Algebraic Basis

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Dong Fang - One of the best experts on this subject based on the ideXlab platform.

  • EUSIPCO - Linear physical layer network coding for multihop wireless networks
    2014
    Co-Authors: Alister G. Burr, Dong Fang
    Abstract:

    We consider linear network coding functions that can be employed at the relays in wireless physical layer network coding, applied to a general multi-hop network topology. We introduce a general model of such a network, and discuss the Algebraic Basis of linear functions, deriving conditions for unambiguous decodability of the source data at the destination. We consider the use of integer rings, integer fields, binary extension fields and the ring of binary matrices as potential Algebraic constructs, and show that the ring constructs provide more flexibility. We use the two-way relay channel and a network containing two sources and two relays to illustrate the concept and to demonstrate the effect of fading of the wireless channels. We show the capacity benefits of the more flexible rings.

  • Linear physical layer network coding for multihop wireless networks
    2014 22nd European Signal Processing Conference (EUSIPCO), 2014
    Co-Authors: Alister Burr, Dong Fang
    Abstract:

    We consider linear network coding functions that can be employed at the relays in wireless physical layer network coding, applied to a general multi-hop network topology. We introduce a general model of such a network, and discuss the Algebraic Basis of linear functions, deriving conditions for unambiguous decodability of the source data at the destination. We consider the use of integer rings, integer fields, binary extension fields and the ring of binary matrices as potential Algebraic constructs, and show that the ring constructs provide more flexibility. We use the two-way relay channel and a network containing two sources and two relays to illustrate the concept and to demonstrate the effect of fading of the wireless channels. We show the capacity benefits of the more flexible rings.

Alister Burr - One of the best experts on this subject based on the ideXlab platform.

  • Linear physical layer network coding for multihop wireless networks
    2014 22nd European Signal Processing Conference (EUSIPCO), 2014
    Co-Authors: Alister Burr, Dong Fang
    Abstract:

    We consider linear network coding functions that can be employed at the relays in wireless physical layer network coding, applied to a general multi-hop network topology. We introduce a general model of such a network, and discuss the Algebraic Basis of linear functions, deriving conditions for unambiguous decodability of the source data at the destination. We consider the use of integer rings, integer fields, binary extension fields and the ring of binary matrices as potential Algebraic constructs, and show that the ring constructs provide more flexibility. We use the two-way relay channel and a network containing two sources and two relays to illustrate the concept and to demonstrate the effect of fading of the wireless channels. We show the capacity benefits of the more flexible rings.

Senhadji Lotfi - One of the best experts on this subject based on the ideXlab platform.

  • Semi-nonnegative joint diagonalization by congruence and semi-nonnegative ICA
    Signal Processing, 2016
    Co-Authors: Coloigner Julie, Albera Laurent, Noury Fanny, Kachenoura Amar, Senhadji Lotfi
    Abstract:

    In this paper, we focus on the Joint Diagonalization by Congruence (JDC) decomposition of a set of matrices, while imposing nonnegative constraints on the joint diagonalizer. The latter will be referred to the semi-nonnegative JDC fitting problem. This problem appears in semi-nonnegative Independent Component Analysis (ICA), say ICA involving nonnegative static mixtures, such as those encountered for instance in image processing and in magnetic resonance spectroscopy. In order to achieve the semi-nonnegative JDC decomposition, we propose two novel algorithms called ELS-ALSexp and CGexp, which optimize an unconstrained problem obtained by means of an exponential change of variable. The proposed methods are based on the line search strategy for which an analytic global plane search procedure has been considered. All derivatives have been jointly calculated in matrix form using the Algebraic Basis for matrix calculus and product operator properties. Our algorithms have been tested on synthetic arrays and the semi-nonnegative ICA problem is illustrated through simulations in magnetic resonance spectroscopy and in image processing. The numerical results show the benefit of using a priori information, such as nonnegativity.

Derek W. Robinson - One of the best experts on this subject based on the ideXlab platform.

