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arxiv: 2605.09655 · v2 · pith:W6W7KI6Znew · submitted 2026-05-10 · 💻 cs.IT · math.CO· math.IT· math.PR

Geometry of R\'enyi Entropy on the Majorization Lattice

Anuj Kumar Yadav , Yanina Y. Shkel This is my paper

Pith reviewed 2026-05-25 06:09 UTC · model grok-4.3

classification 💻 cs.IT math.COmath.ITmath.PR
keywords Rényi entropymajorization latticesubadditivitysupermodularityTsallis entropystochastic orderingprobability distributionscomonotone coupling
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The pith

Rényi entropy is subadditive on the majorization lattice for every order α in [0, ∞].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Rényi entropy is subadditive on the majorization lattice of ordered probability distributions for every α. This follows from a basic relation between the comonotone coupling and the independent coupling of any collection of marginal distributions. The work also identifies the precise range where Rényi entropy becomes supermodular, namely α in {0} union [1, ∞). Parallel results are given for Tsallis entropy, which is subadditive for all α and supermodular for all α.

Core claim

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 ∈ [0,∞], 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 ∈ {0} ∪ [1,∞]. For the Tsallis entropy, we show that it also satisfies subadditivity on the majorization lattice, for every order α ∈ [0,∞). Finally, we show that, unlike the Rényi entropy, the Tsallis entropy is supermodular on the majorization lattice for every α ∈ [0,∞).

What carries the argument

The relation between the comonotone coupling and the independent coupling of marginal distributions, which directly yields the subadditivity of Rényi entropy on the lattice.

If this is right

  • Rényi entropy of the lattice join is at most the sum of the individual entropies for any α.
  • Supermodularity holds exactly when α is zero or at least one.
  • Tsallis entropy is subadditive for every α and supermodular for every α.
  • These lattice properties apply uniformly to the complete lattice of ordered probability distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The subadditivity may supply new upper bounds when combining diversity measures from separate sources.
  • The distinction between Rényi and Tsallis supermodularity regimes could affect which entropy is preferred in lattice-based optimization problems.
  • The coupling relation might extend to other information measures that depend on joint distributions.

Load-bearing premise

The fundamental relation between the comonotone coupling and the independent coupling associated with a collection of marginal distributions.

What would settle it

A concrete collection of marginal distributions for which the Rényi entropy of their comonotone coupling fails to satisfy the inequality that would imply subadditivity on the lattice join.

Figures

Figures reproduced from arXiv: 2605.09655 by Anuj Kumar Yadav, Yanina Y. Shkel.

Figure 1
Figure 1. Figure 1: (Example 1) : Supermodular behavior of the Rényi entropy for order α ∈ (0, 1). 0 1 2 3 4 5 6 7 8 9 10 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 ·10−2 α ∆α(p, q) ∆α(p, q) vs α ∆α(p, q) ∆1(p, q) [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: (Example 1) : Supermodular behavior of the Rényi entropy for order α ∈ (0, 1). 0 1 2 3 4 5 6 7 8 9 10 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 ·10−2 α ∆α(p, q) ∆α(p, q) vs α ∆α(p, q) ∆1(p, q) [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Example 2) : Submodular behavior of the Rényi entropy for order α ∈ (0, 1). Example 1 - Pair with ∆α(p, q) > 0: Let p, q ∈ P3 as, p = (0.6, 0.2, 0.2) q = (0.45, 0.4, 0.15) Consequently, the glb and lub are as follows, p ∧ q = (0.45, 0.35, 0.2) p ∨ q = (0.6, 0.25, 0.15) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Example 2) : Submodular behavior of the Rényi entropy for order α ∈ (0, 1) [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
read the original 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 $\alpha \in \{0\} \,\cup \, [1,\infty]$. For the Tsallis entropy, we show that it also satisfies subadditivity on the majorization lattice, for every order $\alpha \in [0,\infty)$. Finally, we show that, unlike the R\'enyi entropy, the Tsallis entropy is supermodular on the majorization lattice for every $\alpha \in [0,\infty)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that Rényi entropy is subadditive on the majorization lattice for every α ∈ [0,∞] by establishing a relation between the comonotone coupling and the independent coupling of marginal distributions; it further shows supermodularity on the lattice for α ∈ {0} ∪ [1,∞]. Parallel results are given for Tsallis entropy: subadditivity for all α ∈ [0,∞) and supermodularity for all α ∈ [0,∞).

