101 research outputs found
A conditional Entropy Power Inequality for dependent variables
We provide a condition under which a version of Shannon's Entropy Power
Inequality will hold for dependent variables. We provide information
inequalities extending those found in the independent case.Comment: 6 page
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
In noncommutative probability theory independence can be based on free
products instead of tensor products. This yields a highly noncommutative
theory: free probability . Here we show that the classical Shannon's entropy
power inequality has a counterpart for the free analogue of entropy .
The free entropy (introduced recently by the second named author),
consistently with Boltzmann's formula , was defined via volumes of
matricial microstates. Proving the free entropy power inequality naturally
becomes a geometric question.
Restricting the Minkowski sum of two sets means to specify the set of pairs
of points which will be added. The relevant inequality, which holds when the
set of "addable" points is sufficiently large, differs from the Brunn-Minkowski
inequality by having the exponent replaced by . Its proof uses the
rearrangement inequality of Brascamp-Lieb-L\"uttinger
An information-theoretic proof of Nash's inequality
We show that an information-theoretic property of Shannon's entropy power,
known as concavity of entropy power, can be fruitfully employed to prove
inequalities in sharp form. In particular, the concavity of entropy power
implies the logarithmic Sobolev inequality, and Nash's inequality with the
sharp constant
The concavity of R\`enyi entropy power
We associate to the p-th R\'enyi entropy a definition of entropy power, which
is the natural extension of Shannon's entropy power and exhibits a nice
behaviour along solutions to the p-nonlinear heat equation in . We show
that the R\'enyi entropy power of general probability densities solving such
equations is always a concave function of time, whereas it has a linear
behaviour in correspondence to the Barenblatt source-type solutions. We then
shown that the p-th R\'enyi entropy power of a probability density which solves
the nonlinear diffusion of order p, is a concave function of time. This result
extends Costa's concavity inequality for Shannon's entropy power to R\'enyi
entropies
A multivariate generalization of Costa's entropy power inequality
A simple multivariate version of Costa's entropy power inequality is proved.
In particular, it is shown that if independent white Gaussian noise is added to
an arbitrary multivariate signal, the entropy power of the resulting random
variable is a multidimensional concave function of the individual variances of
the components of the signal. As a side result, we also give an expression for
the Hessian matrix of the entropy and entropy power functions with respect to
the variances of the signal components, which is an interesting result in its
own right.Comment: Proceedings of the 2008 IEEE International Symposium on Information
Theory, Toronto, ON, Canada, July 6 - 11, 200
On the Noisy Feedback Capacity of Gaussian Broadcast Channels
It is well known that, in general, feedback may enlarge the capacity region
of Gaussian broadcast channels. This has been demonstrated even when the
feedback is noisy (or partial-but-perfect) and only from one of the receivers.
The only case known where feedback has been shown not to enlarge the capacity
region is when the channel is physically degraded (El Gamal 1978, 1981). In
this paper, we show that for a class of two-user Gaussian broadcast channels
(not necessarily physically degraded), passively feeding back the stronger
user's signal over a link corrupted by Gaussian noise does not enlarge the
capacity region if the variance of feedback noise is above a certain threshold.Comment: 5 pages, 3 figures, to appear in IEEE Information Theory Workshop
2015, Jerusale
On some special cases of the Entropy Photon-Number Inequality
We show that the Entropy Photon-Number Inequality (EPnI) holds where one of
the input states is the vacuum state and for several candidates of the other
input state that includes the cases when the state has the eigenvectors as the
number states and either has only two non-zero eigenvalues or has arbitrary
number of non-zero eigenvalues but is a high entropy state. We also discuss the
conditions, which if satisfied, would lead to an extension of these results.Comment: 12 pages, no figure
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