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69
Kernel dimension reduction in regression
, 2006
"... Acknowledgements. The authors thank the editor and anonymous referees for their helpful comments. The authors also thank Dr. Yoichi Nishiyama for his helpful comments on the uniform convergence of empirical processes. We would like to acknowledge support from JSPS KAKENHI 15700241, ..."
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Cited by 27 (11 self)
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Acknowledgements. The authors thank the editor and anonymous referees for their helpful comments. The authors also thank Dr. Yoichi Nishiyama for his helpful comments on the uniform convergence of empirical processes. We would like to acknowledge support from JSPS KAKENHI 15700241,
Universal kernels
 J. Machine Learning Research
, 2006
"... In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property. ..."
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Cited by 22 (3 self)
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In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property.
S.,Limit theorems for additive functionals of a Markov Chain, version 1
, 2008
"... Abstract. Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstabl ..."
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Cited by 19 (7 self)
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Abstract. Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstable law. A “martingale approximation ” approach and “coupling ” approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N −1/α ∫ Nt 0 V (Xs)ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation. 1.
A RATE OF CONVERGENCE RESULT FOR THE LARGEST EIGENVALUE OF COMPLEX WHITE Wishart Matrices
, 2006
"... It has been recently shown that if X is an n × N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X ∗ X, there exist sequences mn,N and sn,N such that (l1 − mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaus ..."
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Cited by 15 (0 self)
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It has been recently shown that if X is an n × N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X ∗ X, there exist sequences mn,N and sn,N such that (l1 − mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2. In this paper we show that, under the assumption that n/N → γ ∈ (0, ∞), we can find a function M, continuous and nonincreasing, and sequences ˜µn,N and ˜σn,N such that, for all real s0, there exists an integer N(s0,γ)for which, if (n ∧ N) ≥ N(s0,γ), we have, with ln,N = (l1 −˜µn,N) / ˜σn,N, ∀ s ≥ s0 (n ∧ N) 2/3 P(ln,N ≤ s) − F2(s)≤M(s0) exp(−s). The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.
Statistical consistency of kernel canonical correlation analysis
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2007
"... While kernel canonical correlation analysis (CCA) has been applied in many contexts, the convergence of finite sample estimates of the associated functions to their population counterparts has not yet been established. This paper gives a mathematical proof of the statistical convergence of kernel CC ..."
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Cited by 15 (7 self)
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While kernel canonical correlation analysis (CCA) has been applied in many contexts, the convergence of finite sample estimates of the associated functions to their population counterparts has not yet been established. This paper gives a mathematical proof of the statistical convergence of kernel CCA, providing a theoretical justification for the method. The proof uses covariance operators defined on reproducing kernel Hilbert spaces, and analyzes the convergence of their empirical estimates of finite rank to their population counterparts, which can have infinite rank. The result also gives a sufficient condition for convergence on the regularization coefficient involved in kernel CCA: this should decrease as n −1/3, where n is the number of data.
On the orbital (in)stability of spatially periodic stationary solutions of generalized Kortewegde Vries equations. Submitted for publication
, 2010
"... Abstract. In this paper we generalize previous work on the spectral and orbital stability of waves for infinitedimensional Hamiltonian systems to include those cases for which the skewsymmetric operator J is singular. We assume that J restricted to the orthogonal complement of its kernel has a bou ..."
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Cited by 12 (6 self)
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Abstract. In this paper we generalize previous work on the spectral and orbital stability of waves for infinitedimensional Hamiltonian systems to include those cases for which the skewsymmetric operator J is singular. We assume that J restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higherorder pure power nonlinearities.
Optimal control of spatially distributed systems
 IEEE Tran. on Automatic Control, September 2006, accepted. [Online]. Available: http://www.grasp.upenn.edu/ ∼ motee/ TACMoteeJ06SD.pdf
"... Abstract — In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over ar ..."
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Cited by 11 (4 self)
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Abstract — In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over arbitrary graphs. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD) operators. We study the structural properties of infinitehorizon linear quadratic optimal controllers for such systems by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. We prove that the kernel of the optimal feedback of each subsystem decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the system. I.
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
Stationary statistical properties of RayleighBénard convection at large Prandtl number, CPAM
"... This is the third in a sequel in our study of RayleighBénard convection at large Prandtl number. More specifically we investigate if stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to that of the infinite Prandtl number m ..."
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Cited by 9 (6 self)
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This is the third in a sequel in our study of RayleighBénard convection at large Prandtl number. More specifically we investigate if stationary statistical properties of the Boussinesq system for RayleighBénard convection at large Prandtl number are related to that of the infinite Prandtl number model for convection which is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for RayleighBénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for RayleighBénard convection converge to that of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that specific statistical properties such as the Nusselt number for the Boussinesq system is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form Ra 1
Edgeworth expansion of the largest eigenvalue distribution function of GUE and
 ID 61049
"... We derive expansions of the Hermite and Laguerre kernels at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n Laguerre Unitary Ensemble (LUEn), respectively. Using these large n kernel expansions, we prove an Edgeworth type theorem for the largest eigenvalue ..."
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Cited by 7 (0 self)
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We derive expansions of the Hermite and Laguerre kernels at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n Laguerre Unitary Ensemble (LUEn), respectively. Using these large n kernel expansions, we prove an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn and LUEn. In our Edgeworth expansion, the correction terms are expressed in terms of the same Painlevé II function appearing in the leading term, i.e. in the TracyWidom distribution. We conclude with a brief discussion of the universality of these results. 1