On Measures of Dissimilarity on the Majorization Simplex
Abstract
The majorization simplex 𝒫n is the set of probability vectors of length n whose entries are arranged in non-increasing order. Under the majorization order, 𝒫n is a lattice, and it is the natural space for comparing distributions when the labels of the outcomes are not relevant. In this work, we study measures of dissimilarity (and, similarity) on 𝒫n. Our first set of results concerns the Rényi-entropy based distance dα(p,q)=Hα(p)+Hα(q)−2Hα(p∨ q) proposed by Yadav and Shkel (2026), which generalizes a metric of Cicalese, Gargano and Vaccaro (2013). We show that dα is a metric on 𝒫n for α∈[1,∞), while for α∈[0,1)∪{∞}, dα is not a metric in general. Our main result in this part identifies the necessary and sufficient conditions for equality in the triangle inequality of dα when α>1, for which we identify the equality conditions for the supermodularity of the Rényi entropy Hα. Our second set of results concerns the total variation distance and the Hellinger affinity (also called the Bhattacharyya coefficient) on 𝒫n. We obtain a number of bounds on these measures, along with some of their equality conditions. The resulting Hellinger affinity based upper bound on the total variation distance can be smaller than the bounds obtained from Pinsker's and the Bretagnolle--Huber inequalities.
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BibTeX
@misc{hariharan2026measures,
title = {{On Measures of Dissimilarity on the Majorization Simplex}},
author = {Tarun Hariharan and Prasad Krishnan and Abhishek Deshpande},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16687},
eprint = {2610.16687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16687},
}
RIS
TY - UNPB
TI - On Measures of Dissimilarity on the Majorization Simplex
AU - Tarun Hariharan
AU - Prasad Krishnan
AU - Abhishek Deshpande
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16687
UR - https://arxiv.org/abs/2610.16687
AB - The majorization simplex 𝒫_n is the set of probability vectors of length n whose entries are arranged in non-increasing order. Under the majorization order, 𝒫_n is a lattice, and it is the natural space for comparing distributions when the labels of the outcomes are not relevant. In this work, we study measures of dissimilarity (and, similarity) on 𝒫_n. Our first set of results concerns the Rényi-entropy based distance d_α(p,q)=H_α(p)+H_α(q)−2H_α(p∨ q) proposed by Yadav and Shkel (2026), which generalizes a metric of Cicalese, Gargano and Vaccaro (2013). We show that d_α is a metric on 𝒫_n for α∈[1,∞), while for α∈[0,1)∪{∞}, d_α is not a metric in general. Our main result in this part identifies the necessary and sufficient conditions for equality in the triangle inequality of d_α when α>1, for which we identify the equality conditions for the supermodularity of the Rényi entropy H_α. Our second set of results concerns the total variation distance and the Hellinger affinity (also called the Bhattacharyya coefficient) on 𝒫_n. We obtain a number of bounds on these measures, along with some of their equality conditions. The resulting Hellinger affinity based upper bound on the total variation distance can be smaller than the bounds obtained from Pinsker's and the Bretagnolle--Huber inequalities.
ER -