Almost sure localization of the eigenvalues in a Gaussian information plus noise model - Application to the spiked models

By Philippe Loubaton, Pascal Vallet. 2010

64 citations recorded by the source corpus.

Let $S$ be a $M$ times $N$ random matrix defined by $S = B + \sigma W$ where $B$ is a uniformly bounded deterministic matrix and where $W$ is an independent identically distributed complex Gaussian matrix with zero mean and variance $1/N$ entries. The purpose of this paper is to study the almost sure location of the eigenvalues of the Gram matrix $SS^*$ when $M$ and $N$ converge to infinity such that the ratio $M/N$ converges towards a constant $c > 0$. The results are used in order to derive, using an alternative approach, known results concerning the behavior of the largest eigenvalues of $SS^*$ when the rank of $B$ remains fixed and $M$ and $N$ converge to infinity.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ab78aa2be740c363a7267929c6b6de586360e139

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Citations in the corpus

Outliers in the spectrum of large deformed unitarily invariant models (2012, methodology): “See also [29, 4, 5] for sample covariance matrices, [24, 32, 22, 19, 33] for deformed Wigner models and [31] for information-plus-noise models.”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, unknown): “Note that motivated by the seminal article of Baik, Ben Arous and Péché [6], previous works are devoted to the study of the outlier eigenvalues of deformed random Hermitian matrix models, see notably [4, 7, 8, 9, 10, 19, 21, 22, 23, 28, 34, 35, 36, 37, 41, 43, 45, 46].”
A Random Matrix Approach to Neural Networks (2017, methodology): “…ELM_Romain_MSE.tex date: February 20, 2017 Transposed to our present setting, the aforementioned Taylor expansion allows for a consistent approximation Φ̃ of Φ by an information-plusnoise (spiked) random matrix model (see e.g., (Loubaton and Vallet, 2010; Benaych-Georges and Nadakuditi, 2012)).”
Statistical inference for principal components of spiked covariance matrices (2020, methodology): “The limiting behavior of the extreme eigenvalues have also been studied for various related models, such as the finite-rank deformation of Wigner matrices [22, 12, 24, 25, 34, 46, 47, 70, 73], the signal-plus-noise model [13, 53, 26], the general spiked β ensemble [17, 18], and also the finite-rank ”
Analysis of the limiting spectral measure of large random matrices of the separable covariance type (2013, background): “In [14, 10, 21] other separation results as well as statistical estimation algorithms along the lines of [16, 17] were proposed.”