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56
Nonlinear component analysis as a kernel eigenvalue problem

, 1996
"... We describe a new method for performing a nonlinear form of Principal Component Analysis. By the use of integral operator kernel functions, we can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all ..."
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Cited by 1048 (72 self)
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We describe a new method for performing a nonlinear form of Principal Component Analysis. By the use of integral operator kernel functions, we can efficiently compute principal components in highdimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible 5pixel products in 16x16 images. We give the derivation of the method, along with a discussion of other techniques which can be made nonlinear with the kernel approach; and present first experimental results on nonlinear feature extraction for pattern recognition.
On a Kernelbased Method for Pattern Recognition, Regression, Approximation, and Operator Inversion
, 1997
"... We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connection ..."
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Cited by 77 (25 self)
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We present a Kernelbased framework for Pattern Recognition, Regression Estimation, Function Approximation and multiple Operator Inversion. Previous approaches such as ridgeregression, Support Vector methods and regression by Smoothing Kernels are included as special cases. We will show connections between the costfunction and some properties up to now believed to apply to Support Vector Machines only. The optimal solution of all the problems described above can be found by solving a simple quadratic programming problem. The paper closes with a proof of the equivalence between Support Vector kernels and Greene's functions of regularization operators.
Notes on infinite determinants of Hilbert space operators
 Adv. Math
, 1977
"... We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theo ..."
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Cited by 34 (2 self)
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We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theory on the trace ideals, c#~(p < oo). 1.
WKB analysis for nonlinear Schrödinger equations with a potential
 Comm. Math. Phys
"... Abstract. We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. ..."
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Cited by 28 (10 self)
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Abstract. We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity. (1.1) (1.2)
Statistical properties of kernel principal component analysis
 Machine Learning
, 2004
"... The main goal of this paper is to prove inequalities on the reconstruction error for Kernel Principal Component Analysis. With respect to previous work on this topic, our contribution is twofold: (1) we give bounds that explicitly take into account the empirical centering step in this algorithm, and ..."
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Cited by 25 (2 self)
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The main goal of this paper is to prove inequalities on the reconstruction error for Kernel Principal Component Analysis. With respect to previous work on this topic, our contribution is twofold: (1) we give bounds that explicitly take into account the empirical centering step in this algorithm, and (2) we show that a “localized” approach allows to obtain more accurate bounds. In particular, we show faster rates of convergence towards the minimum reconstruction error; more precisely, we prove that the convergence rate can typically be faster than n −1/2. We also obtain a new relative bound on the error. A secondary goal, for which we present similar contributions, is to obtain convergence bounds for the partial sums of the biggest or smallest eigenvalues of the kernel Gram matrix towards eigenvalues of the corresponding kernel operator. These quantities are naturally linked to the KPCA procedure; furthermore these results can have applications to the study of various other kernel algorithms. The results are presented in a functional analytic framework, which is suited to deal rigorously with reproducing kernel Hilbert spaces of infinite dimension. 1
Nonlinear Schrödinger equations with repulsive harmonic potential and applications
 SIAM J. Math. Anal
"... Abstract. We study the Cauchy problem for Schrödinger equations with repulsive quadratic potential and powerlike nonlinearity. The local problem is wellposed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally wellposed, an ..."
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Cited by 13 (5 self)
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Abstract. We study the Cauchy problem for Schrödinger equations with repulsive quadratic potential and powerlike nonlinearity. The local problem is wellposed in the same space as that used when a confining harmonic potential is involved. For a defocusing nonlinearity, it is globally wellposed, and a scattering theory is available, with no long range effect for any superlinear nonlinearity. When the nonlinearity is focusing, we prove that choosing the harmonic potential sufficiently strong prevents blowup in finite time. Thanks to quadratic potentials, we provide a method to anticipate, delay, or prevent wave collapse; this mechanism is explicit for critical nonlinearity. Consider the Schrödinger equation 1.
Nonscattering solutions and blow up at infinity for the critical wave equation, preprint, arXiv: 1201.3258v1
"... Abstract. We consider the critical focusing wave equation (− ∂ 2 t + ∆)u + u 5 = 0 in R 1+3 and prove the existence of energy class solutions which are of the form u(t,x) = t µ 2W(t µ x)+η(t,x) in the forward lightcone {(t,x) ∈ R×R 3: x  ≤ t,t ≫ 1} where W(x) = (1+ 1 3 x2) −1 2 is the ground ..."
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Cited by 12 (9 self)
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Abstract. We consider the critical focusing wave equation (− ∂ 2 t + ∆)u + u 5 = 0 in R 1+3 and prove the existence of energy class solutions which are of the form u(t,x) = t µ 2W(t µ x)+η(t,x) in the forward lightcone {(t,x) ∈ R×R 3: x  ≤ t,t ≫ 1} where W(x) = (1+ 1 3 x2) −1 2 is the ground state soliton, µ is an arbitrary prescribed real number (positive or negative) with µ  ≪ 1, and the error η satisfies ‖∂tη(t,·) ‖ L 2 (Bt) +‖∇η(t,·) ‖ L 2 (Bt) ≪ 1, Bt: = {x ∈ R 3: x  < t} for all t ≫ 1. Furthermore, the kinetic energy of u outside the cone is small. Consequently, depending on the sign of µ, we obtain two new types of solutions which either blow up as t → ∞ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture. 1.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
LINEAR VS. NONLINEAR EFFECTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL
"... Abstract. We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite prec ..."
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Cited by 8 (5 self)
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Abstract. We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semiclassical régimes, as well as finite time blowup and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs. 1.
GLOBAL EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH QUADRATIC POTENTIALS
"... Abstract. We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The p ..."
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Cited by 7 (0 self)
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Abstract. We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to use continuity arguments and to control the nonlinear effects. 1.