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 realistic 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.

翻译：很多深度学习系统与多智能体系统（包括推荐系统）会以向量嵌入的形式学习偏好的空间模型。这些模型通常被假定近似欧几里得结构，即个体更偏好按照欧几里得度量测得的、与其"理想点"距离更近的备选方案。然而，Bogomolnaia与Laslier（2007）指出：当欧几里得空间的维度比个体数或备选方案数少两个时，存在无法用该结构表示的序数偏好剖面。我们将此结论加以推广，证明在实际情况中几乎所有偏好剖面都无法用欧几里得模型表示，并推导出使用欧几里得模型近似非欧几里得偏好剖面时预期误差的理论下界。该结果对向量嵌入的解释与应用具有启示意义——若欲实现任意真实序数关系的精确近似，嵌入向量的维度须达到被表征实体数量的较大比例。