K. Tachibana, E. Hitzer

**Tutorial
Note on Geometric Algebra Neural Networks**

*Tutorial Notes*,
AGACSE 3, Leipzig, