Information Loss in Euclidean Preference Models

Spatial models of preference, in the form of vector embeddings, are learned by many deep learning 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 realistic situations in which almost all preference profiles cannot be represented with the Euclidean model, and derive a theoretical lower bound on the information lost when approximating non-representable preferences with the Euclidean model. Our results have implications for the interpretation and use of vector embeddings, because in some cases close approximation of arbitrary, true preferences is possible only if the dimensionality of the embeddings is a substantial fraction of the number of individuals or alternatives.