The Experts below are selected from a list of 27159 Experts worldwide ranked by ideXlab platform
Hermann Schulzbaldes - One of the best experts on this subject based on the ideXlab platform.
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statistical mechanics of map estimation general replica ansatz
2019Co-Authors: Ali Bereyhi, Ralf R Muller, Hermann SchulzbaldesAbstract:The large-System performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear System with additive white Gaussian noise. The analysis considers the System matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-System analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with $b$ steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued System decouples into a bank of equivalent Decoupled scalar Systems followed by maximum-a-posterior estimators. The structure of the Decoupled System is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For $b$ steps of symmetry breaking, the Decoupled System is found to be an additive System with a non-Gaussian noise term given as the sum of an independent Gaussian random variable with $b$ non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition System described by its corresponding Decoupled System. As an application of our study, we investigate large compressive sensing Systems by considering the $\ell _{p}$ norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for $\ell _{0}$ norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansatze are needed in order to assess the performance precisely.
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statistical mechanics of map estimation general replica ansatz
2016Co-Authors: Ali Bereyhi, Ralf R Muller, Hermann SchulzbaldesAbstract:The large-System performance of MAP estimation is studied considering a general distortion function when the observation vector is received through a linear System with additive white Gaussian noise. The analysis considers the System matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-System analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the Replica Symmetry (RS) breaking ansatz with $b$ steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the MAP decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued System decouples into a bank of equivalent Decoupled linear Systems followed by MAP estimators. The structure of the Decoupled linear System is further studied under both the RS and the Replica Symmetry Breaking (RSB) assumptions. For $b$ steps of RSB, the Decoupled System is found to be an additive System with a noise term given as the sum of an independent Gaussian random variable with $b$ correlated impairment terms. As an application of our study, we investigate large compressive sensing Systems by considering the $\ell_p$ minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for $\ell_0$ norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the RSB ansatze are needed.
Matthias Morzfeld - One of the best experts on this subject based on the ideXlab platform.
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a canonical form of the equation of motion of linear dynamical Systems
2018Co-Authors: Daniel T Kawano, Rubens Goncalves Salsa, Matthias MorzfeldAbstract:The equation of motion of a discrete linear System has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear Systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective Systems. As an important by-product, a damped linear System that possesses three symmetric and positive definite coefficients can always be recast as an undamped and Decoupled System.
Ali Bereyhi - One of the best experts on this subject based on the ideXlab platform.
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statistical mechanics of map estimation general replica ansatz
2019Co-Authors: Ali Bereyhi, Ralf R Muller, Hermann SchulzbaldesAbstract:The large-System performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear System with additive white Gaussian noise. The analysis considers the System matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-System analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with $b$ steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued System decouples into a bank of equivalent Decoupled scalar Systems followed by maximum-a-posterior estimators. The structure of the Decoupled System is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For $b$ steps of symmetry breaking, the Decoupled System is found to be an additive System with a non-Gaussian noise term given as the sum of an independent Gaussian random variable with $b$ non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition System described by its corresponding Decoupled System. As an application of our study, we investigate large compressive sensing Systems by considering the $\ell _{p}$ norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for $\ell _{0}$ norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansatze are needed in order to assess the performance precisely.
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statistical mechanics of map estimation general replica ansatz
2016Co-Authors: Ali Bereyhi, Ralf R Muller, Hermann SchulzbaldesAbstract:The large-System performance of MAP estimation is studied considering a general distortion function when the observation vector is received through a linear System with additive white Gaussian noise. The analysis considers the System matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-System analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the Replica Symmetry (RS) breaking ansatz with $b$ steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the MAP decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued System decouples into a bank of equivalent Decoupled linear Systems followed by MAP estimators. The structure of the Decoupled linear System is further studied under both the RS and the Replica Symmetry Breaking (RSB) assumptions. For $b$ steps of RSB, the Decoupled System is found to be an additive System with a noise term given as the sum of an independent Gaussian random variable with $b$ correlated impairment terms. As an application of our study, we investigate large compressive sensing Systems by considering the $\ell_p$ minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for $\ell_0$ norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the RSB ansatze are needed.
Frank Richard Prieto Medina - One of the best experts on this subject based on the ideXlab platform.
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polar differentiation matrices for the laplace equation in the disk subjected to nonhomogeneous dirichlet neumann and robin boundary conditions and the biharmonic equation subjected to nonhomogeneous dirichlet conditions
2019Co-Authors: Marcela Molina Meyer, Frank Richard Prieto MedinaAbstract:In this paper we present a pseudospectral method in the disk. Unlike the methods known until now, the disk is not duplicated. Moreover, we solve the Laplace equation subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions by only using the elements of the corresponding differentiation matrices. It is worth noting that we don not use any quadrature, do not need to solve any Decoupled System of ordinary differential equations, do not use any pole condition and do not require any lifting. We solve several numerical examples showing that the spectral convergence is being met. The pseudospectral method developed in this paper can be applied to estimate Sherwood numbers integrating the mass flux to the disk and it can be easily implemented to solve Lotka-Volterra Systems and nonlinear problems involving chemical reactions.
Medina, Frank Richard Prieto - One of the best experts on this subject based on the ideXlab platform.
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Polar differentiation matrices for the Laplace equation in the disk subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions
2019Co-Authors: Meyer, Marcela Molina, Medina, Frank Richard PrietoAbstract:In this paper we present a pseudospectral method in the disk. Unlike the methods known until now, the disk is not duplicated. Moreover, we solve the Laplace equation subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions by only using the elements of the corresponding differentiation matrices. It is worth noting that we don not use any quadrature, do not need to solve any Decoupled System of ordinary differential equations, do not use any pole condition and do not require any lifting. We solve several numerical examples showing that the spectral convergence is being met. The pseudospectral method developed in this paper can be applied to estimate Sherwood numbers integrating the mass flux to the disk and it can be easily implemented to solve Lotka-Volterra Systems and nonlinear problems involving chemical reactions.Comment: 17 figures, 32 page
