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A log-Sobolev inequality for the multislice, with applications [article]

Yuval Filmus, Ryan O'Donnell, Xinyu Wu
2018 arXiv   pre-print
We show that the log-Sobolev constant ρ_κ for the chain satisfies (ρ_κ)^-1≤ n ∑_i=1^ℓ12_2(4n/κ_i), which is sharp up to constants whenever ℓ 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  ...  Hypercontractivity, influences, and other preliminaries for our applications In this section we make some further definitions, which will be useful for our applications of the log-Sobolev inequality.  ... 
arXiv:1809.03546v1 fatcat:rq4sw3bsvfgflmqmv3tiup4lde

A Log-Sobolev Inequality for the Multislice, with Applications

Yuval Filmus, Ryan O'Donnell, Xinyu Wu, Michael Wagner
2018 Innovations in Theoretical Computer Science  
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  ...  The lower shadow of A at κ is 2 0 1 9 34: 8 A 98 Log-Sobolev Inequality for the Multislice, with Applications 3 −Θ(n) (for example) must have conductance Ω(n κ ).  ... 
doi:10.4230/lipics.itcs.2019.34 dblp:conf/innovations/FilmusOW19 fatcat:bkvdwesdobe7nab3ajopcduo4q

A sharp log-Sobolev inequality for the multislice

Justin Salez
2021 Annales Henri Lebesgue  
We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant.  ...  Among other applications, we completely quantify the small-set expansion phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.  ...  Acknowledgements The author warmly thanks Jonathan Hermon for his valuable comments on a preliminary version of this work.  ... 
doi:10.5802/ahl.99 fatcat:3pabs2y7xrd45iodmh6fj6y75i

A sharp log-Sobolev inequality for the multislice [article]

Justin Salez
2020 arXiv   pre-print
We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant.  ...  Among other applications, we completely quantify the "small-set expansion" phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.  ...  Acknowledgment The author warmly thanks Jonathan Hermon for his valuable comments on a preliminary version of this work.  ... 
arXiv:2004.05833v1 fatcat:32w23gxehrdevc4n2sve2ndovy

Log-Sobolev inequality for the multislice, with applications

Yuval Filmus, Ryan O'Donnell, Xinyu Wu
2022 Electronic Journal of Probability  
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  ...  Log-Sobolev inequality for the multislice, with applications draw v ∼ π |b in the following unusual way. First, draw u ∼ π |a .  ... 
doi:10.1214/22-ejp749 fatcat:gijdtvufknfihhuixjkbtny2qm

Concentration inequalities on the multislice and for sampling without replacement [article]

Holger Sambale, Arthur Sinulis
2020 arXiv   pre-print
We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials.  ...  Moreover, we give a proof of Talagrand's convex distance inequality for the multislice.  ...  presence of certain discrete log-Sobolev inequalities) and [APS20, Corollary 5.4] (modified log-Sobolev inequalities for Glauber dynamics) for the multislice.  ... 
arXiv:2010.16289v1 fatcat:ksokb42tevg7zowxswbkmpykna

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates [article]

Radosław Adamczak, Bartłomiej Polaczyk, Michał Strzelecki
2020 arXiv   pre-print
We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as p→ 1^+ are equivalent to the modified log Sobolev inequality (previously only one implication  ...  We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations  ...  We would like to thank Franck Barthe, Sergey Bobkov, and Paweł Wolff for discussions concerning the equivalence between various functional inequalities and encouragement to pursue the topics presented  ... 
arXiv:2007.10209v2 fatcat:3j7l5bhouzfz5fn5ovj6udokce

Entropy inequalities for random walks and permutations [article]

Alexandre Bristiel, Pietro Caputo
2022 arXiv   pre-print
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,  ...  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.  ...  One particle problems We start by recalling the definition of the log-Sobolev and modified log-Sobolev constants and their relations with the entropy constant κ(G) defined in (1.3).  ... 
arXiv:2109.06009v3 fatcat:xhbgd2jot5gpzk23g5dxhlaysu

Concentration inequalities for some negatively dependent binary random variables [article]

Radosław Adamczak, Bartłomiej Polaczyk
2021 arXiv   pre-print
Our approach is based on recent results by Hermon and Salez and a general framework involving modified log-Sobolev inequalities on the discrete cube, which is of independent interest.  ...  We also treat in detail the special case of independent Bernoulli random variables conditioned on their sum for which we obtain strengthened estimates, deriving in particular modified log-Sobolev inequalities  ...  the modified log-Sobolev inequality (4.4) is known.  ... 
arXiv:2108.12636v1 fatcat:uhv7b5rxg5fpxn3jpektm33n5u

Boolean constant degree functions on the slice are juntas

Yuval Filmus, Ferdinand Ihringer
2019 Discrete Mathematics  
Moreover, we show that the maximum number of coordinates that a Boolean degree d function can depend on is the same on the slice and on the hypercube.  ...  We show that a Boolean degree d function on the "slice" is a junta (depends on a constant number γ(d) of coordinates), assuming that k, n − k are large enough.  ...  Lee and Yau [12] proved a log Sobolev inequality, which together with classical results of Diaconis and Saloff-Coste [2] implies that the following hypercontractive inequality holds for some constant  ... 
doi:10.1016/j.disc.2019.111614 fatcat:no76y3klbfffrhpfgomczeqjzm

Boolean constant degree functions on the slice are juntas [article]

Yuval Filmus, Ferdinand Ihringer
2018 arXiv   pre-print
Moreover, we show that the maximum number of coordinates that a Boolean degree d function can depend on is the same on the slice and the hypercube.  ...  This generalizes a classical result of Nisan and Szegedy on the hypercube.  ...  Lee and Yau [8] proved a log Sobolev inequality, which together with classical results of Diaconis and Saloff-Coste [1] implies that the following hypercontractive inequality holds for some constant  ... 
arXiv:1801.06338v2 fatcat:toylk6er7jfcppml4mx4awoi5i

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for ε-Product Spaces [article]

Tom Gur, Noam Lifshitz, Siqi Liu
2021 arXiv   pre-print
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  ...  As applications, we obtain Fourier concentration, small-set expansion, and Kruskal-Katona theorems for high dimensional expanders.  ...  A log-Sobolev inequality for the multislice, with applications. In Proceedings of the 10th Innovations in Theoretical Computer Science conference (ITCS’19), 2019. [27] Louis Golowich.  ... 
arXiv:2111.09375v4 fatcat:pd4s5ihsqjcdvox5sikcrqmaoy

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments [article]

Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
2021 arXiv   pre-print
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  ...  have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018).  ...  Hypercontractivity is classically connected to the Log-Sobolev inequality, which gives strong control over the mixing time of its associated random walk.  ... 
arXiv:2111.09444v2 fatcat:glm7cfkvgrbn7ck7knabem7fgq

Generalizations and Applications of Hypercontractivity and Small-Set Expansion

Yu Zhao
2022
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. ?  ...  We present a new kind of "one-block decoupling" with better parameters than the classical results.  ...  Nelson [Nel73] gave the full Hypercontractivity Theorem in the Gaussian setting independently. Gross [Gro75] derived Nelson's result from the Log-Sobolev Inequalities.  ... 
doi:10.1184/r1/19699780 fatcat:zmjgoh5yzbbufezy7lfe76i6la