Infinity Search: Approximate Vector Search with Projections on q-Metric Spaces

TL;DR

This paper introduces a novel approximate vector search method using projections onto q-metric and ultrametric spaces, leveraging learned approximations to balance search speed and accuracy.

Contribution

It proposes a new approach to approximate vector search by transforming dissimilarity functions into ultrametric spaces via learned projections, enabling efficient search.

Findings

Increasing q improves search speed but reduces recall.

The method is competitive with existing vector search techniques.

Learned projections effectively approximate ultrametric distances.

Abstract

An ultrametric space or infinity-metric space is defined by a dissimilarity function that satisfies a strong triangle inequality in which every side of a triangle is not larger than the larger of the other two. We show that search in ultrametric spaces with a vantage point tree has worst-case complexity equal to the depth of the tree. Since datasets of interest are not ultrametric in general, we employ a projection operator that transforms an arbitrary dissimilarity function into an ultrametric space while preserving nearest neighbors. We further learn an approximation of this projection operator to efficiently compute ultrametric distances between query points and points in the dataset. We proceed to solve a more general problem in which we consider projections in $q$-metric spaces -- in which triangle sides raised to the power of $q$ are smaller than the sum of the $q$-powers of the other two. Notice that the use of learned approximations of projected $q$-metric distances renders the search pipeline approximate. We show in experiments that increasing values of $q$ result in faster search but lower recall. Overall, search in q-metric and infinity metric spaces is competitive with existing search methods.