Local Operator Theory, Random Matrices and Banach Spaces

Davidson, Kenneth R.; Szarek, Stanisław J.

doi:10.1016/s1874-5849(01)80010-3

Cited by 359 publications

(368 citation statements)

References 107 publications

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“…The same random matrix also appears in the local theory of Banach spaces or socalled asymptotic convex geometry (see e.g. [10,27]). An important problem that enters these frameworks is the study of some geometric parameters associated to i.i.d.…”

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confidence: 87%

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Pajor

Pastur

2009

Studia Math.

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Abstract. We consider n × n real symmetric and hermitian random matrices Hn that are sums of a non-random matrix H (0) n and of mn rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mn/n → c ∈ [0, ∞) as n → ∞, and the distribution of eigenvalues of H (0) n and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hn converges weakly in probability to the non-random limit, found by Marchenko and Pastur.

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confidence: 87%

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Pajor

Pastur

2009

Studia Math.

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“…Proof: Theorem II.13 in [42] established that suppose Γ is an p × t matrix, whose entries are all i.i.d. N (0, 1) Gaussian variables, then the largest singular value of Γ, denoted by s 1 (Γ), satisfies Pr s 1 (Γ) > √ p + √ t + √ p ∨ tǫ ≤ exp(−(p ∨ t)ǫ 2 /2).…”

Section: Proof Of Theoremmentioning

confidence: 99%

Outlier-Robust PCA: The High-Dimensional Case

Caramanis

Mannor

2013

IEEE Trans. Inform. Theory

107

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Principal Component Analysis plays a central role in statistics, engineering and science. Because of the prevalence of corrupted data in real-world applications, much research has focused on developing robust algorithms. Perhaps surprisingly, these algorithms are unequipped -indeed, unable -to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm that is as efficient as PCA, robust to contaminated points, and easily kernelizable. In particular, our algorithm achieves maximal robustness -it has a breakdown point of 50% (the best possible) while all existing algorithms have a breakdown point of zero. Moreover, our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sub linearly in the dimension.

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“…The simplest idea to get concentration inequalities for the largest eigenvalue of a GUE random matrix is to use Gaussian concentration ; it is a straightforward consequence of the measure concentration phenomenon in the Gaussian space (see [1]) that…”

Section: The Small Deviation Inequalitymentioning

confidence: 99%

“…The answer to the analogous question is known to be positive for the GOE (Gaussian Orthogonal Ensemble), an ensemble of real symmetric matrices defined in a similar way as GUE (see [10] for a precise definition). The argument, due to Gordon, uses a result about comparison of Gaussian processes known as Slepian's lemma (see [1]) and doesn't carry over to the complex setting.…”

Section: The Small Deviation Inequalitymentioning

confidence: 99%

A Sharp Small Deviation Inequality for the Largest Eigenvalue of a Random Matrix

Aubrun

2004

Lecture Notes in Mathematics

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Summary. We prove that the convergence of the largest eigenvalue λ1 of a n × n random matrix from the Gaussian Unitary Ensemble to its Tracy-Widom limit holds in a strong sense, specifically with respect to an appropriate Wasserstein-like distance. This unifying approach allows us both to recover the limiting behaviour and to derive the inequality P(λ1 2+t) C exp(−cnt 3/2 ), valid uniformly for all n and t. This inequality is sharp for "small deviations" and complements the usual "large deviation" inequality obtained from the Gaussian concentration principle. Following the approach by Tracy and Widom, the proof analyses several integral operators, which converge in the appropriate sense to an operator whose determinant can be estimated.

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Local Operator Theory, Random Matrices and Banach Spaces

Cited by 359 publications

References 107 publications

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Outlier-Robust PCA: The High-Dimensional Case

A Sharp Small Deviation Inequality for the Largest Eigenvalue of a Random Matrix

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