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Jean-yves Welschinger - One of the best experts on this subject based on the ideXlab platform.

  • Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
    Journal of the European Mathematical Society, 2016
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.

  • Topology of random real hypersurfaces
    2014
    Co-Authors: Jean-yves Welschinger
    Abstract:

    These are notes of the mini-course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July $2014$. The subject was my joint work with Damien Gayet on the topology of random real hypersurfaces, restricting myself to the case of projective spaces and focusing on our lower estimates. Namely, we estimate from (above and) below the mathematical expectation of all Betti numbers of degree $d$ random real projective hypersurfaces. For any closed connected hypersurface $\Sigma$ of $\R^n$, we actually estimate from below the mathematical expectation of the number of connected components of these degree $d$ random real projective hypersurfaces which are diffeomorphic to $\Sigma$.

  • Lower estimates for the expected Betti numbers of random real hypersurfaces
    Journal of the London Mathematical Society, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a constant which only depends on the dimension n of the manifold with the Kählerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).

  • Expected topology of random real algebraic submanifolds
    Journal of the Institute of Mathematics of Jussieu, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k

  • what is the total betti number of a random real hypersurface
    Crelle's Journal, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve Hormander's theory of peak sections as well as the formula of Poincare-Martinelli.

Nicolas Keriven - One of the best experts on this subject based on the ideXlab platform.

  • Large-Scale High-Dimensional Clustering with Fast Sketching
    2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the k centroids are then retrieved efficiently. However , this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

  • ICASSP - Large-Scale High-Dimensional Clustering with Fast Sketching
    2018 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the $k$ centroids are then retrieved efficiently. However, this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

Damien Gayet - One of the best experts on this subject based on the ideXlab platform.

  • Betti numbers of random real hypersurfaces and determinants of random symmetric matrices
    Journal of the European Mathematical Society, 2016
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from mid-dimensional Betti numbers. In order to get these results, we first establish the equidistribution of the critical points of a given Morse function restricted to the ran- dom real hypersurfaces.

  • Lower estimates for the expected Betti numbers of random real hypersurfaces
    Journal of the London Mathematical Society, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large d-th power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a constant which only depends on the dimension n of the manifold with the Kählerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).

  • Expected topology of random real algebraic submanifolds
    Journal of the Institute of Mathematics of Jussieu, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    Let X be a smooth complex projective manifold of dimension n equipped with an ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0< k

  • what is the total betti number of a random real hypersurface
    Crelle's Journal, 2014
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve Hormander's theory of peak sections as well as the formula of Poincare-Martinelli.

  • What is the total Betti number of a random real hypersurface?
    Journal für die reine und angewandte Mathematik, 2012
    Co-Authors: Damien Gayet, Jean-yves Welschinger
    Abstract:

    We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve Hörmander's theory of peak sections as well as the formula of Poincaré-Martinelli.

Antoine Chatalic - One of the best experts on this subject based on the ideXlab platform.

  • Large-Scale High-Dimensional Clustering with Fast Sketching
    2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the k centroids are then retrieved efficiently. However , this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

  • ICASSP - Large-Scale High-Dimensional Clustering with Fast Sketching
    2018 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the $k$ centroids are then retrieved efficiently. However, this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

Rémi Gribonval - One of the best experts on this subject based on the ideXlab platform.

  • Large-Scale High-Dimensional Clustering with Fast Sketching
    2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the k centroids are then retrieved efficiently. However , this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.

  • ICASSP - Large-Scale High-Dimensional Clustering with Fast Sketching
    2018 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2018
    Co-Authors: Antoine Chatalic, Rémi Gribonval, Nicolas Keriven
    Abstract:

    In this paper, we address the problem of high-dimensional k-means clustering in a large-scale setting, i.e. for datasets that comprise a large number of items. Sketching techniques have already been used to deal with this “large-scale” issue, by compressing the whole dataset into a single vector of random nonlinear generalized moments from which the $k$ centroids are then retrieved efficiently. However, this approach usually scales quadratically with the dimension; to cope with high-dimensional datasets, we show how to use fast structured random matrices to compute the sketching operator efficiently. This yields significant speed-ups and memory savings for high-dimensional data, while the clustering results are shown to be much more stable, both on artificial and real datasets.