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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 ","title":"What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study","url":"https://arxiv.org/abs/2606.11780","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.11780v1 Announce Type: cross \nAbstract: 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 $N$. 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