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Sparse regular random graphs: Spectral density and eigenvectors
"... Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to ..."
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Abstract. We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized. 1.
On asymptotic proximity of distributions
"... Abstract. We consider some general facts concerning convergence Pn − Qn → 0 as n → ∞, where Pn and Qn are probability measures in a complete separable metric space. The main point is that the sequences {Pn} and {Qn} are not assumed to be tight. We compare different possible definitions of the above ..."
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Abstract. We consider some general facts concerning convergence Pn − Qn → 0 as n → ∞, where Pn and Qn are probability measures in a complete separable metric space. The main point is that the sequences {Pn} and {Qn} are not assumed to be tight. We compare different possible definitions of the above convergence, and establish some general properties.
The Isotropic Semicircle Law and Deformation of Wigner Matrices
, 2012
"... We analyse the spectrum of additive finiterank deformations of N ×N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue di of the deformation crosses a critical value ±1. This transition happen ..."
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We analyse the spectrum of additive finiterank deformations of N ×N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue di of the deformation crosses a critical value ±1. This transition happens on the scale di  − 1 ∼ N −1/3. We allow the eigenvalues di ∣ of the deformation to depend on N under the condition ∣ di  − 1 C log log N −1/3 � (log N) N. We make no assumptions on the eigenvectors of the deformation. In the limit N → ∞, we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the isotropic local semicircle law, which establishes optimal highprobability bounds on the quantity 〈 v, ( (H − z) −1 − m(z)1) w 〉 , where m(z) is the Stieltjes transform of Wigner’s semicircle law and v, w are arbitrary deterministic vectors.
18.177 course project: Invariance Principles
"... An invariance principle is a result permitting us to change our underlying probability space—such as occurs in a central limit theorem. For example, we can suggestively state the BerryEssen Theorem in the following way: Theorem 1.1 (BerryEssen) Let X1,..., Xn be i.i.d. random variables with E[Xi] ..."
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An invariance principle is a result permitting us to change our underlying probability space—such as occurs in a central limit theorem. For example, we can suggestively state the BerryEssen Theorem in the following way: Theorem 1.1 (BerryEssen) Let X1,..., Xn be i.i.d. random variables with E[Xi] = 0, E[X2 i] =
Applications of Lindeberg Principle in Communications and Statistical Learning
"... We use a generalization of the Lindeberg principle developed by Sourav Chatterjee to prove universality properties for various problems in communications, statistical learning and random matrix theory. We also show that these systems can be viewed as the limiting case of a properly defined sparse sy ..."
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We use a generalization of the Lindeberg principle developed by Sourav Chatterjee to prove universality properties for various problems in communications, statistical learning and random matrix theory. We also show that these systems can be viewed as the limiting case of a properly defined sparse system. The latter result is useful when the sparse systems are easier to analyze than their dense counterparts. The list of problems we consider is by no means exhaustive. We believe that the ideas can be used in many other problems relevant for information theory.
The convergence of the empirical distribution of canonical correlation coefficients
"... Suppose that {Xjk, j = 1, · · · , p1; k = 1, · · · , n} are independent and identically distributed (i.i.d) real random variables with EX11 = 0 and EX 2 11 = 1, and that {Yjk, j = 1, · · · , p2; k = 1, · · · , n} are i.i.d real random variables with EY11 = 0 and EY 2 11 = 1, and that {Xj ..."
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Suppose that {Xjk, j = 1, · · · , p1; k = 1, · · · , n} are independent and identically distributed (i.i.d) real random variables with EX11 = 0 and EX 2 11 = 1, and that {Yjk, j = 1, · · · , p2; k = 1, · · · , n} are i.i.d real random variables with EY11 = 0 and EY 2 11 = 1, and that {Xjk, j = 1, · · · , p1; k = 1, · · · , n} are independent of {Yjk, j = 1, · · · , p2; k = 1, · · · , n}. This paper investigates the canonical correlation coefficients r1 ≥ r2 ≥ · · · ≥ rp1, whose squares λ1 = r2 1, λ2 = r 2 2, · · · , λp1 = r 2 p1 are the eigenvalues of the matrix where and