Geometry of R\'enyi Entropy on the Majorization Lattice

TL;DR

This paper explores the mathematical properties of Rénnyi entropy within the structure of the majorization lattice, revealing subadditivity and supermodularity regimes across different orders.

Contribution

It establishes a fundamental relation between comonotone and independent couplings, and characterizes the supermodular regime of Rénnyi entropy on the majorization lattice.

Findings

Rénnyi entropy is subadditive for all order α in [0, ∞].

Rénnyi entropy is supermodular for α in {0} ∪ [1, ∞].

The work links coupling concepts with entropy properties on the lattice.

Abstract

Majorization is a stochastic ordering relation that compares the relative diversity of probability distributions with numerous applications in econometrics, spectral theory, and ecology. It is well-known that the majorization partial order forms a complete lattice on the set of ordered probability distributions. In this work, we study the properties of R\'enyi entropy on the majorization lattice. We establish a fundamental relation between the comonotone coupling and the independent coupling associated with a collection of marginal distributions. Consequently, we show that, for every order $ \alpha \in [0,\infty] $, the R\'enyi entropy is subadditive on the majorization lattice. We further characterize the supermodular regime, showing that R\'enyi entropy is supermodular on the majorization lattice for $ \alpha \in \{0\} \cup [1,\infty] $.