[Paper Review] Hypercontractivity and its applications

Punyashloka Biswal|arXiv (Cornell University)|Jan 14, 2011

Complexity and Algorithms in Graphs8 references3 citations

TL;DR

This paper presents a comprehensive survey of hypercontractivity in high-dimensional spaces, establishing its role in analyzing Fourier-analytic inequalities and their applications in theoretical computer science. By leveraging the Bonami-Gross-Beckner noise operator and hypercontractive norm inequalities, the work proves that the noise operator $ T_\rho $ maps $ \ell_1 $ to $ \ell_{1+\rho} $, enabling tight bounds on the integrality gap of the Unique Label Cover SDP, showing that even when no labeling satisfies more than $ 1/2^{\eta k} $ of edges, the SDP can still achieve value $ \geq 1 - \eta $. This demonstrates the power of hypercontractivity in proving strong inapproximability results.

ABSTRACT

Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of hypercontractivity, and also present some relatively recent applications of these techniques in computer science.

Motivation & Objective

• To unify and survey the theory of hypercontractive inequalities across discrete and continuous settings.
• To establish connections between hypercontractivity, Poincaré and logarithmic Sobolev inequalities, and spectral properties of the discrete cube.
• To demonstrate the utility of hypercontractivity in proving strong inapproximability results for combinatorial optimization problems.
• To analyze the soundness of the Khot-Vishnoi Unique Label Cover integrality gap instance using hypercontractive norm estimates.

Proposed method

• Use of the Fourier basis $ \chi_S(x) = \prod_{i \in S} x_i $ on the hypercube $ \{-1,1\}^n $ to represent functions and compute Fourier coefficients.
• Definition of the noise operator $ T_\rho f = \operatorname{BG}_\rho * f $, where $ \widehat{\operatorname{BG}}_\rho(S) = \rho^{|S|} $, modeling random bit flips with probability $ 1 - \rho $.
• Application of the Bonami-Gross-Beckner inequality to bound the $ \ell_2 $-norm of $ T_{\sqrt{\rho}} \phi $ for indicator functions $ \phi $, using hypercontractivity: $ \|T_{\sqrt{\rho}} \phi\|_2 \leq \|\phi\|_{1+\rho} $.
• Derivation of the derivative identity $ \frac{d}{d\rho} \operatorname{BG}_\rho = \frac{1}{\rho} \operatorname{BG}_\rho * \sum_i h_i $ in the Fourier domain to analyze the noise operator’s behavior.
• Construction of a Unique Label Cover instance using cosets of the hypercube group modulo the Fourier basis, with edges defined via $ \rho $-correlated pairs of functions.
• Reduction of the labeling consistency probability to an inner product $ \langle h, T_\rho h \rangle $, bounded via hypercontractivity to $ \|h\|_{1+\rho}^2 = 1/2^{2k/(1+\rho)} \leq 1/2^{\eta k} $.

Experimental results

Research questions

• RQ1 How do hypercontractive inequalities relate to spectral expansion and Poincaré/log-Sobolev constants on the discrete cube?
• RQ2 What is the role of the noise operator $ T_\rho $ in transforming $ \ell_p $ norms, and how does hypercontractivity constrain this transformation?
• RQ3 Can hypercontractivity be used to establish tight bounds on the integrality gap of the Unique Label Cover SDP?
• RQ4 How does the Khot-Vishnoi construction exploit hypercontractivity to achieve a large gap between the SDP optimum and the true optimal labeling?

Key findings

• The Poincaré constant of the discrete cube $ \{-1,1\}^n $ is $ \lambda = 2/n $, and the Log-Sobolev constant is $ \alpha = 1/n $, derived from the Dirichlet form and variance.
• The noise operator $ T_\rho $ satisfies $ \|T_\rho f\|_q \leq \|f\|_p $ for $ q = 1 + \rho(p-1) $, with equality in the $ \ell_1 \to \ell_{1+\rho} $ case, proving hypercontractivity.
• For any indicator function $ \phi $ on the hypercube with $ \mathbb{E}[\phi] = 1/2^k $, the probability $ \Pr_{h,h'}[\phi(h)\phi(h')] \leq 1/2^{\eta k} $ when $ \rho = 1 - 2\eta $, via hypercontractivity.
• The Khot-Vishnoi Unique Label Cover instance has an optimal labeling satisfying at most $ 1/2^{\eta k} $ of the edges, but the SDP relaxation achieves value at least $ 1 - \eta $, demonstrating a super-constant integrality gap.
• The soundness analysis relies on the hypercontractive bound $ \|T_{\sqrt{\rho}} \phi\|_2 \leq \|\phi\|_{1+\rho} $, which yields $ \|\phi\|_{1+\rho}^2 = 1/2^{2k/(1+\rho)} \leq 1/2^{\eta k} $, proving the gap.