SIAM Journal on Scientific Computing, ISSN 1064-8275, 2018, Volume 40, Issue 2, pp. A629 - A652

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for...

High-dimensional approximation | Nonlinear dynamics | Hamilton–Jacobi–Bellman equations | Polynomial approximation | Optimal feedback control | MATHEMATICS, APPLIED | polynomial approximation | high-dimensional approximation | STABILIZATION | optimal feedback control | Hamilton-Jacobi-Bellman equations | nonlinear dynamics

High-dimensional approximation | Nonlinear dynamics | Hamilton–Jacobi–Bellman equations | Polynomial approximation | Optimal feedback control | MATHEMATICS, APPLIED | polynomial approximation | high-dimensional approximation | STABILIZATION | optimal feedback control | Hamilton-Jacobi-Bellman equations | nonlinear dynamics

Journal Article

Constructive Approximation, ISSN 0176-4276, 10/2011, Volume 34, Issue 2, pp. 257 - 280

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional applications. We describe the...

65N35 | Mathematics | FEM | 65F50 | Rank-structured tensor approximation | 65F10 | Analysis | Numerical Analysis | High-dimensional problems | 65F30 | Material sciences | Matrix-valued functions | Quantics folding of vectors | Stochastic modeling | DECOMPOSITION | ALGORITHMS | PRODUCT APPROXIMATION | MATHEMATICS | ELLIPTIC PROBLEMS | OPERATORS | Finite element method

65N35 | Mathematics | FEM | 65F50 | Rank-structured tensor approximation | 65F10 | Analysis | Numerical Analysis | High-dimensional problems | 65F30 | Material sciences | Matrix-valued functions | Quantics folding of vectors | Stochastic modeling | DECOMPOSITION | ALGORITHMS | PRODUCT APPROXIMATION | MATHEMATICS | ELLIPTIC PROBLEMS | OPERATORS | Finite element method

Journal Article

Annals of Statistics, ISSN 0090-5364, 10/2017, Volume 45, Issue 5, pp. 1895 - 1919

We consider the problem of approximating sums of high dimensional stationary time series by Gaussian vectors, using the framework of functional dependence...

Long run covariance matrix | Kolmogorov-Smirnov test | High-dimensional time series | Simultaneous inference | Gaussian approximation | LARGEST ENTRIES | long run covariance matrix | simultaneous inference | STATISTICS & PROBABILITY | ESTIMATOR | SUMS | COVARIANCE-MATRIX | NONPARAMETRIC-ESTIMATION | ASYMPTOTIC-DISTRIBUTION | SAMPLE CORRELATION-MATRICES | high-dimensional time series | CENTRAL-LIMIT-THEOREM | DEVIATIONS

Long run covariance matrix | Kolmogorov-Smirnov test | High-dimensional time series | Simultaneous inference | Gaussian approximation | LARGEST ENTRIES | long run covariance matrix | simultaneous inference | STATISTICS & PROBABILITY | ESTIMATOR | SUMS | COVARIANCE-MATRIX | NONPARAMETRIC-ESTIMATION | ASYMPTOTIC-DISTRIBUTION | SAMPLE CORRELATION-MATRICES | high-dimensional time series | CENTRAL-LIMIT-THEOREM | DEVIATIONS

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Minimax Rates of Entropy Estimation on Large Alphabets via Best Polynomial Approximation

IEEE Transactions on Information Theory, ISSN 0018-9448, 06/2016, Volume 62, Issue 6, pp. 3702 - 3720

Consider the problem of estimating the Shannon entropy of a distribution over k elements from n independent samples. We show that the minimax mean-square error...

