Hypercontractive Inequality Research Articles

Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae’94). Using our hypercontractive inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with t -size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an \((r-\varepsilon)\) -approximation of this value requires \(\Omega (n^{1-2/t})\) quantum space, where r is the alphabet size. We next present a lower bound for locally decodable codes ( \(\mathsf {LDC}\) ) \(\mathbb {Z}_r^n\rightarrow \mathbb {Z}_r^N\) over large alphabets with recoverability probability at least \(1/r + \varepsilon\) . Using hypercontractivity, we give an exponential lower bound \(N= 2^{\Omega (\varepsilon ^4 n/r^4)}\) for 2-query (possibly non-linear) \(\mathsf {LDC}\) s over \(\mathbb {Z}_r\) and using the non-commutative Khintchine inequality we prove an improved lower bound of \(N= 2^{\Omega (\varepsilon ^2 n/r^2)}\) .

An Information-Theoretic Proof of a Hypercontractive Inequality.

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of this inequality begin with two-point space, where some elementary calculus is used and then generalised immediately by introducing another dimension using submultiplicativity (Minkowski's integral inequality). In this paper, we prove this inequality using information theory. We compare the entropy of a pair of correlated vectors in {0,1}n to their separate entropies, analysing them bit by bit (not as a figure of speech, but as the bits are revealed) using the chain rule of entropy.

Hypercontractivity on the symmetric group

Abstract The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$ , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on $S_n$ , which are functions whose $2$ -norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small q-norms, for $q>2$ . As applications, we show the following: 1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$ . 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice and stability versions of the Kruskal–Katona Theorem in some regimes of parameters.

Hypercontractivity and transport-information inequalities for Gaussian ARMA model

Hypercontractivity and transport-information inequalities for Gaussian ARMA model

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker–Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner's inequality.

Hypercontractivity for global functions and sharp thresholds

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut’s junta theorem and the invariance principle of Mossel, O’Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ( 1 / 2 1/2 -biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p p -biased measures. However, simple examples show that when p p is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general p p that applies to ‘global functions’, i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain’s sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain’s theorem, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a p p -biased analogue of the seminal invariance principle of Mossel, O’Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019.

Hypercontractive inequalities for weighted Bergman spaces

AbstractWe obtain sharp hypercontractive inequalities for the weighted Bergman spaces on the unit disk with the usual weights for , thus solving an interesting case of a problem from Janson [Ark. Math. 21 (1983), no. 1, 97–110]. We also give some estimates for .

Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber's-Type Inequalities for the Second Rényi Entropy.

Let , , be the noise operator acting on functions on the boolean cube . Let f be a distribution on and let . We prove tight Mrs. Gerber-type results for the second Rényi entropy of which take into account the value of the Rényi entropy of f. For a general function f on we prove tight hypercontractive inequalities for the norm of which take into account the ratio between and norms of f.

Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamatedmathbb {L}_p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated mathbb {L}_p norms are not uniformly convex for 1le p le 2. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

Improved quantum hypercontractivity inequality for the qubit depolarizing channel

The hypercontractivity inequality for the qubit depolarizing channel Ψt states that ‖Ψt⊗n(X)‖p≤‖X‖q, provided that p ≥ q > 1 and t≥lnp−1q−1. In this paper, we present an improvement of this inequality. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev inequalities and hypercontractivity inequalities to obtain our main result. As applications of these results, we present an asymptotically tight quantum Faber–Krahn inequality on the hypercube and a new quantum Schwartz–Zippel lemma.

Contractivity properties of Ornstein–Uhlenbeck semigroup for mixed q-Araki–Woods von Neumann algebras

We study certain non-tracial von Neumann algebras generated by some self-adjoint operators satisfying mixed q-commutation relations. Such algebras are discussed in the work of Bikram et al. [“Mixed q-deformed Araki-Woods von Neumann algebras,” (submitted)]. We prove the analog of Nelson’s hypercontractivity inequality for the mixed q-Ornstein–Uhlenbeck semigroup. We also show that the mixed q-Ornstein–Uhlenbeck semigroup is ultracontractive.

Hyper-positive definite functions I: Scalar case, branching-type stationary stochastic processes

We propose a definition of branching-type stationary stochastic processes on rooted trees and related definitions of hyper-positivity for functions on the unit circle and functions on the set of non-negative integers. We then obtain (1) a necessary and sufficient condition on a rooted tree for the existence of non-trivial branching-type stationary stochastic processes on it, (2) a complete criterion of the hyper-positive functions in the setting of rooted homogeneous trees in terms of a variant of the classical Herglotz-Bochner Theorem, (3) a prediction theory result for branching-type stationary stochastic processes.As an unexpected application, we obtain natural hypercontractive inequalities for Hankel operators with hyper-positive symbols.

A Moment Ratio Bound for Polynomials and Some Extremal Properties of Krawchouk Polynomials and Hamming Spheres

Let p ≥ 2. We improve the bound ||f|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> / ||f|| <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ≤ (p-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s/2</sup> for a polynomial f of degree s on the boolean cube {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of p and s, which is smaller than (p-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s/2</sup> for any p > 2 and s > 0. We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree s. This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> The Hamming spheres have the following approximate edge-isoperimetric property: For all 1 ≤ s ≤ n/2, and for all even distances 0 ≤ i ≤ 2s(n-s) / n, the Hamming sphere of radius s contains, up to a multiplicative factor of O(i), as many pairs of points at distance i as possible, among sets of the same size (there is a similar, but slightly weaker and somewhat more complicated claim for all distances). This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate. We also describe a family of hypercontractive inequalities for functions on {0,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , which improve on the `usual' “ q → 2” inequality by taking into account the concentration of a function (expressed as the ratio between its <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> norms), and which are nearly tight for characteristic functions of Hamming spheres. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> This has to be interpreted with some care.

Nonlocal Games with Noisy Maximally Entangled States are Decidable

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 15 June 2020Accepted: 09 August 2021Published online: 02 December 2021Keywordsnonlocal games, Fourier analysis, hypercontractive inequality, quantum invariance prinicipleAMS Subject Headings68Q12Publication DataISSN (print): 0097-5397ISSN (online): 1095-7111Publisher: Society for Industrial and Applied MathematicsCODEN: smjcat

A Note on the Probability of Rectangles for Correlated Binary Strings

Consider two sequences of n independent and identically distributed fair coin tosses, X = (X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , . . . , X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) and Y = (Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , . . . , Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ), which are ρ-correlated for each j, i.e. P[X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> = Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> ] = 1+ρ/2 .We study the question of how large (small) the probability P[X ∈ A, Y ∈ B] can be among all sets A, B ⊂ {0, 1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> of a given cardinality. For sets |A|, |B| = Θ(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of |A|, |B| = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Θ</sup> (n). By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize P[X ∈ A, Y ∈ B] in the regime of p → 1. We also prove a similar tight lower bound, i.e. show that for p → 0 the pair of opposite Hamming balls approximately minimizes the probability P[X ∈ A, Y ∈ B].

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q (−1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U() = Γq(e−tI) (−1 ≤ q ≤ 1), where Γq is a q-Gaussian functor.

Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

A supersolutions perspective on hypercontractivity

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.

Moments of random multiplicative functions, II: High moments

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of $\sum_{n \leq x} f(n)$. The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyse this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob's $L^p$ maximal inequality for martingales.

Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

On a stratified Lie group G equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on Lp spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group G. Moreover, if G satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.