H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural networks for signal processing III, Proceedings of the 1993 IEEE-SP Workshop, pages 190--196, New York, 1993a. IEEE Signal Processing Society.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....b i (x) are bell shaped, local functions, whose locality will depend on the choice of the kernel K, on the density of data points, and on the regularization parameter #. This shows that apparently global approximation schemes can be regarded as local, memory based techniques (see equation 3. 29) Mhaskar, 1993a ] 3.2.2 From regression to classification In the particular case that the unknown function takes only two values, i.e. 1 and 1, we have the problem of binary pattern classification, i.e. the case where we are 3 Notice that this duality is di#erent from the one mentioned at the end of ....

H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural networks for signal processing III, Proceedings of the 1993 IEEE-SP Workshop, pages 190--196, New York, 1993a. IEEE Signal Processing Society.

....matter) Again, as in the case of standard SRM, in practice l is finite so H = S n(l,#) i=1 H i is a small space and the solution of this method may have expected risk much larger that the expected risk of the target function. Approximation theory can be used to bound this di#erence [59]. The proposed method is di#cult to implement in practice since it is di#cult to decide the optimal trade o# between empirical error and the bound (21) If we had constructive bounds on the deviation between the empirical and the expected risk like that of theorem 2.1 then we could have a ....

H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural networks for signal processing III, Proceedings of the 1993 IEEE-SP Workshop, pages 190--196, New York, 1993a. IEEE Signal Processing Society.

....b i (x) are bell shaped, local functions, whose locality will depend on the choice of the kernel K, on the density of data points, and on the regularization parameter #. This shows that apparently global approximation schemes can be regarded as local, memory based techniques (see equation 4. 18) [59]. 4.4. From regression to classification So far we only considered the case that the unknown function can take any real values, specifically the case of regression. In the particular case that the unknown function takes only two values, i.e. 1 and 1, we have the problem of binary pattern ....

H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural networks for signal processing III, Proceedings of the 1993 IEEE-SP Workshop, pages 190--196, New York, 1993a. IEEE Signal Processing So- 52 T. Evgeniou et al / Regularization Networks and Support Vector Machines ciety.

....been used by many people for classification (e.g. 20] Recently, they have also become popular for approximation. Farmer and Sidorowich [8] have used local polynomial models for forecasting but have only had success using low order models. 4. Non linear models. One example is splines. Mhaskar [21] has used tensor product b splines for local approximation. Standard results in spline approximation theory can be used. Any non linear model is a candidate for local approximation models and hence we may even use multi layer perceptrons. 4 Simulations In order to investigate the relation between ....

H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural Networks for Signal Processing III: Proceedings of the 1993 IEEE Workshop. IEEE, 1993.

....have to maintain, not in the network architecture itself. Moreover, the network in (4.2) does not require any training at all. In a computer simulation, we may choose m to be a power of two, and the point y can be computed easily by just truncating the binary representation of x. It is argued in [26] that this is a reasonable way to get around the curse of dimensionality when the only knowledge about f is of the form f # W p r,s . If # is the Heaviside function, then it is easy to see that # s (x) # # s # j=1 (#(x j ) #(1 x j ) 2s 1 2 # . 4.3) Thus, the function # s ....

H. N. Mhaskar, Neural networks for localized approximation of real functions, in "Neural Networks for Signal Processing, III", (Kamm, Huhn, Yoon, Chellappa and Kung Eds.), IEEE New York, 1993, pp. 190-196.

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H.N. Mhaskar. Neural networks for localized approximation of real functions. In C.A. Kamm et al., editor, Neural networks for signal processing III, Proceedings of the 1993 IEEE-SP Workshop, pages 190--196, New York, 1993a. IEEE Signal Processing Society.

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