102,495 research outputs found
A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization
No full textFixed-basis and variable-basis approximation schemes are compared for the problems of function approximation and functional optimization (also known as infinite programming). Classes of problems are investigated for which variable-basis schemes with sigmoidal computational
units perform better than fixed-basis ones, in terms of the minimum number of computational units needed to achieve a desired error in function approximation or approximate optimization. Previously known bounds on the accuracy are extended, with better rates, to families o
Error bounds for suboptimal solutions to kernel principal component analysis
No full textSuboptimal 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
Estimates of the Approximation Error Using Rademacher Complexity: Learning Vector-Valued Functions
No full textFor certain families of multivariable vector-valued functions to be approximated, the accuracy of approximation schemes made up of linear combinations of computational units containing adjustable parameters is investigated. Upper bounds on the approximation error are derived that depend on the Rademacher complexities of the families. The estimates exploit possible relationships among the components of the multivariable vector-valued functions. All such components are approximated simultaneously in such a way to use, for a desired approximation accuracy, less computational units than those required by componentwise approximation. An application to -stage optimization problems is discussed
Functional optimization by variable-basis approximation schemes
No full textThis is a summary of the author’s PhD thesis, supervised by Marcello Sanguineti and defended on April 2, 2009 at Università degli Studi di Genova. The thesis is written in English and a copy is available from the author upon request. Functional optimization problems arising in Operations Research are investigated. In such problems, a cost functional Φ has to be minimized over an admissible set S of d-variable functions. As, in general, closed-form solutions cannot be derived, suboptimal solutions are searched for, having the form of variable-basis functions, i.e., elements of the set span n G of linear combinations of at most n elements from a set G of computational units. Upper bounds on inff∈S∩spannGΦ(f)−inff∈SΦ(f) are obtained. Conditions are derived, under which the estimates do not exhibit the so-called “curse of dimensionality” in the number n of computational units, when the number d of variables grows. The problems considered include dynamic optimization, team optimization, and supervised learning from data
Structural properties of optimal coordinate-convex policies for CAC with nonlinearly-constrained feasibility regions
No full textNecessary optimality conditions for Call Admission Control (CAC) problems with nonlinearly-constrained feasibility regions and two classes of users are derived. The policies are restricted to the class of coordinate-convex policies. Two kinds of structural properties of the optimal policies and their robustness with respect to changes of the feasibility region are investigated: 1) general properties not depending on the revenue ratio associated with the two classes of users and 2) more specific properties depending on such a ratio. The results allow one to narrow the search for the optimal policies to a suitable subset of the set of coordinate-convex policies
Metamaterial filter design via surrogate optimization
No full textRecently, an increasing research effort has been dedicated to analyse transmission and dispersion properties of periodic metamaterials containing resonators, and to optimize the amplitude of selected acoustic band gaps between consecutive dispersion curves in the Floquet-Bloch spectrum. Potential novel applications of this research are in the design of passive mechanical filters/diodes. The present work proposes a way to interpolate the objective functions in such band gap optimization problems, using Radial Basis Functions. The study is motivated by the high computational effort often needed for an exact evaluation of the original objective functions, when using iterative optimization algorithms. By replacing such functions with surrogate objective functions, well-performing suboptimal solutions can be obtained with a small computational effort. Numerical results demonstrate the feasibility of the approach
Multi-field asymptotic homogenization approach for Bloch wave propagation in periodic thermodiffusive elastic materials
No full textMulti-field asymptotic homogenization methods are proposed to describe the behaviour of periodic Cauchy materials subject to several physical phenomena, focusing on thermodiffusion. The resulting homogenized models provide the overall constitutive tensors and overall inertial terms. Moreover, they allow one to investigate the complex band structures associated with damped Bloch waves travelling in periodic materials, avoiding the challenging computations needed by the adoption of micromechanical approaches
Minimizing Sequences for a Family of Functional Optimal Estimation Problems
No full textRates of convergence are derived for approximate solutions to optimization problems associated with the design of state estimators for nonlinear dynamic systems. Such problems consist in minimizing the functional given by the worst-case ratio between the ℒ p -norm of the estimation error and the sum of the ℒ p -norms of the disturbances acting on the dynamic system. The state estimator depends on an innovation function, which is searched for as a minimizer of the functional over a subset of a suitably-defined functional space. In general, no closed-form solutions are available for these optimization problems. Following the approach proposed in (Optim. Theory Appl. 134:445–466, 2007), suboptimal solutions are searched for over linear combinations of basis functions containing some parameters to be optimized. The accuracies of such suboptimal solutions are estimated in terms of the number of basis functions. The estimates hold for families of approximators used in applications, such as splines of suitable orders
Approximation and Estimation Bounds for Subsets of Reproducing Kernel Kreǐn Spaces
No full textReproducing kernel Kreın spaces are used in learning from data via kernel methods when the kernel is indefinite. In this paper, a characterization of a subset of the unit ball in
such spaces is provided. Conditions are given, under which upper bounds on the estimation error and the approximation error can be applied simultaneously to such a subset. Finally, it is shown that the hyperbolic-tangent kernel and other indefinite kernels satisfy such conditions
Deeper insights into neural nets with random weights
No full textIn this work, the “effective dimension” of the output of the hidden layer of a one-hidden-layer neural network with random inner weights of its computational units is investigated. To do this, a polynomial approximation of the sigmoidal activation function of each computational unit is used, whose degree is chosen based both on a desired upper bound on the approximation error and on an estimate of the range of the input to that computational unit. This estimate of the range is parameterized by the number of inputs to the network and by an upper bound both on the size of the random inner weights of the network and on the size of its inputs. The results show that the Root Mean Square Error (RMSE) on the training set is influenced by the effective dimension and by the quality of the features associated with the output of the hidden layer
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