基本信息
- 来源: arxiv
- 原始来源: https://arxiv.org/abs/2601.20844v1
- 作者: Zihao Wang, Hang Yin, Lihui Liu, Hanghang Tong, Yangqiu Song, Ginny Wong, Simon See
- 分类: cs.LG
- 论文时间: 2026-01-28T18:45:43Z
- 论文 PDF: https://arxiv.org/pdf/2601.20844v1.pdf
来源摘要/节选
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.
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