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

$\mathbb{R}^{2k}$ is Theoretically Large Enough for Embedding-based Top-$k$ Retrieval

arXiv:2601. 20844v3 Announce Type: replace-cross Abstract: This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison.

A Power Law in Logarithm's Clothing: On the Scalability of Graph-Based Vector Search

arXiv:2609. 02143v1 Announce Type: cross Abstract: Most vector databases rely on graph-based indexes, notably HNSW and Vamana, for approximate nearest neighbor search.

Sharp Capacity Thresholds in Linear Associative Memory: From Top-1 Retrieval to Tail-Average Learning

arXiv:2605. 05189v2 Announce Type: replace-cross Abstract: How many key-value associations can a $d\times d$ linear memory store?

Binary and Scalar Embedding Quantization for Significantly Faster & Cheaper Retrieval

Exact Limits of Random Projections for Preserving Geometry: Distance Recovery, Nearest-Neighbor Rankings, and Covariance Shape in Gaussian Models

arXiv:2609. 02155v1 Announce Type: new Abstract: The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal.

Price of metric universality in vector quantization is at most 0.11 bit

arXiv:2602. 05790v2 Announce Type: replace-cross Abstract: Fast computation of a matrix product $W^\top X$ is a workhorse of modern LLMs.