Geometry of Rényi Entropy on the Majorization Lattice
This provides a fundamental structural property of Rényi entropy on the majorization lattice, relevant to fields like information theory and probability, but the results are theoretical and incremental.
The paper establishes that Rényi entropy is subadditive on the majorization lattice for all orders α ∈ [0,∞], and supermodular for α ∈ {0} ∪ [1,∞], by relating comonotone and independent couplings of marginal distributions.
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 Rényi 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 $ α\in [0,\infty] $, the Rényi entropy is subadditive on the majorization lattice. We further characterize the supermodular regime, showing that Rényi entropy is supermodular on the majorization lattice for $ α\in \{0\} \cup [1,\infty] $.