Is the Chen-Sbert Divergence a Metric?

TL;DR

This paper investigates whether the Chen-Sbert divergence measure qualifies as a metric, providing proofs for specific cases and discussing potential generalizations, aiming to deepen understanding of its mathematical properties.

Contribution

It offers initial proofs for the divergence being a metric in 2-letter cases and discusses pathways for extending these results to more complex alphabets.

Findings

Proved the divergence is a metric for 2-letter alphabet.

Discussed potential proofs for n-letter cases.

Stimulated further research into the divergence's properties.

Abstract

Recently, Chen and Sbert proposed a general divergence measure. This report presents some interim findings about the question whether the divergence measure is a metric or not. It has been postulated that (i) the measure might be a metric when (0 < k <= 1), and (ii) the k-th root of the measure might be a metric when (k > 1). The report shows that for a 2-letter alphabet, postulation (i) can be proved. The possible pathway for obtaining a proof for (i) in n-letter cases is also discussed. The authors hope that the report may stimulate more scholarly effort to study the mathematical properties of this divergence measure.