# THE CAPABILITY OF LOCALIZED NEURAL NETWORK APPROXIMATION

Hahm, Nahmwoo;Hong, Bum Il

• Accepted : 2013.10.16
• Published : 2013.12.25
• 28 7

#### Abstract

In this paper, we investigate a localized approximation of a continuously differentiable function by neural networks. To do this, we first approximate a continuously differentiable function by B-spline functions and then approximate B-spline functions by neural networks. Our proofs are constructive and we give numerical results to support our theory.

#### Keywords

localized approximation;neural network;B-spline

#### References

1. C. K. Chui, X. Li and H. N. Mhaskar, Limitations of the approximation capabilities of neural networks with one hidden layer, Adv. Comput. Math., 5 (1996), 233-243. https://doi.org/10.1007/BF02124745
2. B. Gao and Y. Xu, Univariant approximation by superpositions of a sigmoidal function, J. Math. Anal. Appl., 178 (1993), 221-226. https://doi.org/10.1006/jmaa.1993.1302
3. N. Hahm and B. I. Hong, Extension of localised approximation by neural networks, Bull. Austral. Math. Soc., 59 (1999), 121-131. https://doi.org/10.1017/S0004972700032676
4. N. Hahm and B. I. Hong, An approximation by neural networks with a fixed weight, Comput. Math. Appl., 47 (2004), 1897-1903. https://doi.org/10.1016/j.camwa.2003.06.008
5. N. Hahm and B. I. Hong, Approximation order to a function in $L_{p}$ space by generalized translation networks, Honam Math. J. 28(1) (2006), 125-133.
6. N. Hahm and B. I. Hong, A simultaneous neural network approximation with the squashing function, Honam Math. J. 31(2) (2009), 147-156. https://doi.org/10.5831/HMJ.2009.31.2.147
7. B. I. Hong and N. Hahm, Approximation order to a function in C(R) by superposition of a sigmoidal function, Appl. Math. Lett., 15 (2002), 591-597. https://doi.org/10.1016/S0893-9659(02)80011-8
8. B. L. Kalman and S. C. Kwasny, Why Tanh : Choosing a sigmoidal function, Int. Joint Conf. on Neural Networks 4 (1992), 578-581.
9. M. Leshno, V. Lin, A. Pinkus and S. Schocken, Multilayered feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks, 6 (1993), 61-80.
10. G. Lewicki and G. Marino, Approximation of functions of finite variation by superpositions of a sigmoidal function, Appl. Math. Lett. 17 (2004), 1147-1152. https://doi.org/10.1016/j.aml.2003.11.006
11. H. N. Mhaskar and N. Hahm, Neural networks for functional approximation and system identification, Neural Comput., 9 (1997), 143-159. https://doi.org/10.1162/neco.1997.9.1.143
12. L. L. Schumaker, Spline Functions : Basic Theory, Cambridge University Press, Cambridge, 2007.

#### Acknowledgement

Supported by : Incheon National University