S. Buchholz, K. Tachibana, E. Hitzer

**Introduction
to Theory, Construction and Application
of Clifford Neural Networks**

Tutorial Notes for
*14th International Conf. on Neural Information Processing (ICONIP 2007), Japan, 13 Nov. 2007*,
14 pages.

**Introduction:**
Clifford neural networks is a handle for neural networks in the Clifford algebra domain.
Clifford algebras are named after the mathematician William Kingdon Clifford. They
subsume, for example, the real numbers, the complex numbers and the quaternions. The
latter being a four-dimensional algebra invented byWilliam Rowan Hamilton. In the 1960s
David Hestenes started to study Clifford algebras as a universal language for geometry
for which he coined the term "Geometric algebra". A term that actually dates back to
Clifford himself. Its geometric nature makes Clifford algebra a very interesting framework
for neural computation. These notes are intented to sketch some fundamentals of the
theory of Clifford neural computation.
Fortunately, complex-valued neural networks are a vital topic themself. The recent book
[11] provides an overview about their theory and applications.
Quaternionic neural networks are an emerging topic. Their theory started with the pioneering
work of Arena et al. [1].

[1] P. Arena, L. Fortuna, G. Muscato, and M. G. Xibilia.
Neural Networks in Multidimensional Domains.
Number 234 in LNCIS. Springer, New York, 1998.

[11] A. Hirose.
Complex-Valued Neural Networks.
Number 32 in Studies in Computational Intelligence.
Springer, New York, 2006.

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