Histograms | Additives | Estimation | Mean square error methods | Entropy | Probability distribution | Random variables | Entropy estimation | high-dimensional statistics | best polynomial approximation | functional estimation | large alphabet | NUMBER | COMPLEXITY | COMPUTER SCIENCE, INFORMATION SYSTEMS | CONVERGENCE | UNSEEN | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Approximation theory | Polynomials | Entropy (Information theory) | Estimation theory | Minimax technique | Approximation | Mathematical analysis | Constants | Estimators | Optimization

Histograms | Additives | Estimation | Mean square error methods | Entropy | Probability distribution | Random variables | Entropy estimation | high-dimensional statistics | best polynomial approximation | functional estimation | large alphabet | NUMBER | COMPLEXITY | COMPUTER SCIENCE, INFORMATION SYSTEMS | CONVERGENCE | UNSEEN | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Approximation theory | Polynomials | Entropy (Information theory) | Estimation theory | Minimax technique | Approximation | Mathematical analysis | Constants | Estimators | Optimization

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The numerical approximation of nonlinear functionals and functional differential equations

Physics Reports, ISSN 0370-1573, 02/2018, Volume 732, pp. 1 - 102

The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf...

HIGH-DIMENSIONAL INTEGRATION | SOURCE GALERKIN METHOD | PHYSICS, MULTIDISCIPLINARY | ALTERNATING LEAST-SQUARES | MULTIVARIATE INTERPOLATION | ORTHOGONAL FUNCTIONALS | PROBABILITY DENSITY-FUNCTION | GENERALIZED POLYNOMIAL CHAOS | VARIATIONAL FORMULATION | WIENER-HERMITE EXPANSION | STOCHASTIC COLLOCATION

HIGH-DIMENSIONAL INTEGRATION | SOURCE GALERKIN METHOD | PHYSICS, MULTIDISCIPLINARY | ALTERNATING LEAST-SQUARES | MULTIVARIATE INTERPOLATION | ORTHOGONAL FUNCTIONALS | PROBABILITY DENSITY-FUNCTION | GENERALIZED POLYNOMIAL CHAOS | VARIATIONAL FORMULATION | WIENER-HERMITE EXPANSION | STOCHASTIC COLLOCATION

Journal Article

Artificial Intelligence in Medicine, ISSN 0933-3657, 05/2019, Volume 96, pp. 134 - 141

Omics data usually have ultra-high dimension (p)and small sample size (n). Standard support vector machines (SVMs), which minimize the L norm for the primal...

Ultra-high dimensional data | Feature selection | approximation | Metagenomics sequencing | SVM | Classification | L-0 approximation | PENALTY | MEDICAL INFORMATICS | ENGINEERING, BIOMEDICAL | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

Ultra-high dimensional data | Feature selection | approximation | Metagenomics sequencing | SVM | Classification | L-0 approximation | PENALTY | MEDICAL INFORMATICS | ENGINEERING, BIOMEDICAL | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

Journal Article

SIAM JOURNAL ON NUMERICAL ANALYSIS, ISSN 0036-1429, 2019, Volume 57, Issue 5, pp. 2217 - 2245

Given any domain X subset of R-d and a probability measure rho on X, we study the problem of approximating in L-2(X, rho) a given function u : X -> R, using...

approximation theory | MATHEMATICS, APPLIED | weighted least squares | adaptive approximation | wavelets | convergence rates | high dimensional approximation | multivariate polynomials

approximation theory | MATHEMATICS, APPLIED | weighted least squares | adaptive approximation | wavelets | convergence rates | high dimensional approximation | multivariate polynomials

Journal Article

Mathematical Programming, ISSN 0025-5610, 1/2018, Volume 167, Issue 1, pp. 75 - 97

Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by...

High-dimensional data | 60H10 | Nonconvex optimization | Theoretical, Mathematical and Computational Physics | Stochastic approximation | 68W27 | Mathematics | Online algorithm | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Finite-sample analysis | Stochastic gradient method | Combinatorics | 60H15 | Principal component analysis | 62H25 | EIGENVALUE | MATHEMATICS, APPLIED | POWER | PCA | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Management science | Algorithms | Information management | Analysis | Lower bounds | Data analysis | Minimax technique | Data points | Approximation | Mathematical analysis | Dimensional analysis | Principal components analysis | Estimating techniques

High-dimensional data | 60H10 | Nonconvex optimization | Theoretical, Mathematical and Computational Physics | Stochastic approximation | 68W27 | Mathematics | Online algorithm | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Finite-sample analysis | Stochastic gradient method | Combinatorics | 60H15 | Principal component analysis | 62H25 | EIGENVALUE | MATHEMATICS, APPLIED | POWER | PCA | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Management science | Algorithms | Information management | Analysis | Lower bounds | Data analysis | Minimax technique | Data points | Approximation | Mathematical analysis | Dimensional analysis | Principal components analysis | Estimating techniques

Journal Article

Journal of Machine Learning Research, ISSN 1532-4435, 2010, Volume 11, pp. 411 - 450

We address instance-based learning from a perceptual organization standpoint and present methods for dimensionality estimation, manifold learning and function...