Significance. If the central coupling relation holds, the work supplies a lattice-theoretic characterization of entropy functionals under majorization, distinguishing subadditive and supermodular regimes. This extends classical properties of entropy and may inform bounds in stochastic orders. The paper does not report machine-checked proofs or reproducible code.

major comments (1)
  1. [Section establishing the fundamental coupling relation] The derivation establishing the comonotone-independent coupling relation (invoked to prove subadditivity for the full interval α ∈ [0,∞]): for α < 1 the Rényi functional reverses monotonicity relative to the usual stochastic order, so the manuscript must explicitly verify that the inequality direction required for the join still holds; this step is load-bearing for the claim covering α < 1 and is not addressed by the abstract statement alone.
minor comments (2)
  1. The abstract states the supermodular regime for Rényi entropy but does not indicate whether the proof for α = 0 is handled separately from the α ≥ 1 case.
  2. Notation for the join operation on the majorization lattice should be introduced before its first use in the main results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the inequality direction in the coupling relation when α < 1. We address the major comment below.

read point-by-point responses

  1. Referee: [Section establishing the fundamental coupling relation] The derivation establishing the comonotone-independent coupling relation (invoked to prove subadditivity for the full interval α ∈ [0,∞]): for α < 1 the Rényi functional reverses monotonicity relative to the usual stochastic order, so the manuscript must explicitly verify that the inequality direction required for the join still holds; this step is load-bearing for the claim covering α < 1 and is not addressed by the abstract statement alone.

    Authors: We agree that the reversal of monotonicity for α < 1 requires an explicit check that the comonotone coupling still produces the correct inequality direction for subadditivity on the join. In the revised version we will insert a short dedicated paragraph immediately after the statement of the coupling relation. The paragraph will compute the relevant Rényi expressions for a pair of comonotone and independent couplings when 0 ≤ α < 1 and confirm that the inequality direction remains the one needed to bound the entropy of the join from above. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rest on standard coupling and lattice definitions

full rationale

The central claim establishes a relation between comonotone and independent couplings of marginals, then uses it to bound Rényi entropy on the join of the majorization lattice. This relation is invoked as a derived property of the couplings themselves (not defined via the entropy functional or fitted to data), and the subadditivity and supermodularity results follow from the lattice order and the functional properties of Rényi entropy. No self-citation chains, self-definitional equations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The argument is therefore self-contained against external benchmarks of coupling theory and majorization.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard fact that majorization forms a complete lattice and on the definition of comonotone versus independent couplings; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The majorization partial order forms a complete lattice on the set of ordered probability distributions.
    Explicitly stated as well-known in the abstract.
  • domain assumption Comonotone coupling and independent coupling are well-defined for any collection of marginal distributions.
    Invoked to establish the fundamental relation leading to subadditivity.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Sharma-Mittal Entropy is Subadditive and Supermodular on the Majorization Lattice

    cs.IT 2026-05 unverdicted novelty 7.0

    Sharma-Mittal entropy is proven to be subadditive and supermodular on the majorization lattice of n-dimensional probability distributions.

  2. The Sharma-Mittal Entropy is Subadditive and Supermodular on the Majorization Lattice

    cs.IT 2026-05 unverdicted novelty 7.0

    Sharma-Mittal entropy is proven subadditive and supermodular on the majorization lattice of n-dimensional probability distributions.

