The Experts below are selected from a list of 11673 Experts worldwide ranked by ideXlab platform

Felix Finster - One of the best experts on this subject based on the ideXlab platform.

  • Fermion systems in discrete Space-time – outer symmetries and spontaneous symmetry breaking
    2007
    Co-Authors: Felix Finster
    Abstract:

    A systematic procedure is developed for constructing fermion systems in discrete Space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we prove that fermion systems exist only if the number of particles satisfies certain constraints. When applied to physical systems, this result shows that the permutation symmetry of discrete Space-time is always spontaneously broken by the fermionic projector. 1 Discrete Fermion Systems with Outer Symmetry We briefly recall the mathematical setting of the fermionic projector in discrete Space-time as introduced in [1] (see also [2] or [3]). Let H be a finite-dimensional complex vector Space endowed with a non-degenerate symmetric sesquilinear form <.|.>. We call (H,<.|.>) an indefinite Inner Product Space. To every element x of a finite set M = {1,...,m} we associate a projector Ex. We assume that these projectors are orthogonal and complete, Ex Ey = δxy Ex, Ex = 1, (1) and that the images of the Ex are non-degenerate subSpaces of H. We denote the signatur

  • Fermion systems in discrete Space-time – outer symmetries and spontaneous symmetry breaking
    2007
    Co-Authors: Felix Finster
    Abstract:

    A systematic procedure is developed for constructing fermion systems in discrete Space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we prove that fermion systems exist only if the number of particles satisfies certain constraints. When applied to physical systems, this result shows that the permutation symmetry of discrete Space-time is always spontaneously broken by the fermionic projector. 1 Discrete Fermion Systems with Outer Symmetry We briefly recall the mathematical setting of the fermionic projector in discrete Space-time as introduced in [1] (see also [2] or [3]). Let H be a finite-dimensional complex vector Space endowed with a non-degenerate sesquilinear form <.|.>. We call (H,<.|.>) an indefinite Inner Product Space. To every element x of a finite set M = {1,...,m} we associate a projector Ex. We assume that these projectors are orthogonal and complete, Ex Ey = δxy Ex, Ex = 1, (1) and that the images of the Ex are non-degenerate subSpaces of H. We denote the signatur

Lei Zhang - One of the best experts on this subject based on the ideXlab platform.

  • Log-Euclidean Kernels for Sparse Representation and Dictionary Learning
    2013 IEEE International Conference on Computer Vision, 2013
    Co-Authors: Peihua Li, Qilong Wang, Lei Zhang
    Abstract:

    The symmetric positive definite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the Space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the Space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication defined in the Log-Euclidean framework, is a complete Inner Product Space. We can thus develop a broad family of kernels that satisfies Mercer's condition. These kernels characterize the geodesic distance and can be computed efficiently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian Space instead of in the Euclidean Space. The proposed method is evaluated with various vision problems and shows notable performance gains over state-of-the-arts.

Yuesheng Zhu - One of the best experts on this subject based on the ideXlab platform.

  • robust dictionary learning and sparse coding with riemannian geometry preserving method in symmetric matrices Inner Product Space
    IEEE Access, 2020
    Co-Authors: Yang Zhang, Yuesheng Zhu
    Abstract:

    Existing Dictionary Learning and Sparse Coding (DLSC) algorithms for Symmetric Positive Definite (SPD) matrices usually adopt Reproducing Kernel Hilbert Space as workSpace to perform necessary linear operations. But those methods heavily rely on ideal kernel maps and they are lack of robustness when facing different SPD data, especially in the case of high-sparsity coding. Different from existing methods, we explore a new workSpace called Symmetric Matrices Inner Product Space (SMIPS) for modeling robust DLSC of SPD matrices. SMIPS, which can be treated as the minimal linear expansion Space of SPD manifold, is a linear Space equipped with a pre-defined Inner Product so it supports linear operations. Modelling DLSC in SMIPS is more intuitive, and SPD data can preserve matrix form in SMIPS. Then, in this paper, a Riemannian Geometry Preserving (RGP) method based on graph-regularized is proposed to include the Riemannian geometric information of original SPD data, so that improves the discriminability of sparse codes, and a sparse coding algorithm is developed to acquire sparse codes of SPD matrices in SMIPS efficiently, which extend the feature-sign search to such matrix Space. For dictionary learning, an overall learning strategy utilizing the closed form solution of RGP codes is proposed to speed up the iterative dictionary learning process. Experiments on several computer vision tasks show that our algorithm is more robust even with high-sparsity coding across different datasets and outperforms other comparative algorithms in terms of recognition accuracy and learning efficiency, which also demonstrates the validity of SMIPS as a workSpace.

Peihua Li - One of the best experts on this subject based on the ideXlab platform.

  • Log-Euclidean Kernels for Sparse Representation and Dictionary Learning
    2013 IEEE International Conference on Computer Vision, 2013
    Co-Authors: Peihua Li, Qilong Wang, Lei Zhang
    Abstract:

    The symmetric positive definite (SPD) matrices have been widely used in image and vision problems. Recently there are growing interests in studying sparse representation (SR) of SPD matrices, motivated by the great success of SR for vector data. Though the Space of SPD matrices is well-known to form a Lie group that is a Riemannian manifold, existing work fails to take full advantage of its geometric structure. This paper attempts to tackle this problem by proposing a kernel based method for SR and dictionary learning (DL) of SPD matrices. We disclose that the Space of SPD matrices, with the operations of logarithmic multiplication and scalar logarithmic multiplication defined in the Log-Euclidean framework, is a complete Inner Product Space. We can thus develop a broad family of kernels that satisfies Mercer's condition. These kernels characterize the geodesic distance and can be computed efficiently. We also consider the geometric structure in the DL process by updating atom matrices in the Riemannian Space instead of in the Euclidean Space. The proposed method is evaluated with various vision problems and shows notable performance gains over state-of-the-arts.

Anatolij Dvurecenskij - One of the best experts on this subject based on the ideXlab platform.

  • A Finitely Additive State Criterion for the Completeness of Inner Product Spaces
    Letters in Mathematical Physics, 2003
    Co-Authors: Emmanuel Chetcuti, Anatolij Dvurecenskij
    Abstract:

    We show that an Inner Product Space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subSpaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subSpaces of S. Although it gives only a partial solution to the problem formulated by Ptak on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subSpaces of S.

  • a new algebraic criterion for completeness of Inner Product Spaces
    Letters in Mathematical Physics, 2001
    Co-Authors: Anatolij Dvurecenskij
    Abstract:

    We show that an Inner Product Space S is complete whenever the system E(S) of all splitting subSpaces of S, i.e., of all subSpaces M of S such that M + M ⊥ = S holds, satisfies the σ-Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a σ-complete lattice, and that of the author who required E(S) to be a σ-orthocomplete OMP.

  • regular states and countable additivity on quantum logics
    Proceedings of the American Mathematical Society, 1992
    Co-Authors: Anatolij Dvurecenskij, Tibor Neubrunn, Sylvia Pulmannova
    Abstract:

    We give a counter example of the result of Beaver and Cook concerning a generalization of the Alexandroff theorem for regular, finitely-additive states on quantum logics using states on the system of all splitting subSpaces of an incomplete Inner-Product Space. Moreover, we introduce another type of state regularity which entails countable additivity of states on logics