27 research outputs found
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
Conditional R\'enyi entropy and the relationships between R\'enyi capacities
The analogues of Arimoto's definition of conditional R\'enyi entropy and
R\'enyi 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 R\'enyi divergences,
the relationships between the families of mutual informations defined by
Sibson, Augustin-Csisz\'ar, and Lapidoth-Pfister, as well as the corresponding
capacities, are explored.Comment: 17 pages, 1 figur
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