  • Subcoercivity and subelliptic operators on Lie groups I: Free nilpotent groups
    Potential Analysis, 1994
    Co-Authors: A. F. M. Elst, Derek W. Robinson
    Abstract:

    Let ( χ, G, U ) be a continuous representation of a Lie group G by bounded operators g → U ( g ) on the Banach space χ and let (χ, $$\mathfrak{g}$$ , dU ) denote the representation of the Lie algebra $$\mathfrak{g}$$ obtained by differentiation. If a _1, ..., a _d′ is a Lie algebra Basis of $$\mathfrak{g}$$ , A _i= dU ( a _i) and $$A^\alpha = A_{i_1 } ...A_{i_k } $$ whenever α=( i _1, ..., i _k) we consider the operators $$H = \mathop \sum \limits_{\alpha ;|\alpha | \leqslant 2n} c_\alpha A^\alpha $$ where the c _α are complex coefficients satisfying a subcoercivity condition. This condition is such that the class of operators considered encompasses all the standard second-order subelliptic operators with real coefficients, all operators of the form $$\sum _{i = 1}^{d'} \lambda _i ( - A_i^2 )^n $$ with Re λ_ i >0 together with operators of the form $$H = ( - 1)^n \mathop \sum \limits_{\alpha ;|\alpha | = n} \mathop \sum \limits_{\beta ;|\beta | = n} c_{\alpha ,\beta } A^{\alpha _* } A^\beta $$ where α_*=( i _k, ..., i _1) if α=( i _1, ..., i _k) and the real part of the matrix ( c _α β) is strictly positive. In case the Lie algebra $$\mathfrak{g}$$ is free of step r , where r is the rank of the Algebraic Basis a _1, ..., a _d′, G is connected and U is the left regular representation in G we prove that the closure $$\overline H $$ of H generates a holomorphic semigroup S . Moreover, the semigroup S has a smooth kernel and we derive bounds on the kernel and all its derivatives. This will be a key ingredient for the paper [13] in which the above results will be extended to general groups and representations.

  • Strongly Elliptic and Subelliptic Operators on Lie Groups
    Quantum and Non-Commutative Analysis, 1993
    Co-Authors: Derek W. Robinson
    Abstract:

    Partial differential operators, and in particular elliptic operators, on a Riemannian manifold provide a rich source of problems of non-commutative analysis. In the simplest situation the manifold is a Lie group and one can examine elliptic operators constructed from a vector space Basis of the Lie algebra or subelliptic operators constructed from a Lie Algebraic Basis. In the latter case the choice of Algebraic Basis affects the characteristic geometric features of the subelliptic operators and their description involves an interesting interplay between analysis and geometry.

Coloigner Julie - One of the best experts on this subject based on the ideXlab platform.

  • Semi-nonnegative joint diagonalization by congruence and semi-nonnegative ICA
    Signal Processing, 2016
    Co-Authors: Coloigner Julie, Albera Laurent, Noury Fanny, Kachenoura Amar, Senhadji Lotfi
    Abstract:

    In this paper, we focus on the Joint Diagonalization by Congruence (JDC) decomposition of a set of matrices, while imposing nonnegative constraints on the joint diagonalizer. The latter will be referred to the semi-nonnegative JDC fitting problem. This problem appears in semi-nonnegative Independent Component Analysis (ICA), say ICA involving nonnegative static mixtures, such as those encountered for instance in image processing and in magnetic resonance spectroscopy. In order to achieve the semi-nonnegative JDC decomposition, we propose two novel algorithms called ELS-ALSexp and CGexp, which optimize an unconstrained problem obtained by means of an exponential change of variable. The proposed methods are based on the line search strategy for which an analytic global plane search procedure has been considered. All derivatives have been jointly calculated in matrix form using the Algebraic Basis for matrix calculus and product operator properties. Our algorithms have been tested on synthetic arrays and the semi-nonnegative ICA problem is illustrated through simulations in magnetic resonance spectroscopy and in image processing. The numerical results show the benefit of using a priori information, such as nonnegativity.