High-Dimensional Simplexes for Supermetric Search

TL;DR

This paper introduces a method using high-dimensional simplexes in supermetric spaces to efficiently approximate distances, enabling faster similarity searches with reduced data size and minimal accuracy loss.

Contribution

It presents a novel approach to construct n-dimensional simplexes in supermetric spaces for tight distance bounds, improving search efficiency.

Findings

Significant reduction in data size and metric computation costs.

Maintains high accuracy in similarity search results.

Provides a scalable indexing method for supermetric spaces.

Abstract

In 1953, Blumenthal showed that every semi-metric space that is isometrically embeddable in a Hilbert space has the n-point property; we have previously called such spaces supermetric spaces. Although this is a strictly stronger property than triangle inequality, it is nonetheless closely related and many useful metric spaces possess it. These include Euclidean, Cosine and Jensen-Shannon spaces of any dimension. A simple corollary of the n-point property is that, for any (n+1) objects sampled from the space, there exists an n-dimensional simplex in Euclidean space whose edge lengths correspond to the distances among the objects. We show how the construction of such simplexes in higher dimensions can be used to give arbitrarily tight lower and upper bounds on distances within the original space. This allows the construction of an n-dimensional Euclidean space, from which lower and upper bounds of the original space can be calculated, and which is itself an indexable space with the n-point property. For similarity search, the engineering tradeoffs are good: we show significant reductions in data size and metric cost with little loss of accuracy, leading to a significant overall improvement in search performance.