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

The study establishes theoretical limits on embedding dimensionality for top-$k$ retrieval in dense vector databases under quantization constraints. It shows that while $d = O(k)$ is sufficient for infinite precision, with $B$ bits per coordinate, maintaining perfect top-$k$ retrieval requires $Bd = \Omega(k \ln N)$, indicating that dimension must increase logarithmically with corpus size $N$. This has implications for practitioners, as it suggests that both embedding dimension and quantization precision must scale with the size of the dataset, affecting the design and efficiency of retrieval systems.