The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Marwa Banna - One of the best experts on this subject based on the ideXlab platform.
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Limiting Spectral Distribution for matrices with correlated entries and Bernstein-type inequality
2015Co-Authors: Marwa BannaAbstract:In this thesis, we investigate mainly the limiting Spectral Distribution of random matrices having correlated entries and prove as well a Bernstein-type inequality for the largest eigenvalue of the sum of self-adjoint random matrices that are geometrically absolutely regular. We are interested in the asymptotic Spectral behavior of sample covariance matrices and Wigner-type matrices having correlated entries that are functions of independent random variables. We show that the limiting Spectral Distribution can be obtained by analyzing a Gaussian matrix having the same covariance structure. This approximation approach is valid for both short and long range dependent stationary random processes just having moments of second order. Our approach is based on a blend of a blocking procedure, Lindeberg's method and the Gaussian interpolation technique. We also develop new tools including a concentration inequality for the Spectral measure for matrices having K-dependent rows. This method permits to derive, under mild conditions, an integral equation of the Stieltjes transform of the limiting Spectral Distribution. Applications to matrices whose entries consist of functions of linear processes, ARCH processes or non-linear Volterra-type processes are also given. We also investigate the asymptotic behavior of Gram matrices having correlated entries that are functions of an absolutely regular random process. We give a concentration inequality of the Stieltjes transform and prove that, under an arithmetical decay condition on the absolute regular coefficients, it is almost surely concentrated around its expectation. The study is then reduced to Gaussian matrices, with a close covariance structure, proving then the universality of the limiting Spectral Distribution. Applications to stationary Harris recurrent Markov chains and to dynamical systems are also given. In the last chapter, we prove a Bernstein type inequality for the largest eigenvalue of the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality is an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. (2009) and a generalization, up to a logarithmic term, of Tropp's inequality (2012) by relaxing the independence hypothesis
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Limiting Spectral Distribution for random matrices with correlated entries and Bernstein-type inequality
2015Co-Authors: Marwa BannaAbstract:In this thesis, we investigate mainly the limiting Spectral Distribution of random matrices having correlated entries and prove as well a Bernstein-type inequality for the largest eigenvalue of the sum of self-adjoint random matrices that are geometrically absolutely regular. We are interested in the asymptotic Spectral behavior of sample covariance matrices and Wigner-type matrices having correlated entries that are functions of independent random variables. We show that the limiting Spectral Distribution can be obtained by analysing a Gaussian matrix having the same covariance structure. This approximation approach is valid for both short and long range dependent stationary random processes just having moments of second order. Our approach is based on a blend of a blocking procedure, Lindeberg's method and the Gaussian interpolation technique. We also develop new tools including a concentration inequality for the Spectral measure for matrices having $K$-dependent rows. This method permits to derive, under mild conditions, an integral equation of the Stieltjes transform of the limiting Spectral Distribution. Applications to matrices whose entries consist of functions of linear processes, ARCH processes or non-linear Volterra-type processes are also given. We also investigate the asymptotic behavior of Gram matrices having correlated entries that are functions of an absolutely regular random process. We give a concentration inequality of the Stieltjes transform and prove that, under an arithmetical decay condition on the absolute regular coefficients, it is almost surely concentrated around its expectation. The study is then reduced to Gaussian matrices, with a close covariance structure, proving then the universality of the limiting Spectral Distribution. Applications to stationary Harris recurrent Markov chains and to dynamical systems are also given. In the last chapter, we prove a Bernstein type inequality for the largest eigenvalue of the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality is an extension to the matrix setting of the Bernstein-type inequality obtained by Merlev\`ede et al. (2009) and a generalization, up to a logarithmic term, of Tropp's inequality (2012) by relaxing the independence hypothesis.
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Limiting Spectral Distribution of Gram matrices associated with functionals of β-mixing processes
2014Co-Authors: Marwa BannaAbstract:We give asymptotic Spectral results for Gram matrices of the form n −1 X n X T n where the entries of X n are dependent across both rows and columns and that are functionals of absolutely regular sequences and have only finite second moments. We derive, under mild dependence conditions in addition to an arithmetical decay condition on the β-mixing coefficients, an integral equation of the Stieltjes transform of the limiting Spectral Distribution of n −1 X n X T n in terms of the Spectral density of the underlying process. Applications to examples of positive recurrent Markov chains and dynamical systems are also given.
