\(L^p\) approximation capability of RBF neural networks. (English) Zbl 1169.68578
Summary: \(L^p\) approximation capability of Radial Basis Function (RBF) neural networks is investigated. If \(g: R_{+}^{1} \to R^{1}\) and \(g(\| x\| _{R^n}) \in L _{\text{loc}}^p (R^n)\) with \(1 \leq p < \infty \), then the RBF neural networks with \(g\) as the activation function can approximate any given function in \(L^p(K)\) with any accuracy for any compact set \(K\) in \(R^n\), if and only if \(g(x)\) is not an even polynomial.
MSC:
| 68T05 | Learning and adaptive systems in artificial intelligence |
| 41A20 | Approximation by rational functions |
References:
| [1] | Mhaskar, H. N., Micchelli, A.: Approximation by superposition of sigmoidal and radial basis function. Advances in Applied Mathematics, 13, 350–370 (1992) · Zbl 0763.41015 · doi:10.1016/0196-8858(92)90016-P |
| [2] | Leshno, M., Lin, Y. V., Pinkus, A., Schocen, S.: Multilayer feedforward networks with a non-polynomial activation function can approximate any function. Neural Networks, 6, 861–867 (1993) · doi:10.1016/S0893-6080(05)80131-5 |
| [3] | Chen, T.: Approximate problem on neural networks and its application in system recognise. Science in China, Series A (in Chinese), 24(1), 1–7 (1994) |
| [4] | Chen, T., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans. Neural Networks, 6(4), 918–928 (1995) · doi:10.1109/72.392254 |
| [5] | Chen, T., Chen, H., Liu, R.: Approximation capability in by multilayer feedforward networks and related problems. IEEE Trans. Neural Networks, 6(1), 25–30 (1995) · doi:10.1109/72.363453 |
| [6] | Li, X.: Note on constructing near tight wavelet frames by neural networks. SPIE, 2825, 152–162 (1996) · doi:10.1117/12.255226 |
| [7] | Li, X.: On Simultaneous approximations by radial basis function neural networks. Applied Mathematics and Computation, 95, 75–89 (1998) · Zbl 1085.82503 · doi:10.1016/S0096-3003(97)10089-3 |
| [8] | Jiang, Ch., Chen, T.: Denseness of superposition of one function with its translation and dilation in sobolev space W 2 m (R n). Acta Mathematica Sinica, Chinese Series, 42(3), 495–500 (1999) · Zbl 1018.41007 |
| [9] | Jiang, Ch., Chen, T.: Approximation problems of translation invariant operator in sobolev space W 2 m (R n). Chinese Annals of Mathematics, 20A(4), 499–504 (1999) · Zbl 0954.41010 |
| [10] | Jiang, Ch., Guo, Hb.: Approximate problem on neural networks and its application in system recognize in L p(R n). Annual of Math (in Chinese), 21A(4), 417–422 (2000) |
| [11] | Jiang, Ch.: The approximate problem on neural network. Annual of Math (in Chinese), 19A(3), 295–300 (1998) · Zbl 0929.41021 |
| [12] | Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control, Signals, Syst., 2(4), 303–314 (1989) · Zbl 0679.94019 · doi:10.1007/BF02551274 |
| [13] | Chen, T., Chen, H.: Universal approximation capability of EBF neural networks with arbitrary activation functions, Circuits Systems. Signal Processing, 15(5), 671–683 (1996) · Zbl 0860.68086 |
| [14] | Luo, Y., Shen, S.: L p Approximation of sigma-pi neural networks. IEEE Trans. Neural Networks, 11(6), 1485–1489 (2000) · doi:10.1109/72.883481 |
| [15] | Park, J., Sandberg, I. W.: Universal approximation using radial-basis-function networks. Neural Computation, 3, 246–257 (1991) · doi:10.1162/neco.1991.3.2.246 |
| [16] | Park, J., Sandberg, I. W.: Approximation and radial-basis-function networks. Neural Computation, 5, 305–316 (1993) · doi:10.1162/neco.1993.5.2.305 |
| [17] | Liao, Y., Fang, S., Nuttle, H. L. W.: Relaxed conditions for radial-basis function networks to be universal approximators. Neural Networks, 16, 1019–1028 (2003) · Zbl 1255.65044 · doi:10.1016/S0893-6080(02)00227-7 |
| [18] | Chen, T., Chen, H.: Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks. IEEE Trans. Neural Networks, 6(4), 904–910 (1995) · doi:10.1109/72.392252 |
| [19] | Chen, T., Chen, H.: Denseness of radial-basis functions in L 2(R n) and its applications in neurlan networks. Chinese Annals of Mathematics, 17B(2), 219–226 (1996) · Zbl 0857.41011 |
| [20] | Pinkus, A.: TDI-subspace of C(R d) and some density problems from neural networks. Approx Theory, 85, 269–287 (1996) · Zbl 0858.41023 · doi:10.1006/jath.1996.0042 |
| [21] | Friedlander, F. G.: Introduction to The Theory of Distributions, Cambridge University Press, Cambridge, 1982 · Zbl 0499.46020 |
| [22] | Rudin, W.: Functional Analysis, Springer, New York, 1987 · Zbl 0925.00005 |
| [23] | Hu, S.: Function of Real Variable, Springer, CHEP, 1999 |
| [24] | Friedman, A.: Generalized functions and partial differential equations, Englewood Cliffs, NJ: Prentice-Hall, 1963 · Zbl 0116.07002 |
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