Error in the Euclidean Preference Model

TL;DR

This paper investigates the limitations of Euclidean models in representing preference profiles, demonstrating that many preferences cannot be accurately captured unless the embedding space's dimensionality is sufficiently large.

Contribution

It extends previous results by showing most preference profiles are incompatible with Euclidean models and provides a lower bound on the expected approximation error.

Findings

Most preference profiles cannot be represented with Euclidean models.

A theoretical lower bound on the approximation error is derived.

High-dimensional embeddings are often necessary for accurate preference modeling.

Abstract

Spatial models of preference, in the form of vector embeddings, are learned by many deep learning and multiagent systems, including recommender systems. Often these models are assumed to approximate a Euclidean structure, where an individual prefers alternatives positioned closer to their "ideal point", as measured by the Euclidean metric. However, Bogomolnaia and Laslier (2007) showed that there exist ordinal preference profiles that cannot be represented with this structure if the Euclidean space has two fewer dimensions than there are individuals or alternatives. We extend this result, showing that there are situations in which almost all preference profiles cannot be represented with the Euclidean model, and derive a theoretical lower bound on the expected error when using the Euclidean model to approximate non-Euclidean preference profiles. Our results have implications for the interpretation and use of vector embeddings, because in some cases close approximation of arbitrary, true ordinal relationships can be expected only if the dimensionality of the embeddings is a substantial fraction of the number of entities represented.