New complex-valued activation functions

Nihal Ozgur, Nihal Taş, James Francis Peters


We present a new type of activation functions for a complex-valued neural
network (CVNN). A proposed activation function is constructed such that it
fixes a given ellipse. We obtain an application to a complex-valued Hopfield
neural network (CVHNN) using a special form of the introduced complex
functions as an activation function. Considering the interesting geometric
properties of the plane curve ellipse such as focusing property, we
emphasize that these properties may have possible applications in various
neural networks.


Complex valued neural network; complex-valued Hopfield neural network; activation function; fixed ellipse

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Ceylan, M., Ozbay, Y., Ucan, O.N., Yıldırım, E. (2010). A novel method for lung segmentation on chest CT images: complex-valued artificial neural network with complex wavelet transform. Turk. J. Elec. Eng. and Comp. Sci., 18(4), 613-623.

Ceylan, M., Ya¸sar, H. (2016). A novel approach for automatic blood vessel extraction in retinal images: complex ripplet-I transform and complex valued artificial neural network. Turk. J. Elec. Eng. and Comp. Sci., 24(4), 3212-3227.

Gandal, A.S., Kalra, P.K., Chauhan, D.S. (2007). Performance evaluation of complex valued neural networks using various error functions. International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering, 1(5), 732-737.

Hirose, A. (2009). Complex-valued neural networks: The merits and their origins. Proceedings of the Internatinal Joint Conference on Neural Networks (IJCNN), Atlanta, June 14- 19, 1237-1244.

Jalab, H.A., Ibrahim, R. W. (2011). New activation functions for complex-valued neural network. International Journal of the Physical Sciences, 6(7), 1766-1772.

Oladipo, A.T., Gbolagade, M. (2014). Some subordination results for logistic sigmoid activation function in the space of univalent functions. Advances in Computer Science and Engineering, 12(2), 61-79.

Singh, R.G., Singh, A.P. (2015). Multiple complex extreme learning machine using holomorphic mapping for prediction of wind power generation system. International Journal of Computer Applications, 123(18), 24-33.

Zimmermann, H.G., Minin, A., Kusherbaeva, V. (2011). Comparison of the complex valued and real valued neural networks trained with gradient descent and random search algorithms. ESANN 2011 proceedings, European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning. Bruges (Belgium) 27-29 April 2011.

Kim, T., Adalı, T. (2002) Fully complex multi-layer perceptron network for nonlinear signal processing. J. VLSI Sig. Process., 32, 29-43.

Frantz, M. (1994). A focusing property of the ellipse. Amer. Math. Monthly, 101(3), 250- 258.

Wilker, J.B. (1995). Further thoughts on a focusing property of the ellipse. Bull. Belg. Math. Soc., 2, 153-159.

Connett, J.E. (1992). Trapped reflections?. Amer. Math. Monthly, 99, 178-179.

Di Concilio, A., Guadagni, C., Peters, J.F., Ramanna, S. (2018). Descriptive proximities. Properties and interplay between classical proximities and overlap. Math. Comput. Sci., 12(1), 91-106. arXiv:1609.06246. MR3767897.

Ferrer, S., Hanßmann, H., Palaci´an, J., Yanguas, P. (2002). On perturbed oscillators in 1-1-1 resonance: the case of axially symmetric cubic potentials. J. Geom. Phys., 40(3-4), 320-369.

Grandon, J., Derpich, I. (2011). A Heuristic for the Multi-knapsack Problem. WSEAS Transactions on Mathematics, 10(3), 95-104.

Kanan, H.R., Faez, K., Ezoji, M. (2006). An efficient face recognition system using a new optimized localization method, In Pattern Recognition. ICPR 2006, 18th International Conference on Vol. 3, 564-567.

Kellner, M.A., Hanning, T., Farr, H. (2002). Real-time analysis of the grain on wooden planks, Machine Vision Applications in Industrial Inspection X. Vol. 4664. International Society for Optics and Photonics.

Kobayashi, M., (2013). Hyperbolic Hopfield neural networks. IEEE Trans. Neural Netw. Learn Syst. 24(2), 335-341.

Li, J., Zhang, J. (2004). Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation. Chaos Solitons Fractals 21(4), 889-913.

Peters, J.F. (2018). Proximal Vortex Cycles and Vortex Nerves. Non-Concentric, Nesting, Possibly Overlapping Homology Cell Complexes. Journal of Mathematical Sciences and Modelling, 1 (2), 80-85. arXiv:1805.03998.

Zhang, G., Jayas, D.S., White, N.D. (2005). Separation of touching grain kernels in an image by ellipse fitting algorithm. Biosystems engineering, 92(2), 135-142.

Wolfram Research. (2019). Inc., Mathematica, Version 12.0, Champaign, IL.

Beardon, A.F. (1983). The geometry of discrete groups. Graduate texts in mathematics, vol 91. Springer, New York.

Jones, G.A., Singerman, D. (1987). Complex functions. An algebraic and geometric viewpoint, Cambridge University Press, Cambridge.

Kwok, Y.K. (2010). Applied complex variables for scientists and engineers. Cambridge University Press, New York.

Mandic, D.P. (2000). The use of Mobius transformations in neural networks and signalprocessing. Neural Networks for Signal Processing X, 1, 185-194.

Ozdemir, N., Iskender, B.B., Ozgur, N.Y. (2011). Complex valued neural network with M¨obius activation function. Commun. Nonlinear Sci. Numer. Simul., 16(12), 4698-4703.

Hopfield, J.J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. United States Amer., 79(8), 2554-2558.

Kobayashi, M. (2017). Symmetric complexvalued Hopfield neural networks. IEEE Trans. Neural Netw. Learn Syst., 28(4), 1011-1015.

Khalil, H.K. (1996). Nonlinear systems. 2nd ed. United States of America: Prentice Hall.

Kuroe, Y., Yoshida, M., Mori, T. (2003). On activation functions for complex-valued neural networks - existence of energy functions. In: Kaynak O et al., editors. ICANN/ICONIP 2003. LNCS 2714.



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