Abstract
The approximation of a continuous function on the torus \(\mathbb{T}^{2}\) is an important problem in approximation theory of artificial neural networks. In this work, we investigate the universal approximation capability of one-hidden layer feedforward toroidal approximate identity neural networks. To this end, we present notions of toroidal convolution and toroidal approximate identity. Using these notions, we apply a convolution linear operator approach to prove uniform converges in terms of continuous functions on the torus \(\mathbb{T}^{2}\). Using this result, we also prove a main theorem. The main theorem shows that one-hidden layer feedforward toroidal approximate identity neural networks are universal approximators in the space of continuous functions on the torus \(\mathbb{T}^{2}\).
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References
Marzio, M.D., Panzera, A., Taylor, C.C.: Kernel density estimation on the torus. Journal of Statistical Planning and Inference 141, 2156–2173 (2011)
Taylor, C.C., Mardia, K.V., Marzio, M.D., Panzera, A.: Validating protein structure using kernel density estimates. Journal of Applied Statistics 39, 2379–2388 (2012)
Potts, D.: Approximation of scattered data by trigonometric polynomials on the torus and the 2-sphere. Advances in Computational Mathematics 21, 21–36 (2004)
Kushpel, A., Grandison, C., Ha, M.D.: Optimal sk-splines approximation and reconstruction on the torus and sphere. International Journal of Pure and Applied Mathematics 29, 469–490 (2006)
Nong, J.: Conditions for RBF neural networks to universal approximation and numerical experiments. In: Zhao, M., Sha, J. (eds.) ICCIP 2012, Part II. CCIS, vol. 289, pp. 299–308. Springer, Heidelberg (2012)
Arteaga, C., Marrero, M.: Universal approximation by radial basis function networks of Delsarte translates. Neural Networks 46, 299–305 (2013)
Costarelli, D.: Interpolation by neural network operators activated by ramp function. Journal of Mathematical Analysis and Applications 419, 574–598 (2014)
Lin, S., Rong, Y., Xu, Z.: Multivariate Jackson-type inequality for a new type neural network approximation. Applied Mathematical Modelling (2014), http://dx.doi.org/10.1016/j.apm.2014.05.018
Cao, L., Xi, L., Zhang, Y.: L p estimate of convolution transform of singular measure by approximate identity. Nonlinear Analysis 94, 148–155 (2014)
Zainuddin, Z., Panahian, F. S.: Double approximate identity neural networks universal approximation in real Lebesgue spaces. In: Huang, T., Zeng, Z., Li, C., Leung, C.S. (eds.) ICONIP 2012, Part I. LNCS, vol. 7663, pp. 409–415. Springer, Heidelberg (2012)
Panahian Fard, S., Zainuddin, Z.: On the universal approximation capability of flexible approximate identity neural networks. In: Wong, W.E., Ma, T. (eds.) Emerging Technologies for Information Systems, Computing, and Management. LNEE, vol. 236, pp. 201–207. Springer, Heidelberg (2013)
Panahian Fard, S., Zainuddin, Z.: The universal approximation capabilities of Mellin approximate identity neural networks. In: Guo, C., Hou, Z.-G., Zeng, Z. (eds.) ISNN 2013, Part I. LNCS, vol. 7951, pp. 205–213. Springer, Heidelberg (2013)
Panahian Fard, S., Zainuddin, Z.: The universal approximation capability of double flexible approximate identity neural networks. In: Wong, W.E., Zhu, T. (eds.) Computer Engineering and Networking. LNEE, vol. 277, pp. 125–133. Springer, Heidelberg (2014)
Panahian Fard, S., Zainuddin, Z.: Analyses for L p[a, b]-norm approximation capability of flexible approximate identity neural networks. Neural Computing and Applications 24, 45–50 (2014)
Panahian Fard, S., Zainuddin, Z.: The universal approximation capabilities of “2π-periodic approximate identity” neural networks. In: 2013 International Conference on Information Science and Cloud Computing, pp. 793–798. IEEE Press (2014)
Panahian Fard, S., Zainuddin, Z.: The universal approximation capabilities of “double 2π-periodic approximate identity” neural networks. Soft Computing, doi: 10.1007/s00500-014-1449-8
Zainuddin, Z., Panahian Fard, S.: A study on the universal approximation capability of 2-spherical approximate identity neural networks. In: 5th International Conference on Mathematical Model for Engineering Science, pp. 23–27. WSEAS Press (2014)
Wu, W., Nan, D., Li, Z., Long, J., Wang, J.: Approximation to compact set of functions by feedforward neural networks. In: Proceedings of the 20th International Joint Conference on Neural Networks, pp. 1222–1225 (2007)
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Fard, S.P., Zainuddin, Z. (2014). Toroidal Approximate Identity Neural Networks Are Universal Approximators. In: Loo, C.K., Yap, K.S., Wong, K.W., Teoh, A., Huang, K. (eds) Neural Information Processing. ICONIP 2014. Lecture Notes in Computer Science, vol 8834. Springer, Cham. https://doi.org/10.1007/978-3-319-12637-1_17
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DOI: https://doi.org/10.1007/978-3-319-12637-1_17
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