Reference graph

Works this paper leans on

100 extracted references · 100 canonical work pages · cited by 1 Pith paper

  1. [1]

    Image thresholding using Tsallis entropy , url =

    M. Image thresholding using Tsallis entropy , url =. Pattern Recognition Letters , keywords =. 2004 , bdsk-url-1 =. doi:https://doi.org/10.1016/j.patrec.2004.03.003 , issn =

    work page doi:10.1016/j.patrec.2004.03.003 2004

  2. [2]

    Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics , volume=

    On measures of entropy and information , author=. Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics , volume=. 1961 , organization=

    work page 1961

  3. [3]

    Nonextensive statistical mechanics and economics , url =

    Constantino Tsallis and Celia Anteneodo and Lisa Borland and Roberto Osorio , doi =. Nonextensive statistical mechanics and economics , url =. Physica A: Statistical Mechanics and its Applications , keywords =. 2003 , bdsk-url-1 =

    work page 2003

  4. [4]

    1967 , publisher=

    Economics and Information Theory , author=. 1967 , publisher=

    work page 1967

  5. [5]

    Innovation Representation of Stochastic Processes With Application to Causal Inference , year=

    Painsky, Amichai and Rosset, Saharon and Feder, Meir , journal=. Innovation Representation of Stochastic Processes With Application to Causal Inference , year=

    work page

  6. [6]

    Entropic Causal Inference , url =

    Kocaoglu, Murat and Dimakis, Alexandros and Vishwanath, Sriram and Hassibi, Babak , doi =. Entropic Causal Inference , url =. Proceedings of the AAAI Conference on Artificial Intelligence , month =. 2017 , bdsk-url-1 =

    work page 2017

  7. [7]

    Entropic Causal Inference: Graph Identifiability , url =

    Compton, Spencer and Greenewald, Kristjan and Katz, Dmitriy A and Kocaoglu, Murat , booktitle =. Entropic Causal Inference: Graph Identifiability , url =. 2022 , bdsk-url-1 =

    work page 2022

  8. [8]

    Acta Mathematica Academiae Paedagogicae Ny

    On some properties of Tsallis entropy on majorization lattice , author=. Acta Mathematica Academiae Paedagogicae Ny

    work page

  9. [9]

    , booktitle=

    Yadav, Anuj Kumar and Shkel, Yanina Y. , booktitle=. Approximation Guarantees for Minimum Rényi Entropy Functional Representations , year=

    work page

  10. [10]

    Tsallis and A.R

    C. Tsallis and A.R. Plastino and W.-M. Zheng , doi =. Power-law sensitivity to initial conditions---New entropic representation , url =. Chaos, Solitons & Fractals , number =. 1997 , bdsk-url-1 =

    work page 1997

  11. [11]

    Possible generalization of Boltzmann-Gibbs statistics , url =

    Tsallis, Constantino , date =. Possible generalization of Boltzmann-Gibbs statistics , url =. Journal of Statistical Physics , number =. 1988 , bdsk-url-1 =. doi:10.1007/BF01016429 , id =

    work page doi:10.1007/bf01016429 1988

  12. [12]

    and López-Rosa, S

    Antolín, J. and López-Rosa, S. and Angulo, J. C. and Esquivel, R. O. , title =. The Journal of Chemical Physics , volume =. 2010 , month =. doi:10.1063/1.3298911 , url =

    work page doi:10.1063/1.3298911 2010

  13. [13]

    Jensen--Tsallis divergence for supervised classification under data imbalance , url =

    Squicciarini, Antonio and Trigano, Tom and Luengo, David , date =. Jensen--Tsallis divergence for supervised classification under data imbalance , url =. Machine Learning , number =. 2025 , bdsk-url-1 =. doi:10.1007/s10994-025-06791-4 , id =

    work page doi:10.1007/s10994-025-06791-4 2025

  14. [14]

    Tsallis, R

    Jawad, Abdul and Bamba, Kazuharu and Younas, Muhammad and Qummer, Saba and Rani, Shamaila , doi =. Tsallis, R. Symmetry , number =. 2018 , bdsk-url-1 =

    work page 2018

  15. [15]

    Jensen--Renyi's--Tsallis Fuzzy Divergence Information Measure with its Applications , url =

