The Experts below are selected from a list of 209823 Experts worldwide ranked by ideXlab platform

Jean B. Lasserre - One of the best experts on this subject based on the ideXlab platform.

  • Data analysis from empirical moments and the Christoffel function
    arXiv: Machine Learning, 2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and Approximation Theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and Approximation Theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

  • Spectral analysis of moment data
    2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal poly-nomials and Approximation Theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and Approximation Theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical validation , (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

Martin Uecker - One of the best experts on this subject based on the ideXlab platform.

  • parallel magnetic resonance imaging as Approximation in a reproducing kernel hilbert space
    Inverse Problems, 2015
    Co-Authors: Vivek R Athalye, Michael Lustig, Martin Uecker
    Abstract:

    In magnetic resonance imaging data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sampling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be understood as Approximation of vector-valued functions from the acquired samples and formulated using a reproducing kernel Hilbert space with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between Approximation Theory and parallel imaging. Theoretical tools from Approximation Theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise analysis to both noise amplification and Approximation errors in k-space. This is demonstrated with numerical examples.

  • parallel magnetic resonance imaging as Approximation in a reproducing kernel hilbert space
    arXiv: Medical Physics, 2013
    Co-Authors: Vivek R Athalye, Michael Lustig, Martin Uecker
    Abstract:

    In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. As shown here, reconstruction from samples at arbitrary locations can be understood as Approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between Approximation Theory and parallel imaging. Theoretical tools from Approximation Theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional g-factor noise analysis to both noise amplification and Approximation errors. This is demonstrated with numerical examples.

Edouard Pauwels - One of the best experts on this subject based on the ideXlab platform.

  • Data analysis from empirical moments and the Christoffel function
    arXiv: Machine Learning, 2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and Approximation Theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and Approximation Theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

  • Spectral analysis of moment data
    2018
    Co-Authors: Edouard Pauwels, Mihai Putinar, Jean B. Lasserre
    Abstract:

    Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations we combine ideas from statistics, real algebraic geometry, orthogonal poly-nomials and Approximation Theory for opening new insights relevant for Machine Learning (ML) problems with data supported on singular sets. Refined concepts and results from real algebraic geometry and Approximation Theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical validation , (2) numerical experiments and, (3) connections to real world data as a validation of the stamina of the empirical moment matrix approach.

Vivek R Athalye - One of the best experts on this subject based on the ideXlab platform.

  • parallel magnetic resonance imaging as Approximation in a reproducing kernel hilbert space
    Inverse Problems, 2015
    Co-Authors: Vivek R Athalye, Michael Lustig, Martin Uecker
    Abstract:

    In magnetic resonance imaging data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. To understand and design k-space sampling patterns, a theoretical framework is needed to analyze how well arbitrary sampling patterns reconstruct unsampled k-space using receive coil information. As shown here, reconstruction from samples at arbitrary locations can be understood as Approximation of vector-valued functions from the acquired samples and formulated using a reproducing kernel Hilbert space with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between Approximation Theory and parallel imaging. Theoretical tools from Approximation Theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional image-domain g-factor noise analysis to both noise amplification and Approximation errors in k-space. This is demonstrated with numerical examples.

  • parallel magnetic resonance imaging as Approximation in a reproducing kernel hilbert space
    arXiv: Medical Physics, 2013
    Co-Authors: Vivek R Athalye, Michael Lustig, Martin Uecker
    Abstract:

    In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial frequency domain (k-space), typically by time-consuming line-by-line scanning on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of data using multiple receivers (parallel imaging), and by using more efficient non-Cartesian sampling schemes. As shown here, reconstruction from samples at arbitrary locations can be understood as Approximation of vector-valued functions from the acquired samples and formulated using a Reproducing Kernel Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial sensitivities of the receive coils. This establishes a formal connection between Approximation Theory and parallel imaging. Theoretical tools from Approximation Theory can then be used to understand reconstruction in k-space and to extend the analysis of the effects of samples selection beyond the traditional g-factor noise analysis to both noise amplification and Approximation errors. This is demonstrated with numerical examples.

Weimin Han - One of the best experts on this subject based on the ideXlab platform.

  • spherical harmonics and Approximations on the unit sphere an introduction
    2012
    Co-Authors: Kendall Atkinson, Weimin Han
    Abstract:

    1 Preliminaries.- 2 Spherical Harmonics.- 3 Differentiation and Integration over the Sphere.- 4 Approximation Theory.- 5 Numerical Quadrature.- 6 Applications: Spectral Methods.

  • theoretical numerical analysis a functional analysis framework
    2001
    Co-Authors: Kendall Atkinson, Weimin Han
    Abstract:

    Preface to the Second Edition.- Preface to the First Edition.- Linear Spaces.- Linear Operators on Normed Spaces.- Approximation Theory.- Fourier Analysis and Wavelets.- Nonlinear Equations and Their Solution by Iteration.- Finite Difference Method.- Sobolev Spaces.- Variational Formulations of Elliptic Boundary Value Problems.- The Galerkin Method and Its Variants.- Finite Element Analysis.- Elliptic Variational Inequalities and Their Numerical Approximations.- Numerical Solution of Fredholm Integral Equations of the Second Kind.- Boundary Integral Equations.- References.- Index.

  • theoretical numerical analysis a functional analysis framework
    2001
    Co-Authors: Kendall Atkinson, Weimin Han
    Abstract:

    Preface 1 Linear Spaces 2 Linear Operators on Normed Spaces 3 Approximation Theory 4 Nonlinear Equations and Their Solution by Iteration 5 Finite Difference Method 6 Sobolev Spaces 7 Variational Formulations of Elliptic Boundary Value Problems 8 The Galerkin Method and Its Variants 9 Finite Element Analysis 10 Elliptic Variational Inequalities and Their Numerical Approximations 11 Numerical Solution of Fredholm Integral Equations of the Second Kind 12 Boundary Integral Equations References Index.