# Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth

**Authors:** Yihe Dong, Jean-Baptiste Cordonnier, Andreas Loukas  
**arXiv:** [2103.03404](https://arxiv.org/abs/2103.03404)  
**Date:** 2021-03-05  
**Summary page:** https://www.summarizepaper.com/fr/paper/2103.03404v2/

## Abstract

Attention-based architectures have become ubiquitous in machine learning, yet our understanding of the reasons for their effectiveness remains limited. This work proposes a new way to understand self-attention networks: we show that their output can be decomposed into a sum of smaller terms, each involving the operation of a sequence of attention heads across layers. Using this decomposition, we prove that self-attention possesses a strong inductive bias towards "token uniformity". Specifically, without skip connections or multi-layer perceptrons (MLPs), the output converges doubly exponentially to a rank-1 matrix. On the other hand, skip connections and MLPs stop the output from degeneration. Our experiments verify the identified convergence phenomena on different variants of standard transformer architectures.