Tensor voting | High-dimensional processing | Geodesic distance | Manifold learning | Function approximation | Dimensionality estimation | REGRESSION | CONTOURS | high-dimensional processing | geodesic distance | ALGORITHM | SPARSE | NETWORKS | dimensionality estimation | INFERENCE | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | manifold learning | tensor voting | REDUCTION | function approximation | CURVE | SURFACE | AUTOMATION & CONTROL SYSTEMS | SUPERPOSITIONS

Tensor voting | High-dimensional processing | Geodesic distance | Manifold learning | Function approximation | Dimensionality estimation | REGRESSION | CONTOURS | high-dimensional processing | geodesic distance | ALGORITHM | SPARSE | NETWORKS | dimensionality estimation | INFERENCE | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | manifold learning | tensor voting | REDUCTION | function approximation | CURVE | SURFACE | AUTOMATION & CONTROL SYSTEMS | SUPERPOSITIONS

Journal Article

ESAIM: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, 05/2015, Volume 49, Issue 3, pp. 815 - 837

Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate...

High-dimensional approximation | Approximation theory | Parametric and stochastic PDEs | Polynomial approximation | Least squares | INTERPOLATION | PROJECTION | MATHEMATICS, APPLIED | RANDOM INPUT DATA | COLLOCATION METHOD | polynomial approximation | PARTIAL-DIFFERENTIAL-EQUATIONS | SPACES | least squares | high-dimensional approximation | parametric and stochastic PDEs | Tensors | Approximation | Random sampling | Least squares method | Probability theory | Mathematical models | Hilbert space | Polynomials | Optimization | Optimality criteria | Convergence

High-dimensional approximation | Approximation theory | Parametric and stochastic PDEs | Polynomial approximation | Least squares | INTERPOLATION | PROJECTION | MATHEMATICS, APPLIED | RANDOM INPUT DATA | COLLOCATION METHOD | polynomial approximation | PARTIAL-DIFFERENTIAL-EQUATIONS | SPACES | least squares | high-dimensional approximation | parametric and stochastic PDEs | Tensors | Approximation | Random sampling | Least squares method | Probability theory | Mathematical models | Hilbert space | Polynomials | Optimization | Optimality criteria | Convergence

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Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 02/2015, Volume 103, Issue 2, pp. 400 - 428

The numerical approximation of parametric partial differential equations is a computational challenge when the dimension of the parameter vector is large, due...

Sparse high dimensional interpolation | Parametric PDEs | Curse of dimensionality | Sparse Legendre series | MATHEMATICS | MATHEMATICS, APPLIED | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | ANALYTIC REGULARITY | CONVERGENCE | Anisotropy | Analysis | Numerical Analysis | Mathematics

Sparse high dimensional interpolation | Parametric PDEs | Curse of dimensionality | Sparse Legendre series | MATHEMATICS | MATHEMATICS, APPLIED | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | ANALYTIC REGULARITY | CONVERGENCE | Anisotropy | Analysis | Numerical Analysis | Mathematics

Journal Article

Journal of Fourier Analysis and Applications, ISSN 1069-5869, 10/2012, Volume 18, Issue 5, pp. 915 - 953

We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although...

