Title: Tight Bounds and Achievable Upper Bounds of Minimal Dimensions for Embedding-based Retrieval
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: representation learning, embedding-based retrieval

Abstract: This paper studies the minimal dimension required to embed subset memberships into vector spaces. 
The lower and upper bounds are derived theoretically and supported empirically for various notions of "distances" or "similarities", including $\ell_2$ metric, inner product, and cosine similarity.
Our results suggest no fundamental differences between those metrics in terms of Minimal Embeddable Dimension (MED).
In addition, we conduct experiments in the achievable setting, where we find that we can easily realize the logarithmic dependency between the MED and the number of objects to embed.
Our results also align well with existing practices in large language models, vector databases, and other related fields.