Spreading of Sets in Product Spaces and Hypercontraction of the Markov Operator

By R. Ahlswede, P. Gács. 1976

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Communication lower bounds for statistical estimation problems via a distributed data processing inequality (2015, methodology): “t there we don’t have a completely information theoretic proof of the linear lower bound [CR11]. We will be using the strong data processing inequality for the binary symmetric channel ﬁrst proven by [AG76]. it studies how information decays on a binary symmetric channel. Suppose Xbe a bit distribut”
Information-Theoretic Lower Bounds on Bayes Risk in Decentralized Estimation (2016, positive): “Some explicitly known expressions are: • For binary symmetric channels, η(Bern( (1)2 ),BSC(ε)) = η(BSC(ε)) = (1 − 2ε)2 [22].”
An Information-Theoretic View for Deep Learning (2018, background): “The rst is the strong data processing inequalities (SDPIs) proposed by (Ahlswede and Gács 1976), which states that for a Markov chain U → V → W , if there is information loss on the mapping V → W , then I(U, V ) ≤ ηI(U,W ) and η < 1 is a nonnegative information loss factor.”
Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels (2014, methodology): “In a remarkable paper [1], Ahlswede and Gács have uncovered deep relationships between η(μ, K ) and several other quantities, such as the maximal correlation (see [2] and references therein) and so-called hypercontractivity constants of a certain Markov operator associated to the pair (μ, K ) .”
Extra Samples can Reduce the Communication for Independence Testing (2018, background): “[7], [10], [6], [8], [4], [14]) to relate the measures of rectangles under the joint and the product distributions.”
$f$ -Divergence Inequalities (2015, background): “SUMMARY Since many distance measures of interest fall under the common paradigm of an f -divergence, it is not surprising that bounds on the ratios of various f -divergences are useful in many instances such as proving convergence of probability measures according to various metrics, analysis of rat”
Non-interactive simulation of joint distributions: The Hirschfeld-Gebelein-Rényi maximal correlation and the hypercontractivity ribbon (2012, background): “Using (6), we have For Z satisfying E Z = 0 , E Z 2 1 , The ﬁrst two terms of the expansion on both sides of (24) match.”
Differentially Private Federated Learning: An Information-Theoretic Perspective (2021, unknown): “To account for this, it is customary to consider the contraction coefﬁcient [32] This quantity has been recently in details in [19].”