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Limiting Spectral Distribution of large sample covariance matrices associated with a class of stationary processes
Journal of Theoretical Probability, 2013Co-Authors: Marwa Banna, Florence MerlevèdeAbstract:In this paper we derive an extension of the Marchenko-Pastur theorem to a large class of weak dependent sequences of real random variables having only moment of order 2. Under mild dependence conditions that are easily verifiable in many situations, we derive that the limiting Spectral Distribution of the associated sample covariance matrix is characterised by an explicit equation for its Stieltjes transform, depending on the Spectral density of the underlying process. Applications to linear processes, functions of linear processes and ARCH models are given.
Vahid Tarokh - One of the best experts on this subject based on the ideXlab platform.
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Convergence Rate of Empirical Spectral Distribution of Random Matrices From Linear Codes
IEEE Transactions on Information Theory, 2021Co-Authors: Chin Hei Chan, Vahid Tarokh, Maosheng XiongAbstract:It is known that the empirical Spectral Distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence rate in probability is at least of the order n-1/4 where n is the length of the code.
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convergence rate of empirical Spectral Distribution of random matrices from linear codes
arXiv: Probability, 2019Co-Authors: Chin Hei Chan, Vahid Tarokh, Maosheng XiongAbstract:It is known that the empirical Spectral Distribution of random matrices obtained from linear codes of increasing length converges to the well-known Marchenko-Pastur law, if the Hamming distance of the dual codes is at least 5. In this paper, we prove that the convergence in probability is at least of the order $n^{-1/4}$ where $n$ is the length of the code.
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Spectral Distribution of product of pseudorandom matrices formed from binary block codes
IEEE Transactions on Information Theory, 2013Co-Authors: Behtash Babadi, Vahid TarokhAbstract:Let A ∈ {-1,1}Na ×n and B ∈ {-1,1}Nb ×n be two matrices whose rows are drawn i.i.d. from the codewords of the binary codes Ca and Cb of length n and dual distances d'a and d'b, respectively, under the mapping 0 → 1 and 1 → -1. It is proven that as n → ∞ with ya:=n/Na ∈ (0,∞) and yb:=n/Nb ∈ (0, ∞) fixed, the empirical Spectral Distribution of the matrix A B*/√{Na Nb} resembles a universal Distribution (closely related to the Distribution function of the free multiplicative convolution of two members of the Marchenko-Pastur family of densities) in the sense of the Levy distance, if the asymptotic dual distances of the underlying binary codes are large enough. Moreover, an explicit upper bound on the Levy distance of the two Distributions in terms of ya, yb, d'a, and d'b is given. Under mild conditions, the upper bound is strengthened to the Kolmogorov distance of the underlying Distributions. Numerical studies on the empirical Spectral Distribution of the product of random matrices from BCH and Gold codes are provided, which verify the validity of this result.
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Spectral Distribution of the product of two random matrices based on binary block codes
Allerton Conference on Communication Control and Computing, 2011Co-Authors: Behtash Babadi, Vahid TarokhAbstract:In this paper, we study the Spectral Distribution of the product of two random matrices based on binary block codes, and prove that if the dual distances of the underlying codes are large enough, the asymptotic Spectral Distribution will be close to a deterministic limit in the sense of Le´vy distance. These results extend our previous work on this topic, and strengthen its applications to joint randomness testing.
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Spectral Distribution of random matrices from binary linear block codes
IEEE Transactions on Information Theory, 2011Co-Authors: Behtash Babadi, Vahid TarokhAbstract:Let C be a binary linear block code of length n, dimension k and minimum Hamming distance d over GF(2)n. Let d⊥ denote the minimum Hamming distance of the dual code of C over GF(2)n. Let e:GF(2)n→{-1,1}n be the component-wise mapping e(vi):=(-1)vi, for v=(v1,v2,...,vn) ∈ GF(2)n. Finally, for p <; n, let \mmbΦC be a p × n random matrix whose rows are obtained by mapping a uniformly drawn set of size p of the codewords of C under e. It is shown that for d⊥ large enough and y:=p/n ∈ (0,1) fixed, as n→∞ the empirical Spectral Distribution of the Gram matrix of [1/(√n)]\mmbΦC resembles that of a random i.i.d. Rademacher matrix (i.e., the Marchenko-Pastur Distribution). Moreover, an explicit asymptotic uniform bound on the distance of the empirical Spectral Distribution of the Gram matrix of [1/(√n)]\mmbΦC to the Marchenko-Pastur Distribution as a function of y and d⊥ is presented.