    Kadian, Ratika and Kumar, Satish , date =. Jensen--Renyi's--Tsallis Fuzzy Divergence Information Measure with its Applications , url =. Communications in Mathematics and Statistics , number =. 2022 , bdsk-url-1 =. doi:10.1007/s40304-020-00228-1 , id =

    work page doi:10.1007/s40304-020-00228-1 2022

  16. [16]

    and Vaccaro, U

    Cicalese, F. and Vaccaro, U. , journal=. Supermodularity and subadditivity properties of the entropy on the majorization lattice , year=

    work page

  17. [17]

    Messenger Math

    Some simple inequalities satisfied by convex functions , author=. Messenger Math. , volume=

    work page

  18. [18]

    and Kumar Yadav, Anuj , booktitle=

    Shkel, Yanina Y. and Kumar Yadav, Anuj , booktitle=. Information Spectrum Converse for Minimum Entropy Couplings and Functional Representations , year=

    work page

  19. [19]

    Information theoretic measures of distances and their econometric applications , year=

    Cicalese, Ferdinando and Gargano, Luisa and Vaccaro, Ugo , booktitle=. Information theoretic measures of distances and their econometric applications , year=

    work page

  20. [20]

    Infinite Divisibility of Information , year=

    Li, Cheuk Ting , journal=. Infinite Divisibility of Information , year=

    work page

  21. [21]

    Efficient Approximate Minimum Entropy Coupling of Multiple Probability Distributions , year=

    Li, Cheuk Ting , journal=. Efficient Approximate Minimum Entropy Coupling of Multiple Probability Distributions , year=

    work page

  22. [22]

    2026 , journal=

    Geometry of Rényi Entropy on the Majorization Lattice , author=. 2026 , journal=

    work page 2026

  23. [23]

    An Information Theoretic Approach to Probability Mass Function Truncation , year=

    Cicalese, Ferdinando and Gargano, Luisa and Vaccaro, Ugo , booktitle=. An Information Theoretic Approach to Probability Mass Function Truncation , year=

    work page

  24. [24]

    Minimum-Entropy Couplings and Their Applications , year=

    Cicalese, Ferdinando and Gargano, Luisa and Vaccaro, Ugo , journal=. Minimum-Entropy Couplings and Their Applications , year=

    work page

  25. [25]

    A Tighter Approximation Guarantee for Greedy Minimum Entropy Coupling , year=

    Compton, Spencer , booktitle=. A Tighter Approximation Guarantee for Greedy Minimum Entropy Coupling , year=

    work page

  26. [26]

    Proceedings of The 26th International Conference on Artificial Intelligence and Statistics , pages =

    Minimum-Entropy Coupling Approximation Guarantees Beyond the Majorization Barrier , author =. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics , pages =. 2023 , editor =

    work page 2023

  27. [27]

    Stott and Ram, Prasad , doi =

    Parker, D. Stott and Ram, Prasad , doi =. The Construction of Huffman Codes is a Submodular ("Convex") Optimization Problem Over a Lattice of Binary Trees , url =. 1999 , bdsk-url-1 =. https://doi.org/10.1137/S0097539796311077 , journal =

    work page doi:10.1137/s0097539796311077 1999

  28. [29]

    Extremal elements of a sublattice of the majorization lattice and approximate majorization , url =

    Massri, C and Bellomo, G and Holik, F and Bosyk, G M , doi =. Extremal elements of a sublattice of the majorization lattice and approximate majorization , url =. Journal of Physics A: Mathematical and Theoretical , month =. 2020 , bdsk-url-1 =

    work page 2020

  29. [30]

    1991 , publisher=

    Submodular Functions and Optimization , author=. 1991 , publisher=

    work page 1991

  30. [31]

    Bapat , doi =

    R.B. Bapat , doi =. Majorization and singular values. III , url =. Linear Algebra and its Applications , pages =. 1991 , bdsk-url-1 =

    work page 1991

  31. [32]

    , journal=

    Yeung, R.W. , journal=. A new outlook on Shannon's information measures , year=

    work page

  32. [33]

    Lovász , title=

    L. Lovász , title=. Mathematical Programming The State of the Art , chapter=. 1983 , month=. doi:10.1007/978-3-642-68874-4_10 , url=

    work page doi:10.1007/978-3-642-68874-4_10 1983

  33. [34]

