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Article Details

Copyright Statement: This is an open access article licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, even commercially as long as the original work is properly cited.

Construction of Neural Networks that Do Not Have Critical Points Based on Hierarchical Structure

Author 1: Tohru Nitta

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Digital Object Identifier (DOI) : 10.14569/IJACSA.2013.040911

Article Published in International Journal of Advanced Computer Science and Applications(IJACSA), Volume 4 Issue 9, 2013.

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Abstract: a critical point is a point at which the derivatives of an error function are all zero. It has been shown in the literature that critical points caused by the hierarchical structure of a real-valued neural network (NN) can be local minima or saddle points, although most critical points caused by the hierarchical structure are saddle points in the case of complex-valued neural networks. Several studies have demonstrated that singularity of those kinds has a negative effect on learning dynamics in neural networks. As described in this paper, the decomposition of high-dimensional neural networks into low-dimensional neural networks equivalent to the original neural networks yields neural networks that have no critical point based on the hierarchical structure. Concretely, the following three cases are shown: (a) A 2-2-2 real-valued NN is constructed from a 1-1-1 complex-valued NN. (b) A 4-4-4 real-valued NN is constructed from a 1-1-1 quaternionic NN. (c) A 2-2-2 complex-valued NN is constructed from a 1-1-1 quaternionic NN. Those NNs described above do not suffer from a negative effect by singular points during learning comparatively because they have no critical point based on a hierarchical structure.

Keywords: critical point; singular point; redundancy; complex number; quaternion

Tohru Nitta, “Construction of Neural Networks that Do Not Have Critical Points Based on Hierarchical Structure” International Journal of Advanced Computer Science and Applications(IJACSA), 4(9), 2013. http://dx.doi.org/10.14569/IJACSA.2013.040911

@article{Nitta2013,
title = {Construction of Neural Networks that Do Not Have Critical Points Based on Hierarchical Structure},
journal = {International Journal of Advanced Computer Science and Applications},
doi = {10.14569/IJACSA.2013.040911},
url = {http://dx.doi.org/10.14569/IJACSA.2013.040911},
year = {2013},
publisher = {The Science and Information Organization},
volume = {4},
number = {9},
author = {Tohru Nitta}
}


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