65T50 | Signal, Image and Speech Processing | Sparse Fourier transform | Fourier transform | Mathematics | 15A69 | Abstract Harmonic Analysis | Tensor train format | QTT | Mathematical Methods in Physics | Fourier Analysis | Convolution | Quantum Fourier transform | 15A23 | 65F99 | High-dimensional problems | Approximations and Expansions | Partial Differential Equations | MATHEMATICS, APPLIED | TENSOR APPROXIMATION | ALGORITHM | OPERATORS | VECTORS

65T50 | Signal, Image and Speech Processing | Sparse Fourier transform | Fourier transform | Mathematics | 15A69 | Abstract Harmonic Analysis | Tensor train format | QTT | Mathematical Methods in Physics | Fourier Analysis | Convolution | Quantum Fourier transform | 15A23 | 65F99 | High-dimensional problems | Approximations and Expansions | Partial Differential Equations | MATHEMATICS, APPLIED | TENSOR APPROXIMATION | ALGORITHM | OPERATORS | VECTORS

Journal Article

BERNOULLI, ISSN 1350-7265, 05/2019, Volume 25, Issue 2, pp. 1326 - 1354

We consider the famous Rasch model, which is applied to psychometric surveys when n persons under test answer m questions. The score is given by a realization...

item response model | ASYMPTOTIC EQUIVALENCE | Le Cam distance | high-dimensional central limit theorem | DENSITY-ESTIMATION | psychometrics | NONPARAMETRIC REGRESSION | WHITE-NOISE | STATISTICS & PROBABILITY | PARAMETERS | asymptotic equivalence of statistical experiments

item response model | ASYMPTOTIC EQUIVALENCE | Le Cam distance | high-dimensional central limit theorem | DENSITY-ESTIMATION | psychometrics | NONPARAMETRIC REGRESSION | WHITE-NOISE | STATISTICS & PROBABILITY | PARAMETERS | asymptotic equivalence of statistical experiments

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High-Dimensional Function Approximation With Neural Networks for Large Volumes of Data

IEEE Transactions on Neural Networks and Learning Systems, ISSN 2162-237X, 02/2018, Volume 29, Issue 2, pp. 500 - 508

Approximation of high-dimensional functions is a challenge for neural networks due to the curse of dimensionality. Often the data for which the approximated...

Manifolds | Neurons | function approximation | Linear approximation | Distributed databases | high-dimensional data | neural networks | Approximation error | Big data | manifold mapping | Biological neural networks | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | COMPUTER SCIENCE, THEORY & METHODS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Projection | Approximation | Mathematical analysis | Neural networks | Manifolds (mathematics)

Manifolds | Neurons | function approximation | Linear approximation | Distributed databases | high-dimensional data | neural networks | Approximation error | Big data | manifold mapping | Biological neural networks | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | COMPUTER SCIENCE, THEORY & METHODS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Projection | Approximation | Mathematical analysis | Neural networks | Manifolds (mathematics)

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 03/2017, Volume 86, Issue 304, pp. 661 - 700

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of...

High-dimensional approximation | Compressive sensing | Tensorized Chebyshev polynomial chaos approximation | Affine-parametric operator equations | Parametric diffusion equation | S-term approximation | MATHEMATICS, APPLIED | affine-parametric operator equations | POLYNOMIAL-APPROXIMATION | high-dimensional approximation | SIGNAL RECOVERY | FOURIER | s-term approximation | tensorized Chebyshev polynomial chaos approximation | SPARSE | ANALYTIC REGULARITY | parametric diffusion equation | INTERPOLATION | ADAPTIVE WAVELET METHODS | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE

High-dimensional approximation | Compressive sensing | Tensorized Chebyshev polynomial chaos approximation | Affine-parametric operator equations | Parametric diffusion equation | S-term approximation | MATHEMATICS, APPLIED | affine-parametric operator equations | POLYNOMIAL-APPROXIMATION | high-dimensional approximation | SIGNAL RECOVERY | FOURIER | s-term approximation | tensorized Chebyshev polynomial chaos approximation | SPARSE | ANALYTIC REGULARITY | parametric diffusion equation | INTERPOLATION | ADAPTIVE WAVELET METHODS | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | CONVERGENCE

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2016, Volume 54, Issue 4, pp. 2282 - 2301

The ability to efficiently and accurately approximate an inverse frame operator is critical for establishing the utility of numerical frame approximations....