    The Size of a Share Must Be Large , url =

    Csirmaz, L. The Size of a Share Must Be Large , url =. Journal of Cryptology , number =. 1997 , bdsk-url-1 =. doi:10.1007/s001459900029 , id =

    work page doi:10.1007/s001459900029 1997

  34. [35]

    2011 , isbn =

    Jukna, Stasys , title =. 2011 , isbn =

    work page 2011

  35. [36]

    Polymatroidal dependence structure of a set of random variables , url =

    Satoru Fujishige , doi =. Polymatroidal dependence structure of a set of random variables , url =. Information and Control , number =. 1978 , bdsk-url-1 =

    work page 1978

  36. [37]

    Rényi divergence and majorization , year=

    van Erven, Tim and Harremoës, Peter , booktitle=. Rényi divergence and majorization , year=

    work page

  37. [38]

    Conditions for a Class of Entanglement Transformations , author =. Phys. Rev. Lett. , volume =. 1999 , month =. doi:10.1103/PhysRevLett.83.436 , url =

    work page doi:10.1103/physrevlett.83.436 1999

  38. [39]

    , title =

    Fan, Yanqin and Henry, Marc and Pass, Brendan and Rivero, Jorge A. , title =. The Review of Economics and Statistics , pages =. 2024 , month =. doi:10.1162/rest_a_01539 , url =

    work page doi:10.1162/rest_a_01539 2024

  39. [40]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume =

    Jorda, Vanesa and Sarabia, José María and Jäntti, Markus , title =. Journal of the Royal Statistical Society Series A: Statistics in Society , volume =. 2021 , month =. doi:10.1111/rssa.12702 , url =

    work page doi:10.1111/rssa.12702 2021

  40. [41]

    Quantum majorization and a complete set of entropic conditions for quantum thermodynamics , url =

    Gour, Gilad and Jennings, David and Buscemi, Francesco and Duan, Runyao and Marvian, Iman , date =. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics , url =. Nature Communications , number =. 2018 , bdsk-url-1 =. doi:10.1038/s41467-018-06261-7 , id =

    work page doi:10.1038/s41467-018-06261-7 2018

  41. [42]

    , date =

    Butt, Saad Ihsan and Javed, Iram and Agarwal, Praveen and Nieto, Juan J. , date =. Newton--Simpson-type inequalities via majorization , url =. Journal of Inequalities and Applications , number =. 2023 , bdsk-url-1 =. doi:10.1186/s13660-023-02918-0 , id =

    work page doi:10.1186/s13660-023-02918-0 2023

  42. [43]

    and Jiang, Yi , title =

    Palomar, Daniel P. and Jiang, Yi , title =. Foundations and Trends in Communications and Information Theory , volume =. 2007 , month =. doi:10.1561/0100000018 , url =

    work page doi:10.1561/0100000018 2007

  43. [44]

    Matrix Majorization in Large Samples , year=

    Farooq, Muhammad Usman and Fritz, Tobias and Haapasalo, Erkka and Tomamichel, Marco , journal=. Matrix Majorization in Large Samples , year=

    work page

  44. [45]

    and Nema, Aditya and Strelchuk, Sergii , date =

    Elkouss, David and Maity, Ananda G. and Nema, Aditya and Strelchuk, Sergii , date =. Communications Physics , title =. 2026 , bdsk-url-1 =. doi:10.1038/s42005-026-02583-x , id =

    work page doi:10.1038/s42005-026-02583-x 2026

  45. [46]

    1952 , publisher=

    Inequalities , author=. 1952 , publisher=

    work page 1952

  46. [47]

    M. O. Lorenz , journal =. Methods of Measuring the Concentration of Wealth , urldate =

    work page

  47. [48]

    Sitzungsberichte der Berliner Mathematischen Gesellschaft , volume =

    Schur, Issai , title =. Sitzungsberichte der Berliner Mathematischen Gesellschaft , volume =

    work page

  48. [49]

    2010 , publisher=

    Inequalities: Theory of Majorization and Its Applications , author=. 2010 , publisher=

    work page 2010

  49. [50]

