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

Klochkov, Yegor; Zhivotovskiy, Nikita

doi:10.48550/arxiv.1812.03548

Cited by 3 publications

(6 citation statements)

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“…This inequality provides a concentration bound for quadratic forms of independent random variables and is more general than the aforementioned Bernstein's inequality. In the literature, results of similar flavor have been obtained [3,16,33] but under different assumptions. We will give a brief comparison between our result and a few notable ones in Section 3.…”

Section: Introductionmentioning

confidence: 82%

“…In a more recent paper by Klochkov and Zhivotovskiy [16], the authors proved a uniform version of the Hanson-Wright inequality which, when applying to a single matrix under the same assumption as Corollary 3.1, yields the following bound:…”

Section: Comparison With Other Hanson-wright Inequalitiesmentioning

confidence: 99%

“…To show (16), we need to find a λ such that E exp(λZ 2 /L 2 ) > 2. So by a change of variable t = v 2 L 2 , it suffices to show…”

Section: An Example For Lower Boundmentioning

confidence: 99%

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Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Jeong

LiP

Plan

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to R n . Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Random Gaussian linear mappings have been the object of much study, while the sub-Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub-Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence.We study when a sub-Gaussian matrix can become a near isometry on a set, show that previous best known dependence on the sub-Gaussian norm was sub-optimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein type inequality for sub-exponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables, in both cases improving the bounds in the sub-Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson-Lindenstrauss embeddings, randomized sketches and blind demodulation, whose theoretical guarantees can be improved by our results in the sub-Gaussian case.

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Section: Introductionmentioning

confidence: 82%

Section: Comparison With Other Hanson-wright Inequalitiesmentioning

confidence: 99%

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Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Jeong

LiP

Plan

et al. 2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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“…The next result contains deviation inequalities for suprema of quadratic forms in the spirit of [KZ18] for the weakly dependent case. We set f A (x) := sup A∈A Ax and Σ := sup A∈A A op .…”

Section: This Follows Immediately From Theorem 11 and The Inequality |∇mentioning

confidence: 99%

“…2ρb for some absolute constant c (the same one as in [KZ18]). However, this is at the cost of a weaker denominator in the Gaussian term as compared to (1.2).…”

Section: Introductionmentioning

confidence: 99%

Modified log-Sobolev inequalities and two-level concentration

Sambale,

Sinulis

2019

Preprint

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We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form Ent µ (e f ) ≤ ρ/2 E µ e f Γ(f ) 2 for some difference operator Γ, and show how it implies two-level concentration inequalities akin to the Hanson-Wright or Bernstein inequality. This can be applied to the continuous (e. g. the sphere or bounded perturbations of product measures) as well as discrete setting (finite measures satisfying an approximate tensorization property, stationary measures of Markov chains).

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Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

Shah

Frot

Thanei

et al. 2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We consider the problem of estimating a high dimensional p×p covariance matrix Σ, given n observations of confounded data with covariance Σ+ΓnormalΓsans-serifT, where Γ is an unknown p×q matrix of latent factor loadings. We propose a simple and scalable estimator based on the projection onto the right singular vectors of the observed data matrix, which we call right singular vector projection (RSVP). Our theoretical analysis of this method reveals that, in contrast with approaches based on the removal of principal components, RSVP can cope well with settings where the smallest eigenvalue of normalΓsans-serifTΓ is relatively close to the largest eigenvalue of Σ, as well as when the eigenvalues of normalΓsans-serifTΓ are diverging fast. RSVP does not require knowledge or estimation of the number of latent factors q, but it recovers Σ only up to an unknown positive scale factor. We argue that this suffices in many applications, e.g. if an estimate of the correlation matrix is desired. We also show that, by using subsampling, we can further improve the performance of the method. We demonstrate the favourable performance of RSVP through simulation experiments and an analysis of gene expression data sets collated by the GTEX consortium.

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

Cited by 3 publications

References 0 publications

Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Non-Gaussian Random Matrices on Sets:Optimal Tail Dependence and Applications

Modified log-Sobolev inequalities and two-level concentration

Right Singular Vector Projection Graphs: Fast High Dimensional Covariance Matrix Estimation under Latent Confounding

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