Results 1  10
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314
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
Abstract

Cited by 197 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
On orthogonal and symplectic matrix ensembles
 Commun. Math. Phys
, 1996
"... The focus of this paper is on the probability, Eβ(0; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (β = 1) and Gaussian Symplectic (β = 4) Ensembles and their respective scaling limits both in the bulk and at the edge of the s ..."
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Cited by 151 (11 self)
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The focus of this paper is on the probability, Eβ(0; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (β = 1) and Gaussian Symplectic (β = 4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (β = 2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function. 1 I.
Pseudospectra of linear operators
 SIAM Rev
, 1997
"... Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A ..."
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Cited by 113 (8 self)
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Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
Sum Rules For Jacobi Matrices And Their Applications To Spectral Theory
 Ann. of Math
"... We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for whi ..."
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Cited by 99 (38 self)
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We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J J0 is HilbertSchmidt, and a proof of Nevai's conjecture that the Szegö condition holds if J J0 is trace class.
On the Early History of the Singular Value Decomposition
, 1992
"... This paper surveys the contributions of five mathematicians  Eugenio Beltrami (18351899), Camille Jordan (18381921), James Joseph Sylvester (18141897), Erhard Schmidt (18761959), and Hermann Weyl (18851955)  who were responsible for establishing the existence of the singular value de ..."
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Cited by 82 (1 self)
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This paper surveys the contributions of five mathematicians  Eugenio Beltrami (18351899), Camille Jordan (18381921), James Joseph Sylvester (18141897), Erhard Schmidt (18761959), and Hermann Weyl (18851955)  who were responsible for establishing the existence of the singular value decomposition and developing its theory.
On the Distribution of the Largest Principal Component
 ANN. STATIST
, 2000
"... Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degr ..."
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Cited by 48 (0 self)
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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the TracyWidom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...
Nash Inequalities for Finite Markov Chains
, 1996
"... This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for suc ..."
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Cited by 34 (9 self)
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This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) 2 steps are necessary and suffice to reach stationarity. We consider local Poincar6 inequalities and use them to prove Nash inequalities. These are bounds on (,_norms in terms of Dirichlet forms and l~norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.
Resonance Expansions Of Scattered Waves
 Comm. Pure Appl. Math
"... The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in ..."
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Cited by 30 (10 self)
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The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in time. To avoid the discussion of particular aspects of potential, gravitational or obstacle scattering, the results are stated using the abstract "black box" formalism of Sjöstrand and the second author [19]. When M is a compact Riemannian manifold, and Delta, its Laplacian, then we have a generalized Fourier expansion of the wave group: sin t p p f(x) = X 2 j 2Spec() e i j t w j (x) ; w j = 2 j w j ; (1.1) and the convergence is absolute in the case of smooth data. The simplest case of a noncompact spectral problem is given by taking R^n with the usual Laplacian outside a compact set  we can for instance "glue" any compact Riemannian manifold to R^n or consider the obstacle problem with the Dirichlet Laplacian on R^n\O. Since resonances or scattering poles constitute a natural replacement of discrete spectral data for problems on exterior domains, we expect a similar expansion involving them in place of eigenvalues  this point of view was emphasized early by LaxPhillips [10] (see [18],[32] and [34] for overviews of recent results). The resonances are de ned as poles of the meromorphic continuation of the resolvent or of the scattering matrix but despite the stationary nature of these de nitions, they are fundamentally a dynamical concept: the real part of a resonance describes the rest energy of a state and the imaginary part its rate of decay. Consequently they should be understood in terms of long time behaviour of solutions to evolutio...
The Selberg zeta function for convex cocompact Schottky groups
 Comm. Math. Phys
"... Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the o ..."
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Cited by 30 (8 self)
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Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(Cs  n+1), and it gives new bounds on the number of resonances (scattering poles) of Γ\H n+1. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\H n+1 as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. 1.