Eigen Analysis of Self-Attention and its Reconstruction from Partial Computation

TL;DR

This paper analyzes the eigenstructure of self-attention matrices in transformers, revealing low-dimensional patterns and proposing a partial computation method to reduce complexity while maintaining accuracy.

Contribution

It introduces an eigen analysis of attention scores showing low-dimensional structure and proposes a partial computation approach for efficient attention estimation.

Findings

Most variation in attention scores lies in a low-dimensional eigenspace

Significant overlap of eigenspaces across layers and models

Partial score computation can effectively estimate full attention scores

Abstract

State-of-the-art transformer models use pairwise dot-product based self-attention, which comes at a computational cost quadratic in the input sequence length. In this paper, we investigate the global structure of attention scores computed using this dot product mechanism on a typical distribution of inputs, and study the principal components of their variation. Through eigen analysis of full attention score matrices, as well as of their individual rows, we find that most of the variation among attention scores lie in a low-dimensional eigenspace. Moreover, we find significant overlap between these eigenspaces for different layers and even different transformer models. Based on this, we propose to compute scores only for a partial subset of token pairs, and use them to estimate scores for the remaining pairs. Beyond investigating the accuracy of reconstructing attention scores themselves, we investigate training transformer models that employ these approximations, and analyze the effect on overall accuracy. Our analysis and the proposed method provide insights into how to balance the benefits of exact pair-wise attention and its significant computational expense.