On Affine Homotopy between Language Encoders

TL;DR

This paper investigates affine homotopy as a measure of similarity between language encoders, showing it is informative of extrinsic task performance and helps understand the structure of encoder spaces.

Contribution

It introduces affine alignment as a meaningful, task-independent measure of encoder similarity and explores its implications for understanding encoder relationships.

Findings

Affine alignment is asymmetric but informative of extrinsic similarity.

Affine intrinsic similarity provides bounds on task performance.

Defines an order over pre-trained encoders based on affine similarity.

Abstract

Pre-trained language encoders -- functions that represent text as vectors -- are an integral component of many NLP tasks. We tackle a natural question in language encoder analysis: What does it mean for two encoders to be similar? We contend that a faithful measure of similarity needs to be \emph{intrinsic}, that is, task-independent, yet still be informative of \emph{extrinsic} similarity -- the performance on downstream tasks. It is common to consider two encoders similar if they are \emph{homotopic}, i.e., if they can be aligned through some transformation. In this spirit, we study the properties of \emph{affine} alignment of language encoders and its implications on extrinsic similarity. We find that while affine alignment is fundamentally an asymmetric notion of similarity, it is still informative of extrinsic similarity. We confirm this on datasets of natural language representations. Beyond providing useful bounds on extrinsic similarity, affine intrinsic similarity also allows us to begin uncovering the structure of the space of pre-trained encoders by defining an order over them.