Luca Salasnich - One of the best experts on this subject based on the ideXlab platform.
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Spectral Distribution studies of fp shell nuclei with a modified kuo brown interaction
Physical Review C, 1997Co-Authors: Subir Sarkar, V. R. Manfredi, J M G Gomez, Luca SalasnichAbstract:The structure of nuclei in the lower half of the fp shell is investigated by the Spectral Distribution method using the modified Kuo-Brown interaction. This interaction recently showed success in reproducing observed properties through detailed shell model studies. Spectral Distribution studies avoid explicit diagonalization and hold promise for applications to astrophysics. {copyright} {ital 1997} {ital The American Physical Society}
Yerong Tao - One of the best experts on this subject based on the ideXlab platform.
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correlation between weighted Spectral Distribution and average path length in evolving networks
Chaos, 2016Co-Authors: Bo Jiao, Jianmai Shi, Yuanping Nie, Chengdong Huang, Ying Zhou, Ronghua Guo, Yerong TaoAbstract:The weighted Spectral Distribution (WSD) is a metric defined on the normalized Laplacian spectrum. In this study, synchronic random graphs are first used to rigorously analyze the metric's scaling feature, which indicates that the metric grows sublinearly as the network size increases, and the metric's scaling feature is demonstrated to be common in networks with Gaussian, exponential, and power-law degree Distributions. Furthermore, a deterministic model of diachronic graphs is developed to illustrate the correlation between the slope coefficient of the metric's asymptotic line and the average path length, and the similarities and differences between synchronic and diachronic random graphs are investigated to better understand the correlation. Finally, numerical analysis is presented based on simulated and real-world data of evolving networks, which shows that the ratio of the WSD to the network size is a good indicator of the average path length.
S Gazor - One of the best experts on this subject based on the ideXlab platform.
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Limiting Spectral Distribution of the sample covariance matrix of the windowed array data
EURASIP Journal on Advances in Signal Processing, 2013Co-Authors: Ehsan Yazdian, S Gazor, Mohammad Hassan BastaniAbstract:In this article, we investigate the limiting Spectral Distribution of the sample covariance matrix (SCM) of weighted/windowed complex data. We use recent advances in random matrix theory and describe the Distribution of eigenvalues of the doubly correlated Wishart matrices. We obtain an approximation for the Spectral Distribution of the SCM obtained from windowed data. We also determine a condition on the coefficients of the window, under which the fragmentation of the support of noise eigenvalues can be avoided, in the noise-only data case. For the commonly used exponential window, we derive an explicit expression for the l.s.d of the noise-only data. In addition, we present a method to identify the support of eigenvalues in the general case of signal-plus-noise. Simulations are performed to support our theoretical claims. The results of this article can be directly employed in many applications working with windowed array data such as source enumeration and subspace tracking algorithms.
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Spectral Distribution of the exponentially windowed sample covariance matrix
International Conference on Acoustics Speech and Signal Processing, 2012Co-Authors: Ehsan Yazdian, Mohammad Hassan Bastani, S GazorAbstract:In this paper, we investigate the effect of applying an exponential window on the limiting Spectral Distribution (l.s.d.) of the exponentially windowed sample covariance matrix (SCM) of complex array data. We use recent advances in random matrix theory which describe the Distribution of eigenvalues of the doubly correlated Wishart matrices. We derive an explicit expression for the l.s.d. of the noise-only data. Simulations are performed to support our theoretical claims.
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ICASSP - Spectral Distribution of the exponentially windowed sample covariance matrix
2012 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2012Co-Authors: Ehsan Yazdian, Mohammad Hassan Bastani, S GazorAbstract:In this paper, we investigate the effect of applying an exponential window on the limiting Spectral Distribution (l.s.d.) of the exponentially windowed sample covariance matrix (SCM) of complex array data. We use recent advances in random matrix theory which describe the Distribution of eigenvalues of the doubly correlated Wishart matrices. We derive an explicit expression for the l.s.d. of the noise-only data. Simulations are performed to support our theoretical claims.
