AI RESEARCH
$\mathbb{R}^{2k}$ is Theoretically Large Enough for Embedding-based Top-$k$ Retrieval
arXiv CS.LG
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ArXi:2601.20844v3 Announce Type: replace This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison. Our result shows MED is $\Theta(k)$, independent of $m$, for inner product, Euclidean distance, and cosine similarity. We then consider Robust MED (RMED), where all vectors are unit normed and an $\epsilon$ gap of scores is required.