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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.
