The Experts below are selected from a list of 40809 Experts worldwide ranked by ideXlab platform

Alfred O Hero - One of the best experts on this subject based on the ideXlab platform.

  • geodesic entropic graphs for dimension and entropy estimation in manifold learning
    IEEE Transactions on Signal Processing, 2004
    Co-Authors: Jose A Costa, Alfred O Hero
    Abstract:

    In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional Linear Vector Space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We introduce a novel geometric approach based on entropic graph methods. Although the theory presented applies to this general class of graphs, we focus on the geodesic-minimal-spanning-tree (GMST) to obtaining asymptotically consistent estimates of the manifold dimension and the Re/spl acute/nyi /spl alpha/-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach on standard synthetic manifolds as well as on real data sets consisting of images of faces.

  • manifold learning using euclidean k nearest neighbor graphs image processing examples
    International Conference on Acoustics Speech and Signal Processing, 2004
    Co-Authors: Jose A Costa, Alfred O Hero
    Abstract:

    In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional Linear Vector Space, based on a set of n measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. In previous work, we introduced a geometric probability method called the geodesic minimal spanning tree (GMST) to obtain asymptotically consistent estimates of manifold dimension and entropy. In this paper, we present a simpler method, based on the k-nearest neighbor (k-NN) graph that does not require estimation of geodesic distances on the manifold. The algorithm is applied to standard synthetic manifolds as well as real data sets consisting of images of faces.

Passuello Alberto - One of the best experts on this subject based on the ideXlab platform.

  • Semidefinite programming in combinatorial optimization with applications to coding theory and geometry
    2021
    Co-Authors: Passuello Alberto
    Abstract:

    Une nouvelle borne supérieure sur le cardinal des codes de sous-eSpaces d'un eSpace Vectoriel fini est établie grâce à la méthode de la programmation semidéfinie positive. Ces codes sont d'intérêt dans le cadre du codage de réseau (network coding). Ensuite, par la même méthode, l'on démontre une borne sur le cardinal des ensembles qui évitent une distance donnée dans l'eSpace de Johnson et qui est obtenue par une variante d'un programme de Schrijver. Les résultats numériques permettent d'améliorer les bornes existantes sur le nombre chromatique mesurable de l'eSpace Euclidien. Une hiérarchie de programmes semidéfinis positifs est construite à partir de certaines matrices issues des complexes simpliciaux. Ces programmes permettent d'obtenir une borne supérieure sur le nombre d'indépendance d'un graphe. Aussi, cette hiérarchie partage certaines propriétés importantes avec d'autres hiérarchies classiques. A titre d'exemple, le problème de déterminer le nombre d'indépendance des graphes de Paley est analysé.We apply the semidefinite programming method to obtain a new upper bound on the cardinality of codes made of subSpaces of a Linear Vector Space over a finite field. Such codes are of interest in network coding.Next, with the same method, we prove an upper bound on the cardinality of sets avoiding one distance in the Johnson Space, which is essentially Schrijver semidefinite program. This bound is used to improve existing results on the measurable chromatic number of the Euclidean Space.We build a new hierarchy of semidefinite programs whose optimal values give upper bounds on the independence number of a graph. This hierarchy is based on matrices arising from simplicial complexes. We show some properties that our hierarchy shares with other classical ones. As an example, we show its application to the problem of determining the independence number of Paley graphs

  • Programmation semidéfinie positive dans l’optimisation combinatoire avec applications à la théorie des codes correcteurs et à la géométrie
    2013
    Co-Authors: Passuello Alberto
    Abstract:

    Une nouvelle borne supérieure sur le cardinal des codes de sous-eSpaces d'un eSpace Vectoriel fini est établie grâce à la méthode de la programmation semidéfinie positive. Ces codes sont d'intérêt dans le cadre du codage de réseau (network coding). Ensuite, par la même méthode, l'on démontre une borne sur le cardinal des ensembles qui évitent une distance donnée dans l'eSpace de Johnson et qui est obtenue par une variante d'un programme de Schrijver. Les résultats numériques permettent d'améliorer les bornes existantes sur le nombre chromatique mesurable de l'eSpace Euclidien. Une hiérarchie de programmes semidéfinis positifs est construite à partir de certaines matrices issues des complexes simpliciaux. Ces programmes permettent d'obtenir une borne supérieure sur le nombre d'indépendance d'un graphe. Aussi, cette hiérarchie partage certaines propriétés importantes avec d'autres hiérarchies classiques. A titre d'exemple, le problème de déterminer le nombre d'indépendance des graphes de Paley est analysé.We apply the semidefinite programming method to obtain a new upper bound on the cardinality of codes made of subSpaces of a Linear Vector Space over a finite field. Such codes are of interest in network coding.Next, with the same method, we prove an upper bound on the cardinality of sets avoiding one distance in the Johnson Space, which is essentially Schrijver semidefinite program. This bound is used to improve existing results on the measurable chromatic number of the Euclidean Space.We build a new hierarchy of semidefinite programs whose optimal values give upper bounds on the independence number of a graph. This hierarchy is based on matrices arising from simplicial complexes. We show some properties that our hierarchy shares with other classical ones. As an example, we show its application to the problem of determining the independence number of Paley graphs

  • Programmation semidéfinie positive dans l optimisation combinatoire avec applications à la théorie des codes correcteurs et à la géométrie
    2013
    Co-Authors: Passuello Alberto, Bachoc Christine
    Abstract:

    Une nouvelle borne supérieure sur le cardinal des codes de sous-eSpaces d'un eSpace Vectoriel fini est établie grâce à la méthode de la programmation semidéfinie positive. Ces codes sont d'intérêt dans le cadre du codage de réseau (network coding). Ensuite, par la même méthode, l'on démontre une borne sur le cardinal des ensembles qui évitent une distance donnée dans l'eSpace de Johnson et qui est obtenue par une variante d'un programme de Schrijver. Les résultats numériques permettent d'améliorer les bornes existantes sur le nombre chromatique mesurable de l'eSpace Euclidien. Une hiérarchie de programmes semidéfinis positifs est construite à partir de certaines matrices issues des complexes simpliciaux. Ces programmes permettent d'obtenir une borne supérieure sur le nombre d'indépendance d'un graphe. Aussi, cette hiérarchie partage certaines propriétés importantes avec d'autres hiérarchies classiques. A titre d'exemple, le problème de déterminer le nombre d'indépendance des graphes de Paley est analysé.We apply the semidefinite programming method to obtain a new upper bound on the cardinality of codes made of subSpaces of a Linear Vector Space over a finite field. Such codes are of interest in network coding.Next, with the same method, we prove an upper bound on the cardinality of sets avoiding one distance in the Johnson Space, which is essentially Schrijver semidefinite program. This bound is used to improve existing results on the measurable chromatic number of the Euclidean Space.We build a new hierarchy of semidefinite programs whose optimal values give upper bounds on the independence number of a graph. This hierarchy is based on matrices arising from simplicial complexes. We show some properties that our hierarchy shares with other classical ones. As an example, we show its application to the problem of determining the independence number of Paley graphs.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF

Jose A Costa - One of the best experts on this subject based on the ideXlab platform.

  • geodesic entropic graphs for dimension and entropy estimation in manifold learning
    IEEE Transactions on Signal Processing, 2004
    Co-Authors: Jose A Costa, Alfred O Hero
    Abstract:

    In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional Linear Vector Space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We introduce a novel geometric approach based on entropic graph methods. Although the theory presented applies to this general class of graphs, we focus on the geodesic-minimal-spanning-tree (GMST) to obtaining asymptotically consistent estimates of the manifold dimension and the Re/spl acute/nyi /spl alpha/-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach on standard synthetic manifolds as well as on real data sets consisting of images of faces.

  • manifold learning using euclidean k nearest neighbor graphs image processing examples
    International Conference on Acoustics Speech and Signal Processing, 2004
    Co-Authors: Jose A Costa, Alfred O Hero
    Abstract:

    In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional Linear Vector Space, based on a set of n measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. In previous work, we introduced a geometric probability method called the geodesic minimal spanning tree (GMST) to obtain asymptotically consistent estimates of manifold dimension and entropy. In this paper, we present a simpler method, based on the k-nearest neighbor (k-NN) graph that does not require estimation of geodesic distances on the manifold. The algorithm is applied to standard synthetic manifolds as well as real data sets consisting of images of faces.

