Convergence Analysis of Ensemble Kalman Inversion: The Linear, Noisy Case

We present an analysis of ensemble Kalman inversion, based on the continuous time limit of the algorithm. The analysis of the dynamical behaviour of the ensemble allows us to establish well-posedness and convergence results for a fixed ensemble size. We will build on the results presented in [26] and generalise them to the case of noisy observational data, in particular the influence of the noise on the convergence will be investigated, both theoretically and numerically. We focus on linear inverse problems where a very complete theoretical analysis is possible

A pre-filtering maximum likelihood approach to multiple source direction estimation.

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Optimal quadrature-sparsification for integral operator approximation

The design of sparse quadratures for the approximation of integral operators related to symmetric positive-semidefinite kernels is addressed. Particular emphasis is placed on the approximation of the main eigenpairs of an initial operator and on the assessment of the approximation accuracy. Special attention is drawn to the design of sparse quadratures with support included in fixed finite sets of points (that is, quadrature-sparsification), this framework encompassing the approximation of kernel matrices. For a given kernel, the accuracy of a quadrature approximation is assessed through the squared Hilbert--Schmidt norm (for operators acting on the underlying reproducing kernel Hilbert space) of the difference between the integral operators related to the initial and approximate measures; by analogy with the notion of kernel discrepancy, the underlying criterion is referred to as the squared-kernel discrepancy between the two measures. In the quadrature-sparsification framework, sparsity of the approximate quadrature is promoted through the introduction of an $\ell^{1}$-type penalization, and the computation of a penalized squared-kernel-discrepancy-optimal approximation then consists in a convex quadratic minimization problem; such quadratic programs can in particular be interpreted as the Lagrange dual formulations of distorted one-class support-vector machines related to the squared kernel. Error bounds on the induced spectral approximations are derived, and the connection between penalization, sparsity, and accuracy of the spectral approximation is investigated. Numerical strategies for solving large-scale penalized squared-kernel-discrepancy minimization problems are discussed, and the efficiency of the approach is illustrated by a series of examples. In particular, the ability of the proposed methodology to lead to accurate approximations of the main eigenpairs of kernel matrices related to large-scale datasets is demonstrated

Local, multi-resolution detection of network communities by Markovian dynamics

Complex networks are used to represent systems from many disciplines, including biology, physics, medicine, engineering and the social sciences; Many real-world networks are organised into densely connected communi- ties, whose composition gives some insight into the underlying network. Most approaches for nding such communities do so by partitioning the network into disjoint subsets, at the cost of requiring global information and that nodes belong to exactly one community. In recent years, some effort has been devoted towards the development of local methods, but these are either limited in resolution or ignore relevant network features such as directedness. Here we show that introducing a dynamic process onto the network allows us to de ne a community quality function severability which is inherently multi-resolution, takes into account edge-weight and direction, can accommodate overlapping communities and orphan nodes and crucially does not require global knowledge. Both constructive and real-world examples| drawn from elds as diverse as image segmentation, metabolic networks and word association|are used to illustrate the characteristics of this approach. We envision this approach as a starting point for the future analysis of both evolving networks and networks too large to be readily analysed as a whole (e.g. the World Wide Web).Open Acces

Space time transceiver design over multipath fading channels

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