Concentration of Measure Inequalities in Information Theory, Communications, and Coding

Raginsky, Maxim

doi:10.1561/0100000064

Cited by 118 publications

(10 citation statements)

References 162 publications

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“…The entropy production is also related with the gradient flow structure on well-chosen Riemannian manifolds for those Markov semigroups [12][13][14][15][16][17][18][19]. Furthermore, the validity of log-Sobolev inequalities [20][21][22][23][24][25][26] helps to prove the exponential decay of those entropies under respective Markov semigroups.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Cao

2019

Journal of Mathematical Physics

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We study the entropy production of the sandwiched Rényi divergence under the primitive Lindblad equation with GNS-detailed balance. We prove that the Lindblad equation can be identified as the gradient flow of the sandwiched Rényi divergence of any order α ∈ (0, ∞). This extends a previous result by Carlen and Maas [Journal of Functional Analysis, 273(5), 1810-1869] for the quantum relative entropy (i.e., α = 1). Moreover, we show that the sandwiched Rényi divergence of any order α ∈ (0, ∞) decays exponentially fast under the time-evolution of such a Lindblad equation.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Cao

2019

Journal of Mathematical Physics

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“…Lemma The following philosophy strongly satisfies and holds: The set's variations can be remarkably independent from their expected value ( m e a n ). ()…”

Section: System Description and Problem Formulationmentioning

confidence: 99%

“…The term weakly distributed non‐iid (i.n.i.d) can be interpreted as weaker chaotic behaviour (deterministic chaos ). ()…”

Section: Our Proposed Solutionmentioning

confidence: 99%

“…Proof L ‐Lipschitz is principally interpreted as dependence in L way amount of which is not extremely high ( L =1). () For further discussion, see the following definition in association with the key principle Lipschitz functions .…”

Section: Our Proposed Solutionmentioning

confidence: 99%

“…Proof See Appendix B for the proof; something that is characterised according to the theorem tensorisation of relative entropy . See, for example, the works of Handel and Raginsky and Sason, for further discussion on tensorisation .…”

Section: Our Proposed Solutionmentioning

confidence: 99%

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Probabilistic‐based secrecy rate maximisation for MIMOME wiretap channels: Towards novel convexification procedures—Part II

Zamanipour¹

2018

Trans Emerging Tel Tech

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This paper extends our previous work on multiple‐input–multiple‐output multiantenna eavesdropper–based wiretap channels aimed at fundamentally proposing optimally secure frameworks. This extension is appropriately conducted providing 2 newly asymptotic bounds for the secrecy outage capacity. Such proposed schemes should be necessarily taken into consideration owing to channel uncertainty sets in relation to the eavesdropping fusion center. Our first resultant closed‐form solution, which is mathematically performed according to Hanson‐Wright inequality, is totally tight as well; something that can be technically assigned to other ergodicity‐driven equilibria. Specifically, our solution's outperformance is highlighted with regard to a significantly more adequate secrecy capacity in the low signal‐to‐noise‐ratio regime. Such outperformance is efficiently guaranteed due to assigning optimally convex paradigms regarding the concentration‐of‐measure‐based inequality aforementioned. Additionally, our framework has a markedly acceptable computational complexity compared with our previous work. The latter characteristic is straightforwardly highlighted in parallel with a considerably acceptable computational error as well. Finally, we optimally generalise our framework to dynamically fast fading processes in terms of the secondly closed‐form asymptotic bound. At the end of the discussion, simulation‐based analytical results principally authenticate our contribution's efficiency and tightness.

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Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

Vladimirova

Girard

Nguyen

et al. 2020

Stat

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We propose the notion of sub‐Weibull distributions, which are characterized by tails lighter than (or equally light as) the right tail of a Weibull distribution. This novel class generalizes the sub‐Gaussian and sub‐Exponential families to potentially heavier tailed distributions. Sub‐Weibull distributions are parameterized by a positive tail index θ and reduce to sub‐Gaussian distributions for and to sub‐Exponential distributions for . A characterization of the sub‐Weibull property based on moments and on the moment generating function is provided and properties of the class are studied. An estimation procedure for the tail parameter is proposed and is applied to an example stemming from Bayesian deep learning.

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Concentration of Measure Inequalities in Information Theory, Communications, and Coding

Cited by 118 publications

References 162 publications

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

Probabilistic‐based secrecy rate maximisation for MIMOME wiretap channels: Towards novel convexification procedures—Part II

Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions

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