5,348 research outputs found
Tight Bounds for Symmetric Divergence Measures and a New Inequality Relating -Divergences
Tight bounds for several symmetric divergence measures are introduced, given
in terms of the total variation distance. Each of these bounds is attained by a
pair of 2 or 3-element probability distributions. An application of these
bounds for lossless source coding is provided, refining and improving a certain
bound by Csisz\'ar. A new inequality relating -divergences is derived, and
its use is exemplified. The last section of this conference paper is not
included in the recent journal paper that was published in the February 2015
issue of the IEEE Trans. on Information Theory (see arXiv:1403.7164), as well
as some new paragraphs throughout the paper which are linked to new references.Comment: A final version of the conference paper at the 2015 IEEE Information
Theory Workshop, Jerusalem, Israe
Sharp Bounds on the Entropy of the Poisson Law and Related Quantities
One of the difficulties in calculating the capacity of certain Poisson
channels is that H(lambda), the entropy of the Poisson distribution with mean
lambda, is not available in a simple form. In this work we derive upper and
lower bounds for H(lambda) that are asymptotically tight and easy to compute.
The derivation of such bounds involves only simple probabilistic and analytic
tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and
Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on
the relative entropy D(n, p) between a binomial and a Poisson, thus refining
the work of Harremoes and Ruzankin (2004). Bounds on the entropy of the
binomial also follow easily.Comment: To appear, IEEE Trans. Inform. Theor
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
Tight bounds for LDPC and LDGM codes under MAP decoding
A new method for analyzing low density parity check (LDPC) codes and low
density generator matrix (LDGM) codes under bit maximum a posteriori
probability (MAP) decoding is introduced. The method is based on a rigorous
approach to spin glasses developed by Francesco Guerra. It allows to construct
lower bounds on the entropy of the transmitted message conditional to the
received one. Based on heuristic statistical mechanics calculations, we
conjecture such bounds to be tight. The result holds for standard irregular
ensembles when used over binary input output symmetric channels. The method is
first developed for Tanner graph ensembles with Poisson left degree
distribution. It is then generalized to `multi-Poisson' graphs, and, by a
completion procedure, to arbitrary degree distribution.Comment: 28 pages, 9 eps figures; Second version contains a generalization of
the previous resul
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