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

arXiv:2606.11780v1 Announce Type: cross 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 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.

Source: arXiv cs.AI | 2026-06-11