# Metric properties of homogeneous and spatially inhomogeneous F-divergences

## TL;DR

This paper explores the properties of F-divergences derived from entropy-transport problems, demonstrating how certain choices lead to metric properties and including well-known divergences like Jensen-Shannon.

## Contribution

It introduces the marginal perspective cost function H, analyzes its metric properties, and connects it to classical divergences and the Matusita divergences within the entropy-transport framework.

## Key findings

- H produces symmetric divergences in the entropic case
- Certain F-divergences like Jensen-Shannon are analyzed for metric properties
- For p>1, the induced cost H_p is the square of a metric on a cone space

## Abstract

In this paper I investigate the construction and the properties of the so-called marginal perspective cost $H$, a function related to Optimal Entropy-Transport problems obtained by a minimizing procedure, involving a cost function $c$ and an entropy function. In the pure entropic case, which corresponds to the choice $c=0$, the function $H$ naturally produces a symmetric divergence. I consider various examples of entropies and I compute the induced marginal perspective function, which includes some well-known functionals like the Hellinger distance, the Jensen-Shannon divergence and the Kullback-Liebler divergence. I discuss the metric properties of these functions and I highlight the important role of the so-called Matusita divergences. In the entropy-transport case, starting from the power like entropy $F_p(s)=(s^p-p(s-1)-1)/(p(p-1))$ and the cost $c=d^2$ for a given metric $d$, the main result of the paper ensures that for every $p>1$ the induced marginal perspective cost $H_p$ is the square of a metric on the corresponding cone space.