The Experts below are selected from a list of 279 Experts worldwide ranked by ideXlab platform
Tryphon T. Georgiou - One of the best experts on this subject based on the ideXlab platform.
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An Intrinsic Metric for Power Spectral Density Functions
IEEE Signal Processing Letters, 2007Co-Authors: Tryphon T. GeorgiouAbstract:We present an intrinsic metric that quantifies distances between power Spectral Density functions. The metric was derived by Georgiou as the geodesic distance between Spectral Density functions with respect to a particular pseudo-Riemannian metric motivated by a quadratic prediction problem. We provide an independent verification of the metric inequality and discuss certain key properties of the induced topology.
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Distances and Riemannian Metrics for Spectral Density Functions
IEEE Transactions on Signal Processing, 2007Co-Authors: Tryphon T. GeorgiouAbstract:We introduce a differential-geometric structure for Spectral Density functions of discrete-time random processes. This is quite analogous to the Riemannian structure of information geometry, which is used to study perturbations of probability Density functions, and which is based on the Fisher information metric. Herein, we introduce an analogous Riemannian metric, which we motivate with a problem in prediction theory. It turns out that this problem also provides a prediction theoretic interpretation to the Itakura distortion measure, which relates to our metric. Geodesies and geodesic distances are characterized in closed form and, hence, the geodesic distance between two Spectral Density functions provides an explicit, intrinsic (pseudo)metric on the cone of Density functions. Certain other distortion measures that involve generalized means of Spectral Density functions are shown to lead to the same Riemannian metric. Finally, an alternative Riemannian metric is introduced, which is motivated by an analogous problem involving smoothing instead of prediction.
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What is a natural notion of distance between power Spectral Density functions
2007 European Control Conference (ECC), 2007Co-Authors: Tryphon T. GeorgiouAbstract:We introduce a Riemannian metric on the cone of Spectral Density functions of discrete-time random processes. This is motivated by a problem in prediction theory, and it is analogous to the Fisher information metric on simplices of probability Density functions. Interestingly, in either metric, geodesics and geodesic distances can be characterized in closed form. The goal of this paper is to highlight analogies and differences between the proposed differential-geometric structure of Spectral Density functions and the information geometry of the Fisher metric, and raise the question as to what a natural notion of distance between power Spectral Density functions is.
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Kullback-Leibler approximation of Spectral Density functions
IEEE Transactions on Information Theory, 2003Co-Authors: Tryphon T. Georgiou, Anders LindquistAbstract:We introduce a Kullback-Leibler (1968) -type distance between Spectral Density functions of stationary stochastic processes and solve the problem of optimal approximation of a given Spectral Density /spl Psi/ by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose Spectral Density is sought. In this context, we show (i) that there is a unique Spectral Density /spl Phi/ which minimizes this Kullback-Leibler distance, (ii) that this optimal approximate is of the form /spl Psi//Q where the "correction term" Q is a rational Spectral Density function, and (iii) that the coefficients of Q can be obtained numerically by solving a suitable convex optimization problem. In the special case where /spl Psi/ = 1, the convex functional becomes quadratic and the solution is then specified by linear equations.
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Kullback-Leibler approximation of Spectral Density functions
42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 1Co-Authors: Tryphon T. Georgiou, Anders LindquistAbstract:We introduce a Kullback-Leibler type distance between Spectral Density functions of stationary stochastic processes and solve the problem of optimal approximation of a given Spectral Density /spl Psi/ by one that is consistent with prescribed second-order statistics. In particular, we show (i) that there is a unique Spectral Density /spl Phi/ which minimizes this Kullback-Leibler distance, (ii) that this optimal approximate is of the form /spl Psi//Q where the "correction term" Q is a rational Spectral Density function, and (iii) that the coefficients of Q can be obtained numerically by solving a suitable convex optimization problem. In the special case where /spl Psi/=1, the convex functional becomes quadratic and the solution is then specified by linear equations.
George Kapetanios - One of the best experts on this subject based on the ideXlab platform.
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Estimating the Rank of the Spectral Density Matrix
Journal of Time Series Analysis, 2005Co-Authors: Gonzalo Camba-mendez, George KapetaniosAbstract:The rank of the Spectral Density matrix conveys relevant information in a variety of statistical modelling scenarios. This note shows how to estimate the rank of the Spectral Density matrix at any given frequency. The method presented is valid for any hermitian positive definite matrix estimate that has a normal asymptotic distribution with a covariance matrix the rank of which is known. Copyright 2005 Blackwell Publishing Ltd.
Christian Roumestand - One of the best experts on this subject based on the ideXlab platform.
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Unraveling protein dynamics through fast Spectral Density mapping
Journal of Biomolecular NMR, 2007Co-Authors: Virginie Ropars, Sabine Bouguet-bonnet, Daniel Auguin, Philippe Barthe, Daniel Canet, Christian RoumestandAbstract:Spectral Density mapping at multiple NMR field strengths is probably the best method to describe the dynamical behavior of a protein in solution through the analysis of ^15N heteronuclear relaxation parameters. Nevertheless, such analyses are scarcely reported in the literature, probably because this method is excessively demanding in spectrometer measuring time. Indeed, when using n different magnetic fields and assuming the validity of the high frequency approximation, the discrete sampling of the Spectral Density function with 2 n + 1 points needs the measurement of 3 n ^15N heteronuclear relaxation measurements ( n R _1, n R _2, and n^15N{^1H}NOEs). Based on further approximations, we proposed a new strategy that allows us to describe the Spectral Density with n + 2 points, with the measurement of a total of n + 2 heteronuclear relaxation parameters. Applied to the dynamics analysis of the protein p13^ MTCP 1 at three different NMR fields, this approach allowed us to divide by nearly a factor of two the total measuring time, without altering further results obtained by the “model free” analysis of the resulting Spectral densities. Furthermore, simulations have shown that this strategy remains applicable to any low isotropically tumbling protein ( $$\uptau_{c} > 3$$ ns), and is valid for the types of motion generally envisaged for proteins.
Gonzalo Camba-mendez - One of the best experts on this subject based on the ideXlab platform.
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Estimating the Rank of the Spectral Density Matrix
Journal of Time Series Analysis, 2005Co-Authors: Gonzalo Camba-mendez, George KapetaniosAbstract:The rank of the Spectral Density matrix conveys relevant information in a variety of statistical modelling scenarios. This note shows how to estimate the rank of the Spectral Density matrix at any given frequency. The method presented is valid for any hermitian positive definite matrix estimate that has a normal asymptotic distribution with a covariance matrix the rank of which is known. Copyright 2005 Blackwell Publishing Ltd.
Florent Krzakala - One of the best experts on this subject based on the ideXlab platform.
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Spectral Density of the non backtracking operator on random graphs
EPL, 2014Co-Authors: Alaa Saade, Florent Krzakala, Lenka ZdeborovaAbstract:The non-backtracking operator was recently shown to provide a significant improvement when used for Spectral clustering of sparse networks. In this paper we analyze its Spectral Density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero Spectral Density, that is stable outside a circle of radius , where ρ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero Spectral Density. The fact that this phase transition is absent in the Spectral Density of other matrices commonly used for Spectral clustering provides a physical justification of the performances of the non-backtracking operator in Spectral clustering.
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Spectral Density of the non backtracking operator
arXiv: Disordered Systems and Neural Networks, 2014Co-Authors: Alaa Saade, Florent Krzakala, Lenka ZdeborovaAbstract:The non-backtracking operator was recently shown to provide a significant improvement when used for Spectral clustering of sparse networks. In this paper we analyze its Spectral Density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero Spectral Density, that is stable outside a circle of radius $\sqrt{\rho}$, where $\rho$ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero Spectral Density. That fact that this phase transition is absent in the Spectral Density of other matrices commonly used for Spectral clustering provides a physical justification of the performances of the non-backtracking operator in Spectral clustering.
