Constant distortion embeddings of Symmetric Diversities

TL;DR

This paper proves that symmetric diversities, a generalization of metric spaces where values depend only on set size, can be embedded into L1 space with constant distortion, unlike the logarithmic bounds in metric spaces.

Contribution

It establishes the first constant distortion embedding result for symmetric diversities into L1 spaces, advancing the understanding of diversity embeddings.

Findings

Symmetric diversities can be embedded into L1 with constant distortion.

Optimal distortion bounds for general diversities remain unknown.

The result contrasts with the logarithmic bounds known for metric spaces.

Abstract

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of $L_1$ spaces. In the metric case, it is well known that an $n$-point metric space can be embedded into $L_1$ with $\mathcal{O}(\log n)$ distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in $L_1$ with constant distortion.