Approximation Of Real-valued Functions Research Articles

Metric Approximation of Set-Valued Functions of Bounded Variation by Integral Operators

AbstractWe introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of $${\mathbb {R}}^d$$ R d . All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $$L^1$$ L 1 provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.

Criteria for the Best Approximation by Simple Partial Fractions on Semi-Axis and Axis

We study uniform approximation of real-valued functions f, f(∞) = 0, on ℝ+ and ℝ by real-valued simple partial fractions (the logarithmic derivatives of polynomials). We obtain a criterion for the best approximation on ℝ+ and ℝ in terms of the Chebyshev alternance. This criterion is similar to the known criterion on finite segments. For the problem of approximating odd functions on ℝ we construct an alternance criterion with a weakened condition on the poles of fractions. We present a criterion for the best approximation by simple partial fractions on ℝ+ and ℝ in terms of Kolmogorov. We prove analogs of the de la Vallee-Poussin alternation theorem.

Local adaption for approximation and minimization of univariate functions

Most commonly used adaptive algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a user-specified absolute error tolerance for a cone, C, of non-spiky input functions in the Sobolev space W2,∞[a,b]. Our algorithms automatically determine where to sample the function—sampling more densely where the second derivative is larger. The computational cost of our algorithm for approximating a univariate function f on a bounded interval with L∞-error no greater than ε is O(‖f″‖12/ε) as ε→0. This is the same order as that of the best function approximation algorithm for functions in C. The computational cost of our global minimization algorithm is of the same order and the cost can be substantially less if f significantly exceeds its minimum over much of the domain. Our Guaranteed Automatic Integration Library (GAIL) contains these new algorithms. We provide numerical experiments to illustrate their superior performance.

Resource management and scalability of the XCSF learning classifier system

It has been shown many times that the evolutionary online learning XCS classifier system is a robustly generalizing reinforcement learning system, which also yields highly competitive results in data mining applications. The XCSF version of the system is a real-valued function approximation system, which learns piecewise overlapping local linear models to approximate an iteratively sampled function. While the theory on the binary domain side goes as far as showing that XCS can PAC learn a slightly restricted set of k-DNF problems, theory for XCSF is still rather sparse. This paper takes the theory from the XCS side and projects it onto the real-valued XCSF domain. For a set of functions, in which fitness guidance is given, we even show that XCSF scales optimally with respect to the population size, requiring only a constant overhead to ensure that the evolutionary process can locally optimize the evolving structures. Thus, we provide foundations concerning scalability and resource management for XCSF. Furthermore, we reveal dimensions of problem difficulty for XCSF — and local linear learners in general — showing how structural alignment, that is, alignment of XCSF’s solution representation to the problem structure, can reduce the complexity of challenging problems by orders of magnitude.

Representations and rates of approximation of real-valued Boolean functions by neural networks

We give upper bounds on rates of approximation of real-valued functions of d Boolean variables by one-hidden-layer perceptron networks. Our bounds are of the form c/ n where c depends on certain norms of the function being approximated and n is the number of hidden units. We describe sets of functions where these norms grow either polynomially or exponentially with d.

A Korovkin type theorem for weighted spaces of continuous functions

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

A geometric interpretation of hidden layer units in feedforward neural networks

Feedforward neural networks with several input units, a single output unit and a single hidden layer of sigmoid transfer functions are considered. A method is presented of interpreting the role of a hidden layer unit geometrically, when using such networks for real-valued function approximation problems, in terms of regions of the input space over which it is most important in a particular sense. The relationship between this interpretation and the weight values of the network is highlighted. Then, for the case in which the approximation is of most interest over a bounded region of the input space, it is shown that this interpretation may then be used to check for redundancy among hidden units, and to remove any such units found. Finally, future research issues for which the interpretation may be useful are briefly discussed.

Unicity in the uniform approximation of vector-valued functions

The problem of unicity in the uniform approximation of vector-valued functions is considered. A recent result of Cheney and Wulbert concerning unicity in the uniform approximation of real-valued functions is extended and some “point-wise” criteria for unicity are given.