  • manifold learning using euclidean nearest neighbor graphs
    2004
    Co-Authors: Jose A Costa
    Abstract:

    In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional Linear Vector Space, based on a set of measured sample points on the surface. In this paper we consider the closely related problem of estimating the manifold’s intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. In previous work we introduced a geometric probability method called Geodesic Minimal Spanning Tree (GMST) to obtain asymptotically consistent estimates of manifold dimension and entropy. In this paper we present a simpler method based on the -nearest neighbor ( -NN) graph that does not require estimation of geodesic distances on the manifold. The algorithm is applied to standard synthetic manifolds as well as real data sets consisting of images of faces.

Sayan Mukherjee - One of the best experts on this subject based on the ideXlab platform.

  • bayesian approximate kernel regression with variable selection
    Journal of the American Statistical Association, 2018
    Co-Authors: Lorin Crawford, Kris C Wood, Xiang Zhou, Sayan Mukherjee
    Abstract:

    AbstractNonLinear kernel regression models are often used in statistics and machine learning because they are more accurate than Linear models. Variable selection for kernel regression models is a challenge partly because, unlike the Linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog for each explanatory variable in Bayesian kernel regression models when the kernel is shift-invariant — for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert Spaces (RKHS) to define a Linear Vector Space that: (i) captures nonLinear structure, and (ii) can be projected onto the original explanatory variables. This projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based ...

  • bayesian approximate kernel regression with variable selection
    arXiv: Methodology, 2015
    Co-Authors: Lorin Crawford, Kris C Wood, Xiang Zhou, Sayan Mukherjee
    Abstract:

    NonLinear kernel regression models are often used in statistics and machine learning because they are more accurate than Linear models. Variable selection for kernel regression models is a challenge partly because, unlike the Linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert Spaces (RKHS) to define a Linear Vector Space that: (i) captures nonLinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonLinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and Linear models, respectively. BAKR is the first method that is competitive in both settings.

Lorin Crawford - One of the best experts on this subject based on the ideXlab platform.

  • bayesian approximate kernel regression with variable selection
    Journal of the American Statistical Association, 2018
    Co-Authors: Lorin Crawford, Kris C Wood, Xiang Zhou, Sayan Mukherjee
    Abstract:

    AbstractNonLinear kernel regression models are often used in statistics and machine learning because they are more accurate than Linear models. Variable selection for kernel regression models is a challenge partly because, unlike the Linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog for each explanatory variable in Bayesian kernel regression models when the kernel is shift-invariant — for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert Spaces (RKHS) to define a Linear Vector Space that: (i) captures nonLinear structure, and (ii) can be projected onto the original explanatory variables. This projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based ...

  • bayesian approximate kernel regression with variable selection
    arXiv: Methodology, 2015
    Co-Authors: Lorin Crawford, Kris C Wood, Xiang Zhou, Sayan Mukherjee
    Abstract:

    NonLinear kernel regression models are often used in statistics and machine learning because they are more accurate than Linear models. Variable selection for kernel regression models is a challenge partly because, unlike the Linear regression setting, there is no clear concept of an effect size for regression coefficients. In this paper, we propose a novel framework that provides an effect size analog of each explanatory variable for Bayesian kernel regression models when the kernel is shift-invariant --- for example, the Gaussian kernel. We use function analytic properties of shift-invariant reproducing kernel Hilbert Spaces (RKHS) to define a Linear Vector Space that: (i) captures nonLinear structure, and (ii) can be projected onto the original explanatory variables. The projection onto the original explanatory variables serves as an analog of effect sizes. The specific function analytic property we use is that shift-invariant kernel functions can be approximated via random Fourier bases. Based on the random Fourier expansion we propose a computationally efficient class of Bayesian approximate kernel regression (BAKR) models for both nonLinear regression and binary classification for which one can compute an analog of effect sizes. We illustrate the utility of BAKR by examining two important problems in statistical genetics: genomic selection (i.e. phenotypic prediction) and association mapping (i.e. inference of significant variants or loci). State-of-the-art methods for genomic selection and association mapping are based on kernel regression and Linear models, respectively. BAKR is the first method that is competitive in both settings.