# Index-based, High-dimensional, Cosine Threshold Querying with Optimality Guarantees

arXiv: https://arxiv.org/abs/1812.07695 · HTML: https://mlsanity.com/paper/1812.07695/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2018.

## Built on by (most cited first, 3 of 3)

- [Efficiency and Effectiveness of SPLADE Models on Billion-Scale Web Document Title](https://mlsanity.com/paper/2511.22263/) (2025)
- [Sparse Attention as a Range Searching Problem: Towards an Inference-Efficient Index for KV Cache](https://mlsanity.com/paper/2605.06763/) (2026)
- [Accelerating spherical K-means clustering for large-scale sparse document data](https://mlsanity.com/paper/2411.11300/) (2024)

## Abstract (arXiv)

Given a database of vectors, a cosine threshold query returns all vectors in the database having cosine similarity to a query vector above a given threshold θ. These queries arise naturally in many applications, such as document retrieval, image search, and mass spectrometry. The present paper considers the efficient evaluation of such queries, providing novel optimality guarantees and exhibiting good performance on real datasets. We take as a starting point Fagin's well-known Threshold Algorithm (TA), which can be used to answer cosine threshold queries as follows: an inverted index is first built from the database vectors during pre-processing; at query time, the algorithm traverses the index partially to gather a set of candidate vectors to be later verified for θ-similarity. However, directly applying TA in its raw form misses significant optimization opportunities. Indeed, we first show that one can take advantage of the fact that the vectors can be assumed to be normalized, to obtain an improved, tight stopping condition for index traversal and to efficiently compute it incrementally. Then we show that one can take advantage of data skewness to obtain better traversal strategies. In particular, we show a novel traversal strategy that exploits a common data skewness condition which holds in multiple domains including mass spectrometry, documents, and image databases. We show that under the skewness assumption, the new traversal strategy has a strong, near-optimal performance guarantee. The techniques developed in the paper are quite general since they can be applied to a large class of similarity functions beyond cosine.