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

• URL: http://arxiv.org/abs/2606.11780v1
• Date: Wed, 10 Jun 2026 08:11:41 GMT
• Title: What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study
• Abstract summary: We show that every $k$-subset $S subseteq [N]$ is realizable as a result of top-$k$ retrieval by some query vector.<n>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.
• Score: 0.25782420501870296
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 $Bd = Ω(k \ln N)$; 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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