scholarly journals Integration with respect to a vector measure and function approximation

2000 ◽  
Vol 5 (4) ◽  
pp. 207-226 ◽  
Author(s):  
L. M. García-Raffi ◽  
D. Ginestar ◽  
E. A. Sánchez-Pérez
Keyword(s):  
Hilbert Space ◽  
Function Approximation ◽  
Vector Measure ◽  
Finite Family ◽  
Orthogonal Sequence ◽  
Measurable Set ◽  
Lebesgue Measurable Set ◽  
And Function

The integration with respect to a vector measure may be applied in order to approximate a function in a Hilbert space by means of a finite orthogonal sequence{fi}attending to two different error criterions. In particular, ifΩ∈ℝis a Lebesgue measurable set,f∈L2(Ω), and{Ai}is a finite family of disjoint subsets ofΩ, we can obtain a measureμ0and an approximationf0satisfying the following conditions: (1)f0is the projection of the functionfin the subspace generated by{fi}in the Hilbert spacef∈L2(Ω,μ0). (2) The integral distance betweenfandf0on the sets{Ai}is small.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

PRUNING AND MODEL-SELECTING ALGORITHMS IN THE RBF FRAMEWORKS CONSTRUCTED BY SUPPORT VECTOR LEARNING

2006 ◽  
Vol 16 (04) ◽  
pp. 283-293 ◽  
Author(s):  
PEI-YI HAO ◽  
JUNG-HSIEN CHIANG
Keyword(s):  
Function Approximation ◽  
Synthetic Data ◽  
Support Vector ◽  
Detection Problem ◽  
Data Simulation ◽  
Penalty Term ◽  
Rbf Network ◽  
Sample Classification ◽  
Pruning Algorithms ◽  
And Function

This paper presents the pruning and model-selecting algorithms to the support vector learning for sample classification and function regression. When constructing RBF network by support vector learning we occasionally obtain redundant support vectors which do not significantly affect the final classification and function approximation results. The pruning algorithms primarily based on the sensitivity measure and the penalty term. The kernel function parameters and the position of each support vector are updated in order to have minimal increase in error, and this makes the structure of SVM network more flexible. We illustrate this approach with synthetic data simulation and face detection problem in order to demonstrate the pruning effectiveness.

Decorrelated Hebbian Learning for Clustering and Function Approximation

1995 ◽  
Vol 7 (2) ◽  
pp. 338-348 ◽  
Author(s):  
G. Deco ◽  
D. Obradovic
Keyword(s):  
Cost Function ◽  
Function Approximation ◽  
Hebbian Learning ◽  
Clustering Problem ◽  
New Learning ◽  
Hidden Layer ◽  
And Function ◽  
The Cost ◽  
Winner Take All ◽  
Gaussian Network

This paper presents a new learning paradigm that consists of a Hebbian and anti-Hebbian learning. A layer of radial basis functions is adapted in an unsupervised fashion by minimizing a two-element cost function. The first element maximizes the output of each gaussian neuron and it can be seen as an implementation of the traditional Hebbian learning law. The second element of the cost function reinforces the competitive learning by penalizing the correlation between the nodes. Consequently, the second term has an “anti-Hebbian” effect that is learned by the gaussian neurons without the implementation of lateral inhibition synapses. Therefore, the decorrelated Hebbian learning (DHL) performs clustering in the input space avoiding the “nonbiological” winner-take-all rule. In addition to the standard clustering problem, this paper also presents an application of the DHL in function approximation. A scaled piece-wise linear approximation of a function is obtained in the supervised fashion within the local regions of its domain determined by the DHL. For comparison, a standard single hidden-layer gaussian network is optimized with the initial centers corresponding to the DHL. The efficiency of the algorithm is demonstrated on the chaotic Mackey-Glass time series.

scholarly journals Pointwise density topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Magdalena Górajska
Keyword(s):  
Real Line ◽  
Pointwise Convergence ◽  
Baire Property ◽  
Density Topology ◽  
Borel Set ◽  
Convergence In Measure ◽  
The Real ◽  
Measurable Set ◽  
New Type ◽  
Lebesgue Measurable Set

AbstractThe paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

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