Products of Independent non-Hermitian Random Matrices

O’Rourke, Sean; Soshnikov, Alexander

doi:10.1214/ejp.v16-954

Cited by 66 publications

(61 citation statements)

References 27 publications

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“…[20]. Our result elucidates the transparent analytic structure noted in several papers on the products of random matrices [6][7][8][9]12,[21][22][23][24][28][29][30][31][32][33] and provides a powerful tool for the derivation of similar results for products of some application-designed isotropic random matrices of large (infinite) size. and 1 otherwise.…”

Section: Discussionsupporting

confidence: 83%

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Spectral relations between products and powers of isotropic random matrices

2012

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We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble is equal to the eigenvalue density of nth power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation, one can derive the limiting density of the product of n independent identically distributed non-Hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide evidence that the result holds also for isotropic orthogonal ensembles.

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Section: Discussionsupporting

confidence: 83%

“…This result agrees with that obtained using different methods in Refs. [6,[21][22][23][24], as mentioned in the Introduction of the paper.…”

Section: Applicationsmentioning

confidence: 99%

Spectral relations between products and powers of isotropic random matrices

2012

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“…Therefore, case (b) is closely related to a product of d freely independent random variables; precise results are obtained in [11]. Earlier results [12,22] examine case (b) without explicit use of free probability. The problem of first taking n → ∞ and afterwards taking d → ∞ can also be handled using the tools of free probability in the case of Gaussian matrices, see [27].…”

Section: Connection To Previous Work In Random Matrix Theorymentioning

confidence: 97%

Products of Many Large Random Matrices and Gradients in Deep Neural Networks

Hanin

Nica

2019

Commun. Math. Phys.

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We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the ℓ2 norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn. We also obtain explicit error bounds on the moments of the norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with ReLU activations. This provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with ReLU activations and a given architecture.

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“…Macroscopic properties for eigenvalues of complex (β = 2) matrices have been discussed in the limit of large matrices using diagrammatic methods [4,13,14], while proofs are given * akemann@physik.uni-bielefeld.de † jipsen@math.uni-bielefeld.de ‡ mkieburg@physik.uni-bielefeld.de in [15,16]. The macroscopic behavior of the singular values and their moments have also been discussed in the literature using probabilistic methods [17][18][19] as well as diagrammatic methods [14].…”

Section: Introductionmentioning

confidence: 99%

Products of rectangular random matrices: Singular values and progressive scattering

2013

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We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.

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Products of Independent non-Hermitian Random Matrices

Cited by 66 publications

References 27 publications

Spectral relations between products and powers of isotropic random matrices

Spectral relations between products and powers of isotropic random matrices

Products of Many Large Random Matrices and Gradients in Deep Neural Networks

Products of rectangular random matrices: Singular values and progressive scattering

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