Negative type diversities, a multi-dimensional analogue of negative type metrics

TL;DR

This paper extends the concept of negative type metrics to diversities, providing new characterizations and demonstrating their relevance to embeddings and approximation algorithms in a multi-dimensional setting.

Contribution

It introduces negative type diversities, offers geometric characterizations, and connects them to $L_1$-embeddable diversities, extending metric embedding results to diversities.

Findings

Negative type diversities generalize $L_1$-embeddable diversities.

Geometric characterizations of negative type diversities are provided.

Lower bounds for embeddings into $L_1$ extend from metrics to diversities.

Abstract

Diversities are a generalization of metric spaces in which a non-negative value is assigned to all finite subsets of a set, rather than just to pairs of points. Here we provide an analogue of the theory of negative type metrics for diversities. We introduce negative type diversities, and show that, as in the metric space case, they are a generalization of $L_1$-embeddable diversities. We provide a number of characterizations of negative type diversities, including a geometric characterisation. Much of the recent interest in negative type metrics stems from the connections between metric embeddings and approximation algorithms. We extend some of this work into the diversity setting, showing that lower bounds for embeddings of negative type metrics into $L_1$ can be extended to diversities by using recently established extremal results on hypergraphs.