APM  Vol.3 No.9 A , December 2013
Hausdorff Dimension of Multi-Layer Neural Networks

This elucidation investigates the Hausdorff dimension of the output space of multi-layer neural networks. When the factor map from the covering space of the output space to the output space has a synchronizing word, the Hausdorff dimension of the output space relates to its topological entropy. This clarifies the geometrical structure of the output space in more details.

Cite this paper: J. Ban and C. Chang, "Hausdorff Dimension of Multi-Layer Neural Networks," Advances in Pure Mathematics, Vol. 3 No. 9, 2013, pp. 9-14. doi: 10.4236/apm.2013.39A1002.

[1]   K. Hornik, M. Stinchcombe and H. White, “Multilayer Feedforward Networks Are Universal Approximators,” Neural Networks, Vol. 2, No. 5, 1989, pp. 359-366.

[2]   B. Widrow and M. Lehr, “30 Years of Adaptive Neural Networks: Perceptron Madaline, and Backpropagation,” Proceedings of the IEEE, Vol. 78, No. 9, 1990, pp. 1415-1442.

[3]   Y. A. Alsultanny and M. M. Aqul, “Pattern Recognition Using Multilayer Neural-Genetic Algorithm,” Neurocomputing, Vol. 51, 2003, pp. 237-247.

[4]   B. Widrow, “Layered Neural Nets for Pattern Recognition,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 7, 1962, pp. 1109-1118.

[5]   J. J. Hopeld and D. W. Tank, “Neural Computation of Decisions in Optimization Problems,” Biological Cybernetics, Vol. 52, No. 3, 1985, pp. 141-152.

[6]   C. Peterson and B. Soderberg, “A New Method for Mapping Optimization Problems onto Neural Network,” International Journal of Neural Systems, Vol. 1, 1989, pp. 3-22.

[7]   J.-C. Ban and C.-H. Chang, “The Learning Problem of Multi-Layer Neural Networks,” Neural Networks, Vol. 46, 2013, pp. 116-123.

[8]   J.-C. Ban, C.-H. Chang and S.-S. Lin, “The Structure of Multi-Layer Cellular Neural Networks,” Journal of Differential Equations, Vol. 252, No. 8, 2012, pp. 4563-4597.

[9]   J.-C. Ban, C.-H. Chang, S.-S. Lin and Y.-H. Lin, “Spatial Complexity in Multi-Layer Cellular Neural Networks,” Journal of Differential Equations, Vol. 246, No. 2, 2009, pp. 552-580.

[10]   S.-N. Chow and J. Mallet-Paret, “Pattern Formation and Spatial Chaos in Lattice Dynamical Systems: I and II,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 42, No. 10, 1995, pp. 746-756.

[11]   K. Falconer, “Fractal Geometry: Mathematical Foundations and Application,” 2nd Edition, John Wilet & Sons, New York, London, Sydney, 2003.

[12]   Y. Pesin, “Dimension Theory in Dynamical Systems: Contemporary Views and Application,” The University of Chicago Press, Chicago, 1997.

[13]   J. Juang and S.-S. Lin, “Cellular Neural Networks: Mosaic Pattern and Spatial Chaos,” SIAM Journal on Applied Mathematics, Vol. 60, No. 3, 2000, pp. 891-915.

[14]   D. Lind and B. Marcus, “An Introduction to Symbolic Dynamics and Coding,” Cambridge University Press, Cambridge, 1995.

[15]   B. Kitchens, “Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts,” Springer-Verlag, New York, 1998.

[16]   J.-C. Ban and C.-H. Chang, “On the Structure of Multi-Layer Cellular Neural Networks. Part II: The Complexity between Two Layers,” Submitted, 2012.