# Isotropy, Clusters, and Classifiers

arXiv: https://arxiv.org/abs/2402.03191 · HTML: https://mlsanity.com/paper/2402.03191/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 44 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [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)
- [Measuring the Mixing of Contextual Information in the Transformer](https://mlsanity.com/paper/2203.04212/) (2022)
- [IsoScore: Measuring the Uniformity of Embedding Space Utilization](https://mlsanity.com/paper/2108.07344/) (2021)
- [Towards Opening the Black Box of Neural Machine Translation: Source and Target Interpretations of the Transformer](https://mlsanity.com/paper/2205.11631/) (2022)
- [GlobEnc: Quantifying Global Token Attribution by Incorporating the Whole Encoder Layer in Transformers](https://mlsanity.com/paper/2205.03286/) (2022)
- [A Cluster-based Approach for Improving Isotropy in Contextual Embedding Space](https://mlsanity.com/paper/2106.01183/) (2021)
- [Learning to Remove: Towards Isotropic Pre-trained BERT Embedding](https://mlsanity.com/paper/2104.05274/) (2021)
- [How Does Fine-tuning Affect the Geometry of Embedding Space: A Case Study on Isotropy](https://mlsanity.com/paper/2109.04740/) (2021)
- [How to Dissect a Muppet: The Structure of Transformer Embedding Spaces](https://mlsanity.com/paper/2206.03529/) (2022)
- [Semeval-2022 Task 1: CODWOE -- Comparing Dictionaries and Word Embeddings](https://mlsanity.com/paper/2205.13858/) (2022)
- [Exploring Anisotropy and Outliers in Multilingual Language Models for Cross-Lingual Semantic Sentence Similarity](https://mlsanity.com/paper/2306.00458/) (2023)
- [Isotropic Representation Can Improve Dense Retrieval](https://mlsanity.com/paper/2209.00218/) (2022)
- [Token-wise Decomposition of Autoregressive Language Model Hidden States for Analyzing Model Predictions](https://mlsanity.com/paper/2305.10614/) (2023)
- [Leverage Points in Modality Shifts: Comparing Language-only and Multimodal Word Representations](https://mlsanity.com/paper/2306.02348/) (2023)
- [Why bother with geometry? On the relevance of linear decompositions of Transformer embeddings](https://mlsanity.com/paper/2310.06977/) (2023)

## Built on by (most cited first, 4 of 4)

- [Meaning at the Planck scale? Contextualized word embeddings for doing history, philosophy, and sociology of science](https://mlsanity.com/paper/2411.14073/) (2024)
- [Disentangling Geometry, Performance, and Training in Language Models](https://mlsanity.com/paper/2602.20433/) (2026)
- [Revisiting Anisotropy in Language Transformers: The Geometry of Learning Dynamics](https://mlsanity.com/paper/2604.08764/) (2026)
- [Correcting Mean Bias in Text Embeddings: A Refined Renormalization with Training-Free Improvements on MMTEB](https://mlsanity.com/paper/2511.11041/) (2025)

## Abstract (arXiv)

Whether embedding spaces use all their dimensions equally, i.e., whether they are isotropic, has been a recent subject of discussion. Evidence has been accrued both for and against enforcing isotropy in embedding spaces. In the present paper, we stress that isotropy imposes requirements on the embedding space that are not compatible with the presence of clusters -- which also negatively impacts linear classification objectives. We demonstrate this fact both mathematically and empirically and use it to shed light on previous results from the literature.