Calculus of Variations and Geometric Measure Theory

N. De Ponti

Metric properties of homogeneous and spatially inhomogeneous F-divergences

created by deponti on 19 Feb 2019
modified on 11 Feb 2020


Accepted Paper

Inserted: 19 feb 2019
Last Updated: 11 feb 2020

Journal: IEEE Transactions on Information Theory
Year: 2019
Doi: 10.1109/TIT.2019.2937485


In this paper we investigate the construction and the properties of spatially inhomogeneous divergences, functionals arising from optimal Entropy-Transport problems that are computed in terms of an entropy function $F$ and a cost function. Starting from the power-like entropy $F(s)=(s^p−p(s−1)−1)/(p(p−1))$ and a suitable cost depending on a metric $\mathsf{d}$ on a space $X$, our main result ensures that for every $p>1$ the related inhomogeneous divergence induces a distance on the space of finite measures over $X$. We also study in detail the pure entropic setting, that can be recovered as a particular case when the transport is forbidden. In this situation, corresponding to the classical theory of $F$-divergences, we show that the construction naturally produces a symmetric divergence and we highlight the important role played by the class of Matusita's divergences.