    2017 , publisher=

    Wealth and Welfare , author=. 2017 , publisher=

    work page 2017

  50. [51]

    The Measurement of the Inequality of Incomes , urldate =

    Hugh Dalton , journal =. The Measurement of the Inequality of Incomes , urldate =

    work page

  51. [52]

    Antolín, S

    J. Antolín, S. López-Rosa, J. C. Angulo, and R. O. Esquivel. Jensen–tsallis divergence and atomic dissimilarity for position and momentum space electron densities. The Journal of Chemical Physics , 132(4):044105, 01 2010

    work page 2010

  52. [53]

    R.B. Bapat. Majorization and singular values. iii. Linear Algebra and its Applications , 145:59--70, 1991

    work page 1991

  53. [54]

    Saad Ihsan Butt, Iram Javed, Praveen Agarwal, and Juan J. Nieto. Newton--simpson-type inequalities via majorization. Journal of Inequalities and Applications , 2023(1):16, 2023

    work page 2023

  54. [55]

    On some properties of tsallis entropy on majorization lattice

    PK Bhatia, Surender Singh, and Vinod Kumar. On some properties of tsallis entropy on majorization lattice. Acta Mathematica Academiae Paedagogicae Ny \' regyh \'a ziensis , 31(2):331--340, 2015

    work page 2015

  55. [56]

    Entropic causal inference: Graph identifiability

    Spencer Compton, Kristjan Greenewald, Dmitriy A Katz, and Murat Kocaoglu. Entropic causal inference: Graph identifiability. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research , ...

    work page 2022

  56. [57]

    Information theoretic measures of distances and their econometric applications

    Ferdinando Cicalese, Luisa Gargano, and Ugo Vaccaro. Information theoretic measures of distances and their econometric applications. In 2013 IEEE International Symposium on Information Theory , pages 409--413, 2013

    work page 2013

  57. [58]

    An information theoretic approach to probability mass function truncation

    Ferdinando Cicalese, Luisa Gargano, and Ugo Vaccaro. An information theoretic approach to probability mass function truncation. In 2019 IEEE International Symposium on Information Theory (ISIT) , pages 702--706, 2019

    work page 2019

  58. [59]

    Minimum-entropy couplings and their applications

    Ferdinando Cicalese, Luisa Gargano, and Ugo Vaccaro. Minimum-entropy couplings and their applications. IEEE Transactions on Information Theory , 65(6):3436--3451, 2019

    work page 2019

  59. [60]

    Minimum-entropy coupling approximation guarantees beyond the majorization barrier

    Spencer Compton, Dmitriy Katz, Benjamin Qi, Kristjan Greenewald, and Murat Kocaoglu. Minimum-entropy coupling approximation guarantees beyond the majorization barrier. In Francisco Ruiz, Jennifer Dy, and Jan-Willem van de Meent, editors, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics , volume 206 of Proceedings ...

    work page 2023

  60. [61]

    A tighter approximation guarantee for greedy minimum entropy coupling

    Spencer Compton. A tighter approximation guarantee for greedy minimum entropy coupling. In 2022 IEEE International Symposium on Information Theory (ISIT) , pages 168--173, 2022

    work page 2022

  61. [62]

    The size of a share must be large

    L \'a szl \'o Csirmaz. The size of a share must be large. Journal of Cryptology , 10(4):223--231, 1997

    work page 1997

  62. [63]

    Cicalese and U

    F. Cicalese and U. Vaccaro. Supermodularity and subadditivity properties of the entropy on the majorization lattice. IEEE Transactions on Information Theory , 48(4):933--938, 2002

    work page 2002

  63. [64]

    The measurement of the inequality of incomes

    Hugh Dalton. The measurement of the inequality of incomes. The Economic Journal , 30(119):348--361, 1920

    work page 1920

  64. [65]

    Maity, Aditya Nema, and Sergii Strelchuk

    David Elkouss, Ananda G. Maity, Aditya Nema, and Sergii Strelchuk. A finite sufficient set of conditions for catalytic majorization. Communications Physics , 2026

    work page 2026

  65. [66]

