# Reversal of Rényi Entropy Inequalities Under Log-Concavity

@article{Melbourne2020ReversalOR, title={Reversal of R{\'e}nyi Entropy Inequalities Under Log-Concavity}, author={James Melbourne and Tomasz Tkocz}, journal={IEEE Transactions on Information Theory}, year={2020}, volume={67}, pages={45-51} }

We establish a discrete analog of the Rényi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within <inline-formula> <tex-math notation="LaTeX">$\log e$ </tex-math></inline-formula> of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis, and establish a sharp Rényi version for certain parameters in both the continuous and discrete cases.

## 12 Citations

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

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A reversal of Lyapunov’s inequality for monotone log-concave sequences is established, settling a conjecture of Havrilla-Tkocz and Melbourne-T kocz, and several information theoretic inequalities as consequences are derived.

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Several Schur-convexity type results under fixed variance for weighted sums of independent gamma random variables are established and nonasymptotic bounds on their Rényi entropies are obtained.

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### Moments, Concentration, and Entropy of Log-Concave Distributions

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### Inequalities for Information Potentials and Entropies

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### Concentration inequalities for ultra log-concave distributions

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. We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an…

### A DISCRETE COMPLEMENT OF LYAPUNOV’S INEQUALITY AND ITS INFORMATION THEORETIC CONSEQUENCES

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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…

## References

SHOWING 1-10 OF 43 REFERENCES

### On the Rényi Entropy of Log-Concave Sequences

- Computer Science2020 IEEE International Symposium on Information Theory (ISIT)
- 2020

A discrete analog of the Rényi entropy comparison due to Bobkov and Madiman is established for log-concave variables on the integers with the additional assumption that the variable is monotone, and a sharp bound of loge is obtained.

### Rényi Entropy Power Inequalities via Normal Transport and Rotation

- Computer ScienceEntropy
- 2018

A comprehensive framework for deriving various EPIs for the Rényi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation, and a simple transportation proof of a sharp varentropy bound is obtained.

### A Renyi Entropy Power Inequality for Log-Concave Vectors and Parameters in [0, 1]

- Computer Science2018 IEEE International Symposium on Information Theory (ISIT)
- 2018

A Renyi entropy power inequality for log-concave random vectors when Renyi parameters belong to [0, 1] is derived using a sharp version of the reverse Young inequality and a result due to Fradelizi, Madiman, and Wang.

### On the Entropy Power Inequality for the Rényi Entropy of Order [0, 1]

- Computer ScienceIEEE Transactions on Information Theory
- 2019

The authors derive Rényi entropy power inequalities for log-concave random vectors when Rénery parameters belong to [0, 1] and the estimates are shown to be sharp up to absolute constants.

### Rényi entropy power inequality and a reverse

- Computer ScienceArXiv
- 2017

A refinement of the R\'enyi Entropy Power Inequality recently obtained inBM16 is presented, and a conjecture in BNT15, MMX16 in two cases is confirmed, which largely follows the approach in DCT91 of employing Young's convolution inequalities with sharp constants.

### Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

- Mathematics, Computer ScienceLecture Notes in Mathematics
- 2020

The role of convexity in Renyi entropy power inequalities is investigated and the convergence in the Central Limit Theorem for Renyi entropies of order r ∈ (0, 1) for log-concave densities and for compactly supported, spherically symmetric and unimodal densities is established.

### Entropy jumps for isotropic log-concave random vectors and spectral gap

- Mathematics
- 2012

We prove a quantitative dimension-free bound in the Shannon{Stam en- tropy inequality for the convolution of two log-concave distributions in dimension d in terms of the spectral gap of the density.…

### Convexity/concavity of renyi entropy and α-mutual information

- Computer Science2015 IEEE International Symposium on Information Theory (ISIT)
- 2015

This paper shows the counterpart of this result for the Rényi entropy and the Tsallis entropy, and considers a notion of generalized mutual information, namely α-mutual information, which is defined through the Re⩽i divergence.

### Further Investigations of the Maximum Entropy of the Sum of Two Dependent Random Variables

- Mathematics2018 IEEE International Symposium on Information Theory (ISIT)
- 2018

The authors consider the analogous reversal of recent Renyi Entropy Power Inequalities for random vectors and again show that not only do they hold for s-concave densities, but that s- Conc Cave densities are characterized by satisfying said inequalities.