The largest eigenvalue of small rank perturbations of Hermitian random matrices

Péché, Sandrine

doi:10.1007/s00440-005-0480-1

Cited by 59 publications

(113 citation statements)

References 15 publications

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“…Similar techniques apply to the study of the largest eigenvalue of so called spiked models, which are matrices of the form XT X * with X possessing i.i.d. complex entries and T a diagonal real matrix, all of whose entries except for a finite number equal to 1, and to small rank perturbations of Wigner matrices, see [BaBP05], [Péc06], [FeP07], [Kar07b] and [Ona08].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…This so-called "deformed Wigner ensemble" was studied in [13] and [5], where the authors focused mainly on the problem of the local spacings and in [16] and [8], where they studied the behaviour of the largest eigenvalue. In this framework, our goal in this paper will be to establish a large deviation principle for the largest eigenvalue of X N = W N + A N , that we denote in the sequel by x * N .…”

Section: Introductionmentioning

confidence: 99%

Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles

Maïda

2007

Electron. J. Probab.

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Abstract. We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the largest eigenvalue separates from the bulk when the perturbation is large enough. A large part of the paper is devoted to an auxiliary result on the continuity of spherical integrals in the case when one of the matrix is of rank one, as studied in [9].

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“…One natural approach for this problem is PCA: that is, diagonalize X and use its leading eigenvector v as an estimate of v. The threshold at which this algorithm succeeds can be computed using the theory of random matrices with rank-one perturbations [8,42,11]:…”

Section: Sparse Pcamentioning

confidence: 99%

Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization

Banks

Moore

Vershynin

et al. 2018

IEEE Trans. Inform. Theory

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We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. For sparse PCA and submatrix localization, we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse; for the clustering problem, our bounds differ by a factor of √ 2 when the number of clusters is large. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information-theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.

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The largest eigenvalue of small rank perturbations of Hermitian random matrices

Cited by 59 publications

References 15 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles

Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization

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