Metric distances derived from cosine similarity and Pearson and Spearman correlations

TL;DR

This paper explores how to transform cosine similarity and correlation measures into metric distances using metric-preserving functions, enabling better analysis of object relationships in data.

Contribution

It introduces two classes of transformations, including a novel sine-based metric, expanding the toolkit for converting correlation measures into metric distances.

Findings

The first class separates anti-correlated objects maximally.

The second class groups correlated and anti-correlated objects.

The sine function provides a valid metric distance for centered data.

Abstract

We investigate two classes of transformations of cosine similarity and Pearson and Spearman correlations into metric distances, utilising the simple tool of metric-preserving functions. The first class puts anti-correlated objects maximally far apart. Previously known transforms fall within this class. The second class collates correlated and anti-correlated objects. An example of such a transformation that yields a metric distance is the sine function when applied to centered data.