Majorization and Rényi entropy inequalities via Sperner theory

Madiman, Mokshay; Wang, Liyao; Woo, Jae Oh

doi:10.1016/j.disc.2019.03.002

Cited by 20 publications

(9 citation statements)

References 65 publications

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“…There are also interesting upper bounds on cardinalities of sumsets in the discrete setting that have similar combinatorial structure, see, e.g., [42,8,54] and references therein. Furthermore, as discussed in the introduction for the continuous domain, there are also discrete entropy analogues of these cardinality inequalities, explored in depth in [68,83,52,8,54,43,88,59,58] and references therein. We do not discuss discrete analogues further in this paper.…”

Section: Monotonicity Of Volume Deficit In Dimension One and For Cartmentioning

confidence: 99%

The convexification effect of Minkowski summation

Fradelizi

Madiman

Marsiglietti

et al. 2018

EMS Surv. Math. Sci.

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Let us define for a compact set A ⊂ R n the sequence A(k) = a 1 + · · · + a k k : a 1 , . . . , a k ∈ A = 1 k A + · · · + A k times.It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to ∞. We explore in this survey how exactly A(k) approaches the convex hull of A, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on R n , the volume deficit (the difference of volumes), a nonconvexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets A with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k), and both the Hausdorff distance and effective standard deviation are eventually monotone (once k exceeds n). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), falsify a conjecture of Dyn and Farkhi (2004), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

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Section: Monotonicity Of Volume Deficit In Dimension One and For Cartmentioning

confidence: 99%

The convexification effect of Minkowski summation

Fradelizi

Madiman

Marsiglietti

et al. 2018

EMS Surv. Math. Sci.

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“…This effort fits within a more general pursuit, developing discrete analogs for the continuous convexity theory, which in recent investigation has connected information theory and convex geometry (see [25] for background). One instantiation is the effort to understand the behavior of the entropy of discrete variables under independent summation, see [19,27,33,24,7]. Another is the pursuit of discrete Brunn-Minkowski type inequalities [15,35,30,20,16,14,40,17].…”

Section: Introductionmentioning

confidence: 99%

A discrete complement of Lyapunov's inequality and its information theoretic consequences

Melbourne¹,

Palafox-Castillo²

2021

Preprint

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We establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter example. We also derive several information theoretic inequalities as consequences. In particular sharp bounds are derived for the varentropy, Rényi entropies, and the concentration of information of monotone logconcave random variables. Moreover, the majorization approach utilized in the proof of the main theorem, is applied to derive analogous information theoretic results in the symmetric setting, where the Lyapunov reversal is known to fail.

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“…Rényi’s information measures are also fundamental– indeed, they are (for

) just monotone functions of

-norms, whose relevance or importance in any field that relies on analysis need not be justified. Furthermore, they show up in probability theory, PDE, functional analysis, additive combinatorics, and convex geometry (see, e.g., [ 3 , 4 , 5 , 6 , 7 , 8 , 9 ]), in ways where understanding them as information measures instead of simply as monotone functions of

-norms is fruitful. For example, there is an intricate story of parallels between entropy power inequalities (see, e.g., [ 10 , 11 , 12 ]), Brunn-Minkowski-type volume inequalities (see, e.g., [ 13 , 14 , 15 ]) and sumset cardinality (see, e.g., [ 16 , 17 , 18 , 19 , 20 ]), which is clarified by considering logarithms of volumes and Shannon entropies as members of the larger class of Rényi entropies.…”

Section: Introductionmentioning

confidence: 99%

Conditional Rényi Entropy and the Relationships between Rényi Capacities

Aishwarya

Madiman

2020

Entropy

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The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

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Majorization and Rényi entropy inequalities via Sperner theory

Cited by 20 publications

References 65 publications

The convexification effect of Minkowski summation

The convexification effect of Minkowski summation

A discrete complement of Lyapunov's inequality and its information theoretic consequences

Conditional Rényi Entropy and the Relationships between Rényi Capacities

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