Numerical frame approximation | High-dimensional frames | Localized frames | Fourier frames | Inverse frame operator | Nonuniform fourier data | MATHEMATICS, APPLIED | MRI | high-dimensional frames | RECONSTRUCTION | ALGORITHM | ROSETTE TRAJECTORIES | inverse frame operator | localized frames | numerical frame approximation | nonuniform Fourier data

Numerical frame approximation | High-dimensional frames | Localized frames | Fourier frames | Inverse frame operator | Nonuniform fourier data | MATHEMATICS, APPLIED | MRI | high-dimensional frames | RECONSTRUCTION | ALGORITHM | ROSETTE TRAJECTORIES | inverse frame operator | localized frames | numerical frame approximation | nonuniform Fourier data

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 05/2018, Volume 87, Issue 311, pp. 1415 - 1450

This work proposes and analyzes a compressed sensing approach to polynomial approximation of complex-valued functions in high dimensions. In this context, the...

Downward closed (lower) sets | Convex optimization | Polynomial approximation | Compressed sensing | High-dimensional methods | MATHEMATICS, APPLIED | RECONSTRUCTION | EXPANSIONS | ALGORITHMS | downward closed (lower) sets | INTERPOLATION | RECOVERY | STOCHASTIC COLLOCATION METHOD | polynomial approximation | PARTIAL-DIFFERENTIAL-EQUATIONS | UNCERTAINTY | high-dimensional methods | convex optimization | CONVERGENCE | Mathematics - Numerical Analysis

Downward closed (lower) sets | Convex optimization | Polynomial approximation | Compressed sensing | High-dimensional methods | MATHEMATICS, APPLIED | RECONSTRUCTION | EXPANSIONS | ALGORITHMS | downward closed (lower) sets | INTERPOLATION | RECOVERY | STOCHASTIC COLLOCATION METHOD | polynomial approximation | PARTIAL-DIFFERENTIAL-EQUATIONS | UNCERTAINTY | high-dimensional methods | convex optimization | CONVERGENCE | Mathematics - Numerical Analysis

Journal Article

SBORNIK MATHEMATICS, ISSN 1064-5616, 04/2019, Volume 210, Issue 4, pp. 565 - 588

Consider the parametric elliptic problem - div (a(y)(x)del u(y)(x)) = f(x), x is an element of D, y is an element of I-infinity, u vertical bar(partial...

MATHEMATICS | HYPERBOLIC CROSS APPROXIMATION | parametric and stochastic elliptic PDEs | POLYNOMIAL-APPROXIMATION | affine dependence of the diffusion coefficients | PARTIAL-DIFFERENTIAL-EQUATIONS | linear collective collocation approximation | high-dimensional problems

MATHEMATICS | HYPERBOLIC CROSS APPROXIMATION | parametric and stochastic elliptic PDEs | POLYNOMIAL-APPROXIMATION | affine dependence of the diffusion coefficients | PARTIAL-DIFFERENTIAL-EQUATIONS | linear collective collocation approximation | high-dimensional problems

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Sparse support vector machines with L0 approximation for ultra-high dimensional omics data

Artificial Intelligence In Medicine, ISSN 0933-3657, 05/2019, Volume 96, pp. 134 - 141

Omics data usually have ultra-high dimension ( ) and small sample size ( ). Standard support vector machines (SVMs), which minimize the norm for the primal...

Ultra-high dimensional data | Feature selection | Metagenomics sequencing | SVM | L0 approximation | Classification

Ultra-high dimensional data | Feature selection | Metagenomics sequencing | SVM | L0 approximation | Classification

Journal Article

Structural and Multidisciplinary Optimization, ISSN 1615-147X, 5/2019, Volume 59, Issue 5, pp. 1439 - 1454

In recent years, the importance of computationally efficient surrogate models has been emphasized as the use of high-fidelity simulation models increases....

Engineering | Computational Mathematics and Numerical Analysis | Surrogate model | Limited data | Gaussian process regression | Engineering Design | Theoretical and Applied Mechanics | High-dimensional problem | Variable selection | DESIGN | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | SENSITIVITY-ANALYSIS

Engineering | Computational Mathematics and Numerical Analysis | Surrogate model | Limited data | Gaussian process regression | Engineering Design | Theoretical and Applied Mechanics | High-dimensional problem | Variable selection | DESIGN | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | SENSITIVITY-ANALYSIS

Journal Article

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