Accession Number : ADA260100


Title :   Convergence Rates of Approximation by Translates


Descriptive Note : Memorandum rept.


Corporate Author : MASSACHUSETTS INST OF TECH CAMBRIDGE ARTIFICIAL INTELLIGENCE LAB


Personal Author(s) : Girosi, Federico ; Anzellotti, Gabriele


Full Text : https://apps.dtic.mil/dtic/tr/fulltext/u2/a260100.pdf


Report Date : Feb 1992


Pagination or Media Count : 18


Abstract : In this paper, the authors consider the problem of approximating a function belonging to some function space PHI by a linear combination of n translates of a given function G. Using a lemma by Jones (1990) and Barron (1991), they show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the error is O(1 over the square root of n) in any number of dimensions. The apparent avoidance of the curse of dimensionality is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev type, in which the number of weak derivatives is required to be larger than the number of dimensions. They give results both for approximation in the L(sub 2) norm and in the L(sub infinity) norm. The interesting feature of these results is that, thanks to the constructive nature of Jones' and Barron's lemma, an iterative procedure is defined that can achieve this rate.


Descriptors :   *APPROXIMATION(MATHEMATICS) , *CONVERGENCE , *FUNCTIONS(MATHEMATICS) , *FUNCTIONAL ANALYSIS , FOURIER TRANSFORMATION , BESSEL FUNCTIONS , DERIVATIVES(MATHEMATICS) , FOURIER ANALYSIS , KERNEL FUNCTIONS , ARTIFICIAL INTELLIGENCE , ITERATIONS


Subject Categories : Numerical Mathematics
      Cybernetics


Distribution Statement : APPROVED FOR PUBLIC RELEASE