10,600 research outputs found
A Simple Proof of the Entropy-Power Inequality via Properties of Mutual Information
While most useful information theoretic inequalities can be deduced from the
basic properties of entropy or mutual information, Shannon's entropy power
inequality (EPI) seems to be an exception: available information theoretic
proofs of the EPI hinge on integral representations of differential entropy
using either Fisher's information (FI) or minimum mean-square error (MMSE). In
this paper, we first present a unified view of proofs via FI and MMSE, showing
that they are essentially dual versions of the same proof, and then fill the
gap by providing a new, simple proof of the EPI, which is solely based on the
properties of mutual information and sidesteps both FI or MMSE representations.Comment: 5 pages, accepted for presentation at the IEEE International
Symposium on Information Theory 200
Information Theoretic Proofs of Entropy Power Inequalities
While most useful information theoretic inequalities can be deduced from the
basic properties of entropy or mutual information, up to now Shannon's entropy
power inequality (EPI) is an exception: Existing information theoretic proofs
of the EPI hinge on representations of differential entropy using either Fisher
information or minimum mean-square error (MMSE), which are derived from de
Bruijn's identity. In this paper, we first present an unified view of these
proofs, showing that they share two essential ingredients: 1) a data processing
argument applied to a covariance-preserving linear transformation; 2) an
integration over a path of a continuous Gaussian perturbation. Using these
ingredients, we develop a new and brief proof of the EPI through a mutual
information inequality, which replaces Stam and Blachman's Fisher information
inequality (FII) and an inequality for MMSE by Guo, Shamai and Verd\'u used in
earlier proofs. The result has the advantage of being very simple in that it
relies only on the basic properties of mutual information. These ideas are then
generalized to various extended versions of the EPI: Zamir and Feder's
generalized EPI for linear transformations of the random variables, Takano and
Johnson's EPI for dependent variables, Liu and Viswanath's
covariance-constrained EPI, and Costa's concavity inequality for the entropy
power.Comment: submitted for publication in the IEEE Transactions on Information
Theory, revised versio
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
Yet Another Proof of the Entropy Power Inequality
Yet another simple proof of the entropy power inequality is given, which
avoids both the integration over a path of Gaussian perturbation and the use of
Young's inequality with sharp constant or R\'enyi entropies. The proof is based
on a simple change of variables, is formally identical in one and several
dimensions, and easily settles the equality case
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
Hessian and concavity of mutual information, differential entropy, and entropy power in linear vector Gaussian channels
Within the framework of linear vector Gaussian channels with arbitrary
signaling, closed-form expressions for the Jacobian of the minimum mean square
error and Fisher information matrices with respect to arbitrary parameters of
the system are calculated in this paper. Capitalizing on prior research where
the minimum mean square error and Fisher information matrices were linked to
information-theoretic quantities through differentiation, closed-form
expressions for the Hessian of the mutual information and the differential
entropy are derived. These expressions are then used to assess the concavity
properties of mutual information and differential entropy under different
channel conditions and also to derive a multivariate version of the entropy
power inequality due to Costa.Comment: 33 pages, 2 figures. A shorter version of this paper is to appear in
IEEE Transactions on Information Theor
Bounds on Information Combining With Quantum Side Information
"Bounds on information combining" are entropic inequalities that determine
how the information (entropy) of a set of random variables can change when
these are combined in certain prescribed ways. Such bounds play an important
role in classical information theory, particularly in coding and Shannon
theory; entropy power inequalities are special instances of them. The arguably
most elementary kind of information combining is the addition of two binary
random variables (a CNOT gate), and the resulting quantities play an important
role in Belief propagation and Polar coding. We investigate this problem in the
setting where quantum side information is available, which has been recognized
as a hard setting for entropy power inequalities.
Our main technical result is a non-trivial, and close to optimal, lower bound
on the combined entropy, which can be seen as an almost optimal "quantum Mrs.
Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the
concavity of von Neumann entropy, which is tight in the regime of low pairwise
state fidelities; (2) the quantitative improvement of strong subadditivity due
to Fawzi-Renner, in which we manage to handle the minimization over recovery
maps; (3) recent duality results on classical-quantum-channels due to Renes et
al. We furthermore present conjectures on the optimal lower and upper bounds
under quantum side information, supported by interesting analytical
observations and strong numerical evidence.
We finally apply our bounds to Polar coding for binary-input
classical-quantum channels, and show the following three results: (A) Even
non-stationary channels polarize under the polar transform. (B) The blocklength
required to approach the symmetric capacity scales at most sub-exponentially in
the gap to capacity. (C) Under the aforementioned lower bound conjecture, a
blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction
Sumset and Inverse Sumset Inequalities for Differential Entropy and Mutual Information
The sumset and inverse sumset theories of Freiman, Pl\"{u}nnecke and Ruzsa,
give bounds connecting the cardinality of the sumset of two discrete sets , to the cardinalities (or the finer
structure) of the original sets . For example, the sum-difference bound of
Ruzsa states that, , where the difference set . Interpreting the differential entropy of a
continuous random variable as (the logarithm of) the size of the effective
support of , the main contribution of this paper is a series of natural
information-theoretic analogs for these results. For example, the Ruzsa
sum-difference bound becomes the new inequality, , for any pair of independent continuous random variables and .
Our results include differential-entropy versions of Ruzsa's triangle
inequality, the Pl\"{u}nnecke-Ruzsa inequality, and the
Balog-Szemer\'{e}di-Gowers lemma. Also we give a differential entropy version
of the Freiman-Green-Ruzsa inverse-sumset theorem, which can be seen as a
quantitative converse to the entropy power inequality. Versions of most of
these results for the discrete entropy were recently proved by Tao,
relying heavily on a strong, functional form of the submodularity property of
. Since differential entropy is {\em not} functionally submodular, in the
continuous case many of the corresponding discrete proofs fail, in many cases
requiring substantially new proof strategies. We find that the basic property
that naturally replaces the discrete functional submodularity, is the data
processing property of mutual information.Comment: 23 page
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