A note on concentration for polynomials in the Ising model

Adamczak, Radosław; Kotowski, Michał; Polaczyk, Bartłomiej; Strzelecki, Michał

doi:10.1214/19-ejp280

Cited by 16 publications

(38 citation statements)

References 40 publications

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“…Remark 2.5. In the case that A = {A} our result implies the upper tail of the recent concentration inequality proved in Adamczak et al (2018a) (see Theorem 2.2 and Example 2.5). To show this fact (denoting σ = σ − Eσ) we observe that…”

Section: Uniform Concentration Results In the Ising Modelsupporting

confidence: 67%

“…Furthermore, we apply our techniques to obtain a uniform concentration result similar to Theorem 1.1 in a particular case of non-independent components. We consider the Ising model under Dobrushin's condition, a setting that has been studied recently by Adamczak et al (2018a) and Götze et al (2018). The question we study was raised by Marton (2003) in a closely related scenario.…”

Section: Introductionmentioning

confidence: 99%

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Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

Klochkov¹,

Zhivotovskiy²

2020

Electron. J. Probab.

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This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector X ∈ R n with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of X itself. Our results recover, in particular, the classic uniform bound of Talagrand (1996) for Rademacher chaoses and the more recent uniform result of Adamczak (2015) which holds under certain rather strong assumptions on the distribution of X. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici (2014). Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin's condition. A closely related question was posed by Marton (2003).

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Section: Uniform Concentration Results In the Ising Modelsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

Klochkov¹,

Zhivotovskiy²

2020

Electron. J. Probab.

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“…In particular, these works prove concentration bounds (which are qualitatively stronger than variance bounds) for multilinear functions of arbitrary degree d (rather than just bilinear functions, which are of degree d = 2). Further works sharpen these results both quantitatively (by narrowing the radius of concentration and weight of the tails) and qualitatively (with multilevel concentration results) [GSS18,AKPS18].…”

Section: Identity On Forestsmentioning

confidence: 68%

Testing Ising Models

Daskalakis¹,

Dikkala²,

Kamath³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Given samples from an unknown multivariate distribution p, is it possible to distinguish whether p is the product of its marginals versus p being far from every product distribution? Similarly, is it possible to distinguish whether p equals a given distribution q versus p and q being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters [BFF + 01, Pan08, VV17, ADK15, DK16]. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity.Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.

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Section: Testing Problemmentioning

confidence: 89%

Testing Ising Models

Daskalakis

Dikkala

Kamath

2019

IEEE Trans. Inform. Theory

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Given samples from an unknown multivariate distribution p, is it possible to distinguish whether p is the product of its marginals versus p being far from every product distribution? Similarly, is it possible to distinguish whether p equals a given distribution q versus p and q being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity.Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.

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A note on concentration for polynomials in the Ising model

Cited by 16 publications

References 40 publications

Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

Testing Ising Models

Testing Ising Models

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