    Matrix majorization in large samples

    Muhammad Usman Farooq, Tobias Fritz, Erkka Haapasalo, and Marco Tomamichel. Matrix majorization in large samples. IEEE Transactions on Information Theory , 70(5):3118--3144, 2024

    work page 2024

  66. [67]

    Yanqin Fan, Marc Henry, Brendan Pass, and Jorge A. Rivero. Multidimensional inequality measurement via optimal transport. The Review of Economics and Statistics , pages 1--45, 11 2024

    work page 2024

  67. [68]

    Polymatroidal dependence structure of a set of random variables

    Satoru Fujishige. Polymatroidal dependence structure of a set of random variables. Information and Control , 39(1):55--72, 1978

    work page 1978

  68. [69]

    Fujishige

    S. Fujishige. Submodular Functions and Optimization . Annals of Discrete Mathematics. North Holland, 1991

    work page 1991

  69. [70]

    The geometry of coalition power: Majorization, lattices, and displacement in multiwinner elections

    Qian Guo, Yidan Hu, and Rui Zhang. The geometry of coalition power: Majorization, lattices, and displacement in multiwinner elections. arXiv preprint arXiv:2601.16723 , 2026

    work page arXiv 2026

  70. [71]

    Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

    Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics. Nature Communications , 9(1):5352, 2018

    work page 2018

  71. [72]

    Some simple inequalities satisfied by convex functions

    Godfrey H Hardy. Some simple inequalities satisfied by convex functions. Messenger Math. , 58:145--152, 1929

    work page 1929

  72. [73]

    Hardy, J.E

    G.H. Hardy, J.E. Littlewood, and G. P \'o lya. Inequalities . Cambridge Mathematical Library. Cambridge University Press, 1952

    work page 1952

  73. [74]

    Tsallis, r \'e nyi and sharma-mittal holographic dark energy models in loop quantum cosmology

    Abdul Jawad, Kazuharu Bamba, Muhammad Younas, Saba Qummer, and Shamaila Rani. Tsallis, r \'e nyi and sharma-mittal holographic dark energy models in loop quantum cosmology. Symmetry , 10(11), 2018

    work page 2018

  74. [75]

    Inequality measurement with grouped data: Parametric and non-parametric methods

    Vanesa Jorda, José María Sarabia, and Markus Jäntti. Inequality measurement with grouped data: Parametric and non-parametric methods. Journal of the Royal Statistical Society Series A: Statistics in Society , 184(3):964--984, 07 2021

    work page 2021

  75. [76]

    Extremal Combinatorics: With Applications in Computer Science

    Stasys Jukna. Extremal Combinatorics: With Applications in Computer Science . Springer Publishing Company, Incorporated, 2nd edition, 2011

    work page 2011

  76. [77]

    Entropic causal inference

    Murat Kocaoglu, Alexandros Dimakis, Sriram Vishwanath, and Babak Hassibi. Entropic causal inference. Proceedings of the AAAI Conference on Artificial Intelligence , 31(1), Feb. 2017

    work page 2017

  77. [78]

    Jensen--renyi's--tsallis fuzzy divergence information measure with its applications

    Ratika Kadian and Satish Kumar. Jensen--renyi's--tsallis fuzzy divergence information measure with its applications. Communications in Mathematics and Statistics , 10(3):451--482, 2022

    work page 2022

  78. [79]

    Efficient approximate minimum entropy coupling of multiple probability distributions

    Cheuk Ting Li. Efficient approximate minimum entropy coupling of multiple probability distributions. IEEE Transactions on Information Theory , 67(8):5259--5268, 2021

    work page 2021

  79. [80]

    Infinite divisibility of information

    Cheuk Ting Li. Infinite divisibility of information. IEEE Transactions on Information Theory , 68(7):4257--4271, 2022

    work page 2022

  80. [81]

    M. O. Lorenz. Methods of measuring the concentration of wealth. Publications of the American Statistical Association , 9(70):209--219, 1905

    work page 1905

Showing first 80 references.