningBERT: An Easy Unsupervised Sentence Embedding Approach](https://mlsanity.com/paper/2104.01767/) (2021)
- [Outliers Dimensions that Disrupt Transformers Are Driven by Frequency](https://mlsanity.com/paper/2205.11380/) (2022)
- [IsoScore: Measuring the Uniformity of Embedding Space Utilization](https://mlsanity.com/paper/2108.07344/) (2021)
- [Exploring Anisotropy and Outliers in Multilingual Language Models for Cross-Lingual Semantic Sentence Similarity](https://mlsanity.com/paper/2306.00458/) (2023)

## Abstract (arXiv)

The standard way to compare two text embeddings is cosine similarity. Scattered studies report that a different metric does better, but never pin down the geometric condition that decides when, or why. We settle both with a comprehensive empirical study: nineteen parameter-free similarity metrics on nineteen encoders, from compact sentence transformers up to seven-billion-parameter large language models, across seven datasets. The answer is geometric. When an encoder spreads its variance evenly across directions, cosine is the best parameter-free choice and no other metric helps by a usable margin. When the variance concentrates into a few dominant directions, a property known as anisotropy, rank-based and L1-type metrics beat cosine by a clear margin. The absolute gain is modest, but because cosine starts low on these encoders it is a sizable relative improvement, around twenty percent on average and largest where cosine is weakest. What decides this is the geometry of the embedding space, not how the model was trained: where the two disagree, the metric follows the geometry. One number, the fraction of variance held by the single most dominant dimension, predicts how much the alternatives help across all nineteen encoders, with a rank correlation of 0.86 and a linear correlation of 0.95. To test this as the cause rather than a correlate, we project out the dominant directions: cosine recovers and the advantage of the other metrics nearly vanishes, but only on the encoders that were anisotropic to begin with. The effect is directional, not magnitude based, since it survives normalizing every vector to unit length. Among parameter-free metrics, then, cosine is the right tool wherever an encoder is well spread, which includes the fine-tuned embedders commonly deployed for retrieval, and we give a one-number diagnostic for when it is not.