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FINITE SAMPLE APPROXIMATION RESULTS FOR PRINCIPAL COMPONENT ANALYSIS: A MATRIX PERTURBATION APPROACH
"... Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite ..."
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Cited by 62 (15 self)
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Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, to those of the limiting population PCA as n → ∞. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit p, n → ∞, with p/n = c. We present a matrix perturbation view of the “phase transition phenomenon”, and a simple linearalgebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite p, n where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size n, the eigenvector of sample PCA may exhibit a sharp ”loss of tracking”, suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.
Geometrically Induced Spectrum in Curved Leaky Wires
 J. Phys. A34
, 2001
"... Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side ..."
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Cited by 42 (17 self)
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Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side we note that quantum mechanics of electrons confined to narrow tubelike regions has attracted a considerable interest, because such systems represent a natural model for semiconductor "quantum wires". In some examples the region in question is a strip or tube with hard walls  see, e.g., [DE] and references therein  while other treatments assume even stronger localization to a curve 1 or a graph  a rich bibliography to such models can be found in [KS]. Various interesting spectral effects were found in such a setting related to the geometry and topology of the underlying restricted configuration space. One of th
Numerical Exterior Algebra and the Compound Matrix Method
 Numer. Math
, 2000
"... The compound matrix method, which has been proposed by Ng & Reid for numerically integrating systems of dierential equations in hydrodynamic stability on k = 2; 3 dimensional subspaces, is reformulated in terms of exterior algebra. This formulation leads to a general framework for deriving th ..."
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Cited by 34 (2 self)
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The compound matrix method, which has been proposed by Ng & Reid for numerically integrating systems of dierential equations in hydrodynamic stability on k = 2; 3 dimensional subspaces, is reformulated in terms of exterior algebra. This formulation leads to a general framework for deriving the induced systems, and leads to several new results including: the role of Hodge duality in constructing systems, adjoints and boundary conditions, the role of analyticity for systems on unbounded domains, general formulation of induced boundary conditions, and the role of geometric integrators for preserving the manifold of k dimensional subspaces. The formulation is presented for kdimensional subspaces of systems on C with k and n arbitrary, and detailed examples are given for the case k = 2 and n = 4, with an indication of implementation details for systems of larger dimension. The theory is then applied to two examples: 2D boundarylayer ow past a compliant surface and the instability of jetlike pro les.
Homogenization and influence of fragmentation in a biological invasion model
"... In this paper, some properties of the minimal speeds of pulsating FisherKPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the sp ..."
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Cited by 25 (9 self)
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In this paper, some properties of the minimal speeds of pulsating FisherKPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.
The longtime dynamics of Dirac particles in the KerrNewman black hole geometry
 Adv. Theor. Math. Phys
, 2003
"... We consider the Cauchy problem for the massive Dirac equation in the nonextreme KerrNewman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar’s separation of variables. It is proved that ..."
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Cited by 23 (11 self)
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We consider the Cauchy problem for the massive Dirac equation in the nonextreme KerrNewman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar’s separation of variables. It is proved that for initial data in L ∞ loc near the event horizon with L 2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. 1
Asymptotics of repeated interaction quantum systems
"... Prépublication de l’Institut Fourier n o 682 (2005) wwwfourier.ujfgrenoble.fr/prepublications.html A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling be ..."
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Cited by 21 (5 self)
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Prépublication de l’Institut Fourier n o 682 (2005) wwwfourier.ujfgrenoble.fr/prepublications.html A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between S and E. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ–periodic in time and does not depend on the initial state. If the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.
Nonselfadjoint harmonic oscillator, compact semigroups and pseudospectra
 J. Operator Theory
"... We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. ..."
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Cited by 20 (2 self)
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We provide new information concerning the pseudospectra of the complex harmonic oscillator. Our analysis illustrates two different techniques for getting resolvent norm estimates. The first uses the JWKB method and extends for this particular potential some results obtained recently by E.B. Davies. The second relies on the fact that the bounded holomorphic semigroup generated by the complex harmonic oscillator is of HilbertSchmidt type in a maximal angular region. In order to show this last property, we deduce a nonselfadjoint version of the classical Mehler’s formula.
Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds
"... We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to P ..."
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Cited by 19 (2 self)
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We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly’s inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.
Rank one perturbations with infinitesimal coupling
"... Abstract. We consider a positive selfadjoint operator A and formal rank one pertubrations B = A + α(ϕ, ·)ϕ where ϕ ∈H−2(A) but ϕ /∈H−1(A), with Hs(A) the usual scale of spaces. We show that B can be defined for such ϕ and what are essentially negative infinitesimal values of α. Ina sense we’ll make ..."
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Cited by 18 (2 self)
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Abstract. We consider a positive selfadjoint operator A and formal rank one pertubrations B = A + α(ϕ, ·)ϕ where ϕ ∈H−2(A) but ϕ /∈H−1(A), with Hs(A) the usual scale of spaces. We show that B can be defined for such ϕ and what are essentially negative infinitesimal values of α. Ina sense we’ll make precise, every rank one perturbation is one of three forms: (i) ϕ ∈H−1(A), α ∈ R; (ii) ϕ ∈H−1, α = ∞; or (iii) the new type we consider here. There has recently been considerable interest in the study of rankone perturbations of positive selfadjoint operators (see [11] and references therein). Let A ≥ 0 on a Hilbert space H and consider B = A + α(ϕ, ·)ϕ. (1.1)