On Reverse Hypercontractivity

By Elchanan Mossel, K. Oleszkiewicz, Arnab Sen. Geometric and Functional Analysis, 2011

94 citations recorded by the source corpus.

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Stroock–Varopoulos inequality. A consequence of our analysis is that all simple operators $${L = Id - \mathbb{E}}$$ as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all q < p < 1 and every positive valued function f for $${t \geq \log \frac{1-q}{1-p}}$$ we have $${\| e^{-tL}f\|_{q} \geq \| f\|_{p}}$$. This should be contrasted with the case of hypercontractive inequalities for simple operators where t is known to depend not only on p and q but also on the underlying space. The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with m-sided dice.

Subfields: Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ca6e8f8bfd3790aae63dd54ac605d8103c29c520

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Citations in the corpus

Non-interactive simulation of joint distributions: The Hirschfeld-Gebelein-Rényi maximal correlation and the hypercontractivity ribbon (2012, methodology): “As in [3], we define for any non-negative random variable X, the function Ent(X) := EX logX − EX · logEX, where by convention 0 log 0 := 0.”
Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels (2014, methodology): “We now particularize the above general results to two speciﬁc types of functional inequalities: These inequalities are well-known in functional analysis and probability theory (see, e.g., [8], [24]–[27]).”
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.”