Quasi-metric spaces with measure

TL;DR

This paper extends the concept of measure concentration from metric spaces to quasi-metric spaces with probability measures, motivated by biological sequence comparison, and shows high-dimensional pq-spaces resemble mm-spaces.

Contribution

It introduces the pq-space framework, generalizing mm-spaces to quasi-metrics, and demonstrates their similarity in high dimensions, with applications to biological data.

Findings

Many biological sequence similarity measures can be converted to quasi-metrics.

High-dimensional pq-spaces are nearly mm-spaces.

The framework bridges biological sequence analysis and geometric measure theory.

Abstract

The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a Borel measure, also called an mm-space. We extend some of the mm-space concepts to the setting of a quasi-metric space with probability measure (pq-space). Our motivation comes from biological sequence comparison: we show that many common similarity measures on biological sequences can be converted to quasi-metrics. We show that a high dimensional pq-space is very close to being an mm-space.