On the spectral norm of Gaussian random matrices

Handel, Ramon van

doi:10.1090/tran/6922

Cited by 39 publications

(50 citation statements)

References 9 publications

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“…Let us note that, on the face of it, this conclusion is quite striking: it is trivial that the operator norm of a matrix must be large if the matrix possesses a row with large Euclidean norm; what we have shown is that for symmetric Gaussian matrices with independent centered entries, this is the only reason why the operator norm can be large, regardless of the variance pattern of the matrix entries. For some further discussion and a different probabilistic interpretation, see [17]. It turns out that the above probabilistic formulation can be developed in much greater generality.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Unfortunately, this bound already yields the optimal result that could be obtained for the operator norm by the moment method. In order to surmount this obstacle, the second author developed in [17] an entirely different method to bound the operator norm of nonhomogeneous Gaussian matrices through the geometry of random processes. This approach made it possible to prove the following dimension-free bound, cf.…”

Section: 2mentioning

confidence: 99%

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The dimension-free structure of nonhomogeneous random matrices

2018

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Let X be a symmetric random matrix with independent but nonidentically distributed centered Gaussian entries. We show thatwhere Sp denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case p = ∞, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on ℓ 2 . Along the way, we obtain optimal dimension-free bounds on the moments (E X p Sp ) 1/p that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.2000 Mathematics Subject Classification. 60B20; 46B09; 46L53; 15B52.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

The dimension-free structure of nonhomogeneous random matrices

2018

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“…Note that even the most trivial of examples from the random matrix perspective, such as the case where X is a diagonal matrix with i.i.d. Gaussian entries on the diagonal, require already a delicate multiscale net to obtain sharp results; see, e.g., [30].…”

Section: Remark 27mentioning

confidence: 99%

Structured Random Matrices

Handel

2017

The IMA Volumes in Mathematics and Its Applications

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Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.

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“…Unfortunately, this is not the case in general; a simple counterexample was found by Seginer [54], see [8,Remark 4.8]. Nevertheless, it is an open conjecture of Latala that Seginer's theorem does hold if M has independent Gaussian entries, see the papers [52,56] and the survey [57].…”

Section: )mentioning

confidence: 99%

Concentration of Random Graphs and Application to Community Detection

Levina

Vershynin

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Random matrix theory has played an important role in recent work on statistical network analysis. In this paper, we review recent results on regimes of concentration of random graphs around their expectation, showing that dense graphs concentrate and sparse graphs concentrate after regularization. We also review relevant network models that may be of interest to probabilists considering directions for new random matrix theory developments, and random matrix theory tools that may be of interest to statisticians looking to prove properties of network algorithms. Applications of concentration results to the problem of community detection in networks are discussed in detail.2010 Mathematics Subject Classification. 05C80 (primary) and 05C85, 60B20 (secondary).

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On the spectral norm of Gaussian random matrices

Cited by 39 publications

References 9 publications

The dimension-free structure of nonhomogeneous random matrices

The dimension-free structure of nonhomogeneous random matrices

Structured Random Matrices

Concentration of Random Graphs and Application to Community Detection

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