Geometry of Rényi Entropy on the Majorization Lattice
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918518496165888 |
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| author | Yadav, Anuj Kumar Shkel, Yanina Y. |
| author_facet | Yadav, Anuj Kumar Shkel, Yanina Y. |
| contents | 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 $α\in \{0\} \,\cup \, [1,\infty]$. For the Tsallis entropy, we show that it also satisfies subadditivity on the majorization lattice, for every order $α\in [0,\infty)$. Finally, we show that, unlike the Rényi entropy, the Tsallis entropy is supermodular on the majorization lattice for every $α\in [0,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09655 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry of Rényi Entropy on the Majorization Lattice Yadav, Anuj Kumar Shkel, Yanina Y. Information Theory Combinatorics Probability 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 $α\in \{0\} \,\cup \, [1,\infty]$. For the Tsallis entropy, we show that it also satisfies subadditivity on the majorization lattice, for every order $α\in [0,\infty)$. Finally, we show that, unlike the Rényi entropy, the Tsallis entropy is supermodular on the majorization lattice for every $α\in [0,\infty)$. |
| title | Geometry of Rényi Entropy on the Majorization Lattice |
| topic | Information Theory Combinatorics Probability |
| url | https://arxiv.org/abs/2605.09655 |