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Register a new userThe Convexity and Concavity of Envelopes of the Minimum-Relative-Entropy Region for the DSBS
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Added by
Lei Yu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Lei Yu
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In this paper, we prove that for the doubly symmetric binary distribution, the lower increasing envelope and the upper envelope of the minimum-relative-entropy region are respectively convex and concave. We also prove that another function induced the minimum-relative-entropy region is concave. These two envelopes and this function were previously used to characterize the optimal exponents in strong small-set expansion problems and strong Brascamp--Lieb inequalities. The results in this paper, combined with the strong small-set expansion theorem derived by Yu, Anantharam, and Chen (2021), and the strong Brascamp--Lieb inequality derived by Yu (2021), confirm positively Ordentlich--Polyanskiy--Shayevitzs conjecture on the strong small-set expansion (2019) and Polyanskiys conjecture on the strong Brascamp--Lieb inequality (2016). The proofs in this paper are based on the equivalence between the convexity of a function and the convexity of the set of minimizers of its Lagrangian dual.
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The relative entropy and chi-squared divergence are fundamental divergence measures in information theory and statistics. This paper is focused on a study of integral relations between the two divergences, the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of $f$-divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong~data-processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains.
A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csiszar and Talata. It is further extended to an upper bound on the Renyi divergence of an arbitrary non-negative order (including $infty$) as a function of the total variation distance.
Using a sharp version of the reverse Young inequality, and a Renyi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors are able to derive Renyi entropy power inequalities for log-concave random vectors when Renyi parameters belong to $(0,1)$. Furthermore, the estimates are shown to be sharp up to absolute constants.
An extension of the entropy power inequality to the form $N_r^alpha(X+Y) geq N_r^alpha(X) + N_r^alpha(Y)$ with arbitrary independent summands $X$ and $Y$ in $mathbb{R}^n$ is obtained for the Renyi entropy and powers $alpha geq (r+1)/2$.
In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted $mathscr{I}_{alpha}$) were studied. Such minimizers were called forward $mathscr{I}_{alpha}$-projections. Here, a complementary class of minimization problems leading to the so-called reverse $mathscr{I}_{alpha}$-projections are studied. Reverse $mathscr{I}_{alpha}$-projections, particularly on log-convex or power-law families, are of interest in robust estimation problems ($alpha >1$) and in constrained compression settings ($alpha <1$). Orthogonality of the power-law family with an associated linear family is first established and is then exploited to turn a reverse $mathscr{I}_{alpha}$-projection into a forward $mathscr{I}_{alpha}$-projection. The transformed problem is a simpler quasiconvex minimization subject to linear constraints.
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