Reverse hypercontractivity using information measures

Kamath, Sudeep

doi:10.1109/allerton.2015.7447063

Cited by 23 publications

(15 citation statements)

References 24 publications

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“…While there have been many applications of (the forward) hypercontractivity, mostly in the converse proofs (see e.g., [17], [26], [31], [32], [50]), the applications of the reverse hypercontractivity have been very rare; see the discussion [33] (see however, an application in [37,Section 4]). Besides the applications in operational problems, some pure mathematical aspects of hypercontractivity and its reverse also received some recent attentions in the information theory community [6]- [8], [18], [24], [25], [34].…”

Section: Introductionmentioning

confidence: 99%

Dispersion Bound for the Wyner-Ahlswede-Körner Network via a Semigroup Method on Types

Liu

2021

IEEE Trans. Inform. Theory

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We revisit the Wyner-Ahlswede-Körner network, focusing especially on the converse part of the dispersion analysis, which is known to be challenging. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the secondorder behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of inf P U X ØcHÔY UÕ IÔU; XÕÙ with respect to Q XY , the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the c-dispersion as the variance of cı Y U ÔY UÕ ıU;XÔU; XÕ, which improves the existing upper bounds but has a gap to the aforementioned lower bound. Our converse analysis should be immediately extendable to other distributed source-type problems, such as distributed source coding, common randomness generation, and hypothesis testing with communication constraints. We further present improved bounds for the general image-size problem via our semigroup technique.

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Section: Introductionmentioning

confidence: 99%

Dispersion Bound for the Wyner-Ahlswede-Körner Network via a Semigroup Method on Types

Liu

2021

IEEE Trans. Inform. Theory

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“…While these works produce stronger bounds, they do not necessarily converge to the optimal limit and indeed basic questions about simulation remain open. For instance, till our work, even the following question was open [Kam15]: If P is the uniform disribution on {(0, 0), (0, 1), (1, 0)} and Q = DSBS(.49) (i.e. U, V are uniformly ±1, with E[U V ] = .49), can P simulate Q arbitrarily well?…”

Section: Estimating Binary Correlations: Previous Work and Our Resultmentioning

confidence: 96%

Decidability of Non-interactive Simulation of Joint Distributions

Ghazi

Kamath

Sudan

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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“…The equivalent formulations of reverse hypercontractivity were observed in [59], where the proof is based on the method of types.…”

Section: Reverse Hypercontractivity (Positive Parameters)mentioning

confidence: 97%

A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

Liu

Courtade

Cuff³

et al. 2018

Entropy

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Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality that unifies both the Brascamp-Lieb inequality and Barthe's inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.

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Reverse hypercontractivity using information measures

Abstract: We provide an equivalent description of reverse hypercontractivity using inequalities among information measures.

Cited by 23 publications

References 24 publications

Dispersion Bound for the Wyner-Ahlswede-Körner Network via a Semigroup Method on Types

Dispersion Bound for the Wyner-Ahlswede-Körner Network via a Semigroup Method on Types

Decidability of Non-interactive Simulation of Joint Distributions

A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

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