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arXiv · 2606.11780

What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study

Koki Okajima·Tsukasa Yoshida
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Abstract

We establish conditions for embedding a corpus of $N$ documents as $d$-dimensional vectors such that every $k$-subset $S \subseteq [N]$ is realizable as a result of top-$k$ retrieval by some query vector. Recent work shows that $d = O(k)$ suffices for such embeddings to exist in $\mathbb{R}^d$, independently of $N$. We theoretically prove that this corpus-independent bound is specific to infinite precision. With $B$ bits per coordinate, perfect top-$k$ retrieval requires $Bd = \Omega(k \ln N)$; thus, at any fixed precision, the dimension must grow at least logarithmically with $N$. Specializing to a $\ell_2$-normalized $B$-bit uniform scalar quantization model, we also identify a threshold on the precision $B^{*} = O(\ln \ln N)$ below which no dimension suffices, together with two further regimes that bound the feasible $(B, d)$ pairs. Our result implies that in practical vector databases and dense retrieval systems where quantization is standard, the embedding dimension and possibly the precision must grow with the corpus size.

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Koki Okajima, Tsukasa Yoshida. 2026-06-10. What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study. https://arxiv.org/abs/2606.11780

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