This paper studies the minimal dimension required to embed subset memberships ($m$ elements and ${m\choose k}$ subsets of at most $k$ elements) into vector spaces, denoted as Minimal Embeddable Dimension (MED). The tight bounds of MED are derived theoretically and supported empirically for various notions of "distances" or "similarities," including the $\ell_2$ metric, inner product, and cosine similarity. In addition, we conduct numerical simulation in a more achievable setting, where the ${m\choose k}$ subset embeddings are chosen as the centroid of the embeddings of the contained elements. Our simulation easily realizes a logarithmic dependency between the MED and the number of elements to embed. These findings imply that embedding-based retrieval limitations stem primarily from learnability challenges, not geometric constraints, guiding future algorithm design.

翻译：本文研究了将子集隶属关系（$m$个元素以及至多包含$k$个元素的${m\choose k}$个子集）嵌入向量空间所需的最小维度，称为最小可嵌入维度（MED）。我们理论上推导了MED的紧致界，并针对多种“距离”或“相似度”度量（包括$\ell_2$度量、内积和余弦相似度）进行了实证验证。此外，我们在一个更易实现的设定下进行了数值模拟，其中${m\choose k}$个子集的嵌入被选为其包含元素嵌入的质心。我们的模拟轻松实现了MED与待嵌入元素数量之间的对数依赖关系。这些发现表明，基于嵌入的检索限制主要源于可学习性挑战，而非几何约束，从而为未来算法设计提供了指导。