TY - JOUR

T1 - The spectrum of random inner-product kernel matrices

AU - Cheng, Xiuyuan

AU - Singer, Amit

N1 - Funding Information:
The authors were partially supported by Award Number DMS-0914892 from the NSF and by Award Number R01GM090200 from the NIGMS. A. Singer was partially supported by Award Numbers FA9550-12-1-0317 and FA9550-13-1-0076 from AFOSR and by Award Number LTR DTD 06-05-2012 from the Simons Foundation.

PY - 2013/10/1

Y1 - 2013/10/1

N2 - We consider n × n matrices whose (i, j)th entry is f(X-{i}-{T}X-{j}), where X1, ..., Xn are i.i.d. standard Gaussian in p, and f is a real-valued function. The weak limit of the eigenvalue distribution of these random matrices is studied at the limit when p, n → ∞ and p/n = γwhich is a constant. We show that, under certain conditions on the kernel function f, the limiting spectral density exists and is dictated by a cubic equation involving its Stieltjes transform. The parameters of this cubic equation are decided by a Hermite-like expansion of the rescaled kernel function. While the case that f is differentiable at the origin has been previously resolved by El Karoui [The spectrum of kernel random matrices, Ann. Statist.38 (2010) 1-50], our result is applicable to non-smooth f, such as the Sign function and the hard thresholding operator of sample covariance matrices. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marčenko-Pastur (M.P.) distribution and Wigner's semi-circle distribution as special cases.

AB - We consider n × n matrices whose (i, j)th entry is f(X-{i}-{T}X-{j}), where X1, ..., Xn are i.i.d. standard Gaussian in p, and f is a real-valued function. The weak limit of the eigenvalue distribution of these random matrices is studied at the limit when p, n → ∞ and p/n = γwhich is a constant. We show that, under certain conditions on the kernel function f, the limiting spectral density exists and is dictated by a cubic equation involving its Stieltjes transform. The parameters of this cubic equation are decided by a Hermite-like expansion of the rescaled kernel function. While the case that f is differentiable at the origin has been previously resolved by El Karoui [The spectrum of kernel random matrices, Ann. Statist.38 (2010) 1-50], our result is applicable to non-smooth f, such as the Sign function and the hard thresholding operator of sample covariance matrices. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marčenko-Pastur (M.P.) distribution and Wigner's semi-circle distribution as special cases.

KW - Hermite polynomials

KW - Kernel matrices

KW - Stieltjes transform

KW - limiting spectrum

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U2 - 10.1142/S201032631350010X

DO - 10.1142/S201032631350010X

M3 - Article

AN - SCOPUS:84958556046

VL - 2

JO - Random Matrices: Theory and Application

JF - Random Matrices: Theory and Application

SN - 2010-3263

IS - 4

M1 - 1350010

ER -