6,095 research outputs found
R\'enyi Entropy Power Inequalities via Normal Transport and Rotation
Following a recent proof of Shannon's entropy power inequality (EPI), a
comprehensive framework for deriving various EPIs for the R\'enyi entropy is
presented that uses transport arguments from normal densities and a change of
variable by rotation. Simple arguments are given to recover the previously
known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with
constant c and a modification with exponent {\alpha} of previous works. In
particular, for log-concave densities, we obtain a simple transportation proof
of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea
Two remarks on generalized entropy power inequalities
This note contributes to the understanding of generalized entropy power
inequalities. Our main goal is to construct a counter-example regarding
monotonicity and entropy comparison of weighted sums of independent identically
distributed log-concave random variables. We also present a complex analogue of
a recent dependent entropy power inequality of Hao and Jog, and give a very
simple proof.Comment: arXiv:1811.00345 is split into 2 papers, with this being on
Displacement convexity of generalized relative entropies
We investigate the -relative entropy, which stems from the Bregman
divergence, on weighted Riemannian and Finsler manifolds. We prove that the
displacement -convexity of the -relative entropy is equivalent to the
combination of the nonnegativity of the weighted Ricci curvature and the
-convexity of the weight function. We use this to show appropriate variants
of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the
concentration of measures. We also prove that the gradient flow of the
-relative entropy produces a solution to the porous medium equation or the
fast diffusion equation.Comment: 43page
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