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Jun Kawabe - One of the best experts on this subject based on the ideXlab platform.
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MDAI - Convergence in Measure Theorems of the Choquet Integral Revisited.
Modeling Decisions for Artificial Intelligence, 2019Co-Authors: Jun KawabeAbstract:The validity of the Monotone Convergence Theorem, the Fatou and the reverse Fatou lemmas, and the dominated Convergence Theorem of the Choquet integral of measurable functions converging in measure are fully characterized by the conditional versions of the Monotone autocontinuity and the autocontinuity. In those Theorems the nonadditive measure may be infinite and the functions may be unbounded. The dual measure forms and the extension to symmetric and asymmetric Choquet integrals are also discussed.
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A Unified Approach to Convergence Theorems of Nonlinear Integrals
Advances in Mathematical Economics, 2018Co-Authors: Jun KawabeAbstract:There are several types of nonlinear integrals with respect to nonadditive measures, such as the Choquet, Sipos, Sugeno, and Shilkret integrals. In order to put those integrals into practical use and aim for application to various fields, it is indispensable to establish Convergence Theorems of such nonlinear integrals. However, they have individually been discussed for each of the integrals up to the present. In this article, several important Convergence Theorems of nonlinear integrals, such as the Monotone Convergence Theorem, the bounded Convergence Theorem, and the Vitali Convergence Theorem, are formulated in a unified way regardless of the types of integrals.
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A unified approach to the Monotone Convergence Theorem for nonlinear integrals
Fuzzy Sets and Systems, 2016Co-Authors: Jun KawabeAbstract:Abstract We present a unified approach to the Monotone Convergence Theorem for nonlinear integrals such as the Choquet, the Sipos, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of Monotone Convergence Theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ -additive.
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The Choquet integral in Riesz space
Fuzzy Sets and Systems, 2008Co-Authors: Jun KawabeAbstract:A comprehensive discussion of the theory of Choquet integration in a Riesz space is given. In particular, it is proved that the Monotone Convergence Theorem, the Fatou lemma, and the dominated Convergence Theorem are still valid for Riesz space-valued non-additive measures if we assume that the Riesz space has a new property concerning the cardinality of the set of points of discontinuity of a Monotone function.
Yuemin Zhu - One of the best experts on this subject based on the ideXlab platform.
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Generic Half-Quadratic Optimization for Image Reconstruction
SIAM Journal on Imaging Sciences, 2015Co-Authors: Marc C. Robini, Yuemin ZhuAbstract:We study the global and local Convergence of a generic half-quadratic optimization algorithm inspired from the dual energy formulation of Geman and Reynolds [IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 367--383]. The target application is the minimization of $C^{1}$ convex and nonconvex objective functionals arising in regularized image reconstruction. Our global Convergence proofs are based on a Monotone Convergence Theorem of Meyer [J. Comput. System Sci., 12 (1976), pp. 108--121]. Compared to existing results, ours extend to a larger class of objectives and apply under weaker conditions; in particular, we cover the case where the set of stationary points is not discrete. Our local Convergence results use a majorization-minimization interpretation to derive an insightful characterization of the basins of attraction; this new perspective grounds a formal description of the intuitive water-flooding analogy. We conclude with image restoration experiments to illustrate the efficiency of the algorithm under various nonconvex scenarios.Read More: http://epubs.siam.org/doi/abs/10.1137/140987845
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Generic Half-Quadratic Optimization for Image Reconstruction
SIAM Journal on Imaging Sciences, 2015Co-Authors: Marc C. Robini, Yuemin ZhuAbstract:We study the global and local Convergence of a generic half-quadratic optimization algorithm inspired from the dual energy formulation of Geman and Reynolds [IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 367--383]. The target application is the minimization of $C^{1}$ convex and nonconvex objective functionals arising in regularized image reconstruction. Our global Convergence proofs are based on a Monotone Convergence Theorem of Meyer [J. Comput. System Sci., 12 (1976), pp. 108--121]. Compared to existing results, ours extend to a larger class of objectives and apply under weaker conditions; in particular, we cover the case where the set of stationary points is not discrete. Our local Convergence results use a majorization-minimization interpretation to derive an insightful characterization of the basins of attraction; this new perspective grounds a formal description of the intuitive water-flooding analogy. We conclude with image restoration experiments to illustrate the efficiency of the algo...
Xuekun Ren - One of the best experts on this subject based on the ideXlab platform.
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A note on the sendograph metric of fuzzy numbers
Information Sciences, 2009Co-Authors: Xuekun RenAbstract:In this paper, by studying the attainable properties of sendograph metric in fuzzy number space, we generalize Monotone Convergence Theorem and Nested Theorem of intervals from the space of real numbers to the space of fuzzy numbers.
Marc C. Robini - One of the best experts on this subject based on the ideXlab platform.
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Generic Half-Quadratic Optimization for Image Reconstruction
SIAM Journal on Imaging Sciences, 2015Co-Authors: Marc C. Robini, Yuemin ZhuAbstract:We study the global and local Convergence of a generic half-quadratic optimization algorithm inspired from the dual energy formulation of Geman and Reynolds [IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 367--383]. The target application is the minimization of $C^{1}$ convex and nonconvex objective functionals arising in regularized image reconstruction. Our global Convergence proofs are based on a Monotone Convergence Theorem of Meyer [J. Comput. System Sci., 12 (1976), pp. 108--121]. Compared to existing results, ours extend to a larger class of objectives and apply under weaker conditions; in particular, we cover the case where the set of stationary points is not discrete. Our local Convergence results use a majorization-minimization interpretation to derive an insightful characterization of the basins of attraction; this new perspective grounds a formal description of the intuitive water-flooding analogy. We conclude with image restoration experiments to illustrate the efficiency of the algorithm under various nonconvex scenarios.Read More: http://epubs.siam.org/doi/abs/10.1137/140987845
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Generic Half-Quadratic Optimization for Image Reconstruction
SIAM Journal on Imaging Sciences, 2015Co-Authors: Marc C. Robini, Yuemin ZhuAbstract:We study the global and local Convergence of a generic half-quadratic optimization algorithm inspired from the dual energy formulation of Geman and Reynolds [IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 367--383]. The target application is the minimization of $C^{1}$ convex and nonconvex objective functionals arising in regularized image reconstruction. Our global Convergence proofs are based on a Monotone Convergence Theorem of Meyer [J. Comput. System Sci., 12 (1976), pp. 108--121]. Compared to existing results, ours extend to a larger class of objectives and apply under weaker conditions; in particular, we cover the case where the set of stationary points is not discrete. Our local Convergence results use a majorization-minimization interpretation to derive an insightful characterization of the basins of attraction; this new perspective grounds a formal description of the intuitive water-flooding analogy. We conclude with image restoration experiments to illustrate the efficiency of the algo...
Horst Osswald - One of the best experts on this subject based on the ideXlab platform.
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Nonstandard Integration Theory in Topological
1997Co-Authors: Vector Lattices, Peter A. Loeb, Horst OsswaldAbstract:This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functional I on an internal lattice of vector valued functions, we produce a nonstandard hull valued integral J satisfying the Monotone Convergence Theorem. Nonstandard hulls form a natural extension of infinite dimensional spaces and are equivalent to Banach space ultrapower constructions. The first application of our integral is a construction of Banach limits for bounded, vector valued sequences. The second example yields an integral representation for bounded and quasibounded harmonic functions similar to that of the Martin boundary. The third application uses our general integral to extend the Bochner integral.
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Nonstandard Integration Theory in Topological Vector Lattices.
Monatshefte f�r Mathematik, 1997Co-Authors: Peter A. Loeb, Horst OsswaldAbstract:This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functionalI on an internal lattice of vector valued functions, we produce a nonstandard hull valued integralJ satisfying the Monotone Convergence Theorem. Nonstandard hulls form a natural extension of infinite dimensional spaces and are equivalent to Banach space ultrapower constructions. The first application of our integral is a construction of Banach limits for bounded, vector valued sequences. The second example yields an integral representation for bounded and quasibounded harmonic functions similar to that of the Martin boundary. The third application uses our general integral to extend the Bochner integral.