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1.
Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation. 相似文献
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Meccanica - Traditional winglets are designed as fixed devices attached at the tips of the wings. The primary purpose of the winglets is to reduce the lift-induced drag, therefore improving... 相似文献
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A variational norm that plays a role in functional optimization and learning from data is investigated. For sets of functions
obtained by varying some parameters in fixed-structure computational units (e.g., Gaussians with variable centers and widths),
upper bounds on the variational norms associated with such units are derived. The results are applied to functional optimization
problems arising in nonlinear approximation by variable-basis functions and in learning from data. They are also applied to
the construction of minimizing sequences by an extension of the Ritz method. 相似文献
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For Principal Component Analysis in Reproducing Kernel Hilbert Spaces (KPCA), optimization over sets containing only linear
combinations of all n-tuples of kernel functions is investigated, where n is a positive integer smaller than the number of data. Upper bounds on the accuracy in approximating the optimal solution,
achievable without restrictions on the number of kernel functions, are derived. The rates of decrease of the upper bounds
for increasing number n of kernel functions are given by the summation of two terms, one proportional to n
−1/2 and the other to n
−1, and depend on the maximum eigenvalue of the Gram matrix of the kernel with respect to the data. Primal and dual formulations
of KPCA are considered. The estimates provide insights into the effectiveness of sparse KPCA techniques, aimed at reducing
the computational costs of expansions in terms of kernel units. 相似文献
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Mauro Gaggero Giorgio Gnecco Marcello Sanguineti 《Computational Optimization and Applications》2014,58(1):31-85
Stochastic optimization problems with an objective function that is additive over a finite number of stages are addressed. Although Dynamic Programming allows one to formally solve such problems, closed-form solutions can be derived only in particular cases. The search for suboptimal solutions via two approaches is addressed: approximation of the value functions and approximation of the optimal decision policies. The approximations take on the form of linear combinations of basis functions containing adjustable parameters to be optimized together with the coefficients of the combinations. Two kinds of basis functions are considered: Gaussians with varying centers and widths and sigmoids with varying weights and biases. The accuracies of such suboptimal solutions are investigated via estimates of the error propagation through the stages. Upper bounds are derived on the differences between the optimal value of the objective functional and its suboptimal values corresponding to the use at each stage of approximate value functions and approximate policies. Conditions under which the number of basis functions required for a desired approximation accuracy does not grow “too fast” with respect to the dimensions of the state and random vectors are provided. As an example of application, a multidimensional problem of optimal consumption under uncertainty is investigated, where consumers aim at maximizing a social utility function. Numerical simulations are provided, emphasizing computational pros and cons of the two approaches (i.e., value-function approximation and optimal-policy approximation) using the above-mentioned two kinds of basis functions. To investigate the dependencies of the performances on dimensionality, the numerical analysis is performed for various numbers of consumers. In the simulations, discretization techniques exploiting low-discrepancy sequences are used. Both theoretical and numerical results give insights into the possibility of coping with the curse of dimensionality in stochastic optimization problems whose decision strategies depend on large numbers of variables. 相似文献
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Complexity of Gaussian-radial-basis-function networks, with varying widths, is investigated. Upper bounds on rates of decrease of approximation errors with increasing number of hidden units are derived. Bounds are in terms of norms measuring smoothness (Bessel and Sobolev norms) multiplied by explicitly given functions a(r,d) of the number of variables d and degree of smoothness r. Estimates are proven using suitable integral representations in the form of networks with continua of hidden units computing scaled Gaussians and translated Bessel potentials. Consequences on tractability of approximation by Gaussian-radial-basis function networks are discussed. 相似文献
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Sanguineti A Monguzzi A Vaccaro G Meinardi F Ronchi E Moret M Cosentino U Moro G Simonutti R Mauri M Tubino R Beverina L 《Physical chemistry chemical physics : PCCP》2012,14(18):6452-6455
A new oxyiminopyrazole-based ytterbium chelate enables NIR emission upon UV excitation in colorless single layer luminescent solar concentrators for building integrated photovoltaics. 相似文献
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Suboptimal solutions to kernel principal component analysis are considered. Such solutions take on the form of linear combinations of all n-tuples of kernel functions centered on the data, where n is a positive integer smaller than the cardinality m of the data sample. Their accuracy in approximating the optimal solution, obtained in general for n = m, is estimated. The analysis made in Gnecco and Sanguineti (Comput Optim Appl 42:265–287, 2009) is extended. The estimates derived therein for the approximation of the first principal axis are improved and extensions to the successive principal axes are derived. 相似文献
10.
The approximation of the optimal policy functions is investigated for dynamic optimization problems with an objective that
is additive over a finite number of stages. The distance between optimal and suboptimal values of the objective functional
is estimated, in terms of the errors in approximating the optimal policy functions at the various stages. Smoothness properties
are derived for such functions and exploited to choose the approximating families. The approximation error is measured in
the supremum norm, in such a way to control the error propagation from stage to stage. Nonlinear approximators corresponding
to Gaussian radial-basis-function networks with adjustable centers and widths are considered. Conditions are defined, guaranteeing
that the number of Gaussians (hence, the number of parameters to be adjusted) does not grow “too fast” with the dimension
of the state vector. The results help to mitigate the curse of dimensionality in dynamic optimization. An example of application
is given and the use of the estimates is illustrated via a numerical simulation. 相似文献