IA Scholar Query: A Log-Sobolev Inequality for the Multislice, with Applications.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 10 May 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Entropy inequalities for random walks and permutations
https://scholar.archive.org/work/pykololvpfafbhixn6s3bexsr4
We consider a new functional inequality controlling the rate of relative entropy decay for random walks, the interchange process and more general block-type dynamics for permutations. The inequality lies between the classical logarithmic Sobolev inequality and the modified logarithmic Sobolev inequality, roughly interpolating between the two as the size of the blocks grows. Our results suggest that the new inequality may have some advantages with respect to the latter well known inequalities when multi-particle processes are considered. We prove a strong form of tensorization for independent particles interacting through synchronous updates. Moreover, for block dynamics on permutations we compute the optimal constants in all mean field settings, namely whenever the rate of update of a block depends only on the size of the block. Along the way we establish the independence of the spectral gap on the number of particles for these mean field processes. As an application of our entropy inequalities we prove a new subadditivity estimate for permutations, which implies a sharp upper bound on the permanent of arbitrary matrices with nonnegative entries, thus resolving a well known conjecture.Alexandre Bristiel, Pietro Caputowork_pykololvpfafbhixn6s3bexsr4Tue, 10 May 2022 00:00:00 GMTGeneralizations and Applications of Hypercontractivity and Small-Set Expansion
https://scholar.archive.org/work/4rliulq3gbcwrpkbwrni6nm32m
Hypercontractive inequalities and small-set expansion are two fundamental topics closely related to each other and play important roles in many fields, including hardness of approximation, probability theory, social choice theory, information theory, and cryptography. This thesis studies generalizations and applications of hypercontractivity and small-set expansion in the following areas: ? The recent breakthrough proof of the 2-to-2 games conjecture was completed by showing a pseudorandom-set expansion result on Grassmann graphs [KMS18]. A similar property has also been shown on Johnson graphs [KMMS18]. These results can be seen as an improved version of small-set expansion on pseudorandom sets. We prove the pseudorandom-set expansion result on general product probability spaces, with a very clean and short proof. A key step in the proof involves a new hypercontractive-style inequality. ? The communication-assisted agreement distillation problem is about two parties with noisy private randomness trying to extract a common random string via communication. We give the optimal upper bound on the amount of communication necessary for achieving constant success probability for this problem. In addition, we calculate the optimal communication for the reverse binary erasure channel case by studying properties of extreme points in its hypercontractivity region. The proof technique is highly related to the equivalence of hypercontractivity and small-set expansion. ? "Decoupling" refers to the idea of analyzing a complicated random sum involving dependent random variables by comparing it to a simpler random sum where some independence is introduced between the variables. We present a new kind of "one-block decoupling" with better parameters than the classical results. We use decoupling and hypercontractivity to show tight tail bounds of low-degree Boolean functions and tight versions of several theorems from [DFKO07]. ? A probability distribution over f {-1,1}n is k-wise uniform if its marginal distribution on [...]Yu Zhaowork_4rliulq3gbcwrpkbwrni6nm32mWed, 04 May 2022 00:00:00 GMTLog-Sobolev inequality for the multislice, with applications
https://scholar.archive.org/work/egegdv4ekfbj7bi7ryfjbc4af4
Let κ ∈ N + satisfy κ1 + • • • + κ = n, and let Uκ denote the multislice of all strings u ∈ [ ] n having exactly κi coordinates equal to i, for all i ∈ [ ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant κ for the chain satisfies which is sharp up to constants whenever is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal-Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan-Szegedy Theorem.Yuval Filmus, Ryan O'Donnell, Xinyu Wuwork_egegdv4ekfbj7bi7ryfjbc4af4Sat, 01 Jan 2022 00:00:00 GMTHypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for ε-Product Spaces
https://scholar.archive.org/work/tliaeydynnhnbas6fhzowbql44
We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we obtain Fourier concentration, small-set expansion, and Kruskal-Katona theorems for high dimensional expanders. Our techniques rely on a new approximate Efron-Stein decomposition for high dimensional link expanders.Tom Gur, Noam Lifshitz, Siqi Liuwork_tliaeydynnhnbas6fhzowbql44Fri, 24 Dec 2021 00:00:00 GMTHypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments
https://scholar.archive.org/work/juqra2mn7nfajaxawnkeagsvj4
Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the p-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets. Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovettwork_juqra2mn7nfajaxawnkeagsvj4Wed, 24 Nov 2021 00:00:00 GMT