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

  • linear embeddings of low dimensional subsets of a hilbert Space to r m
    European Signal Processing Conference, 2015
    Co-Authors: Gilles Puy, Michael Davies, Remi Gribonval
    Abstract:

    We consider the problem of embedding a low-dimensional set, M, from an inFinite-Dimensional Hilbert Space, H, to a Finite-Dimensional Space. Defining appropriate random linear projections, we propose two constructions of linear maps that have the restricted isometry property (RIP) on the secant set of M with high probability. The first one is optimal in the sense that it only needs a number of projections essentially proportional to the intrinsic dimension of M to satisfy the RIP. The second one, which is based on a variable density sampling technique, is computationally more efficient, while potentially requiring more measurements.

  • linear embeddings of low dimensional subsets of a hilbert Space to mathbb r m
    SPARS15 - Signal Processing with Adaptive Sparse Structured Representations, 2015
    Co-Authors: Gilles Puy, Michael Davies, Remi Gribonval
    Abstract:

    We consider the problem of embedding a low-dimensional set, M, from an inFinite-Dimensional Hilbert Space to a Finite-Dimensional Space. Defining appropriate random linear projections, we construct a linear map which has the restricted isometry property on the secant set of M, with high probability for a number of projections essentially proportional to the intrinsic dimension of M.

Vivek S. Borkar - One of the best experts on this subject based on the ideXlab platform.

  • Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert Space
    Automatica, 2009
    Co-Authors: Ankur A. Kulkarni, Vivek S. Borkar
    Abstract:

    This paper presents a finite dimensional approach to stochastic approximation in infinite dimensional Hilbert Space. The problem was motivated by applications in the field of stochastic programming wherein we minimize a convex function defined on a Hilbert Space. We define a finite dimensional approximation to the Hilbert Space minimizer. A justification is provided for this finite dimensional approximation. Estimates of the dimensionality needed are also provided. The algorithm presented is a two time-scale Newton-based stochastic approximation scheme that lives in this finite dimensional Space. Since the finite dimensional problem can be prohibitively large dimensional, we operate our Newton scheme in a projected, randomly chosen smaller dimensional subSpace.

Gilles Puy - One of the best experts on this subject based on the ideXlab platform.

  • linear embeddings of low dimensional subsets of a hilbert Space to r m
    European Signal Processing Conference, 2015
    Co-Authors: Gilles Puy, Michael Davies, Remi Gribonval
    Abstract:

    We consider the problem of embedding a low-dimensional set, M, from an inFinite-Dimensional Hilbert Space, H, to a Finite-Dimensional Space. Defining appropriate random linear projections, we propose two constructions of linear maps that have the restricted isometry property (RIP) on the secant set of M with high probability. The first one is optimal in the sense that it only needs a number of projections essentially proportional to the intrinsic dimension of M to satisfy the RIP. The second one, which is based on a variable density sampling technique, is computationally more efficient, while potentially requiring more measurements.

  • linear embeddings of low dimensional subsets of a hilbert Space to mathbb r m
    SPARS15 - Signal Processing with Adaptive Sparse Structured Representations, 2015
    Co-Authors: Gilles Puy, Michael Davies, Remi Gribonval
    Abstract:

    We consider the problem of embedding a low-dimensional set, M, from an inFinite-Dimensional Hilbert Space to a Finite-Dimensional Space. Defining appropriate random linear projections, we construct a linear map which has the restricted isometry property on the secant set of M, with high probability for a number of projections essentially proportional to the intrinsic dimension of M.

Qingyun Shi - One of the best experts on this subject based on the ideXlab platform.

  • matrix factorizations for reversible integer mapping
    IEEE Transactions on Signal Processing, 2001
    Co-Authors: Pengwei Hao, Qingyun Shi
    Abstract:

    Reversible integer mapping is essential for lossless source coding by transformation. A general matrix factorization theory for reversible integer mapping of invertible linear transforms is developed. Concepts of the integer factor and the elementary reversible matrix (ERM) for integer mapping are introduced, and two forms of ERM-triangular ERM (TERM) and single-row ERM (SERM)-are studied. We prove that there exist some approaches to factorize a matrix into TERMs or SERMs if the transform is invertible and in a Finite-Dimensional Space. The advantages of the integer implementations of an invertible linear transform are (i) mapping integers to integers, (ii) perfect reconstruction, and (iii) in-place calculation. We find that besides a possible permutation matrix, the TERM factorization of an N-by-N nonsingular matrix has at most three TERMs, and its SERM factorization has at most N+1 SERMs. The elementary structure of ERM transforms is the ladder structure. An executable factorization algorithm is also presented. Then, the computational complexity is compared, and some optimization approaches are proposed. The error bounds of the integer implementations are estimated as well. Finally, three ERM factorization examples of DFT, DCT, and DWT are given.

Qinghua Qin - One of the best experts on this subject based on the ideXlab platform.

  • post buckling solutions of hyper elastic beam by canonical dual finite element method
    Mathematics and Mechanics of Solids, 2014
    Co-Authors: Kun Cai, David Yang Gao, Qinghua Qin
    Abstract:

    The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional Space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified ...

  • post buckling solutions of hyper elastic beam by canonical dual finite element method
    arXiv: Computational Engineering Finance and Science, 2013
    Co-Authors: Kun Cai, David Yang Gao, Qinghua Qin
    Abstract:

    Post buckling problem of a large deformed beam is analyzed using canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional Space. Based on the canonical duality theory and the associated triality theorem, a primal-dual algorithm is proposed, which can be used to find all possible solutions of this nonconvex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam. While the local extrema of the pure-complementary energy present unstable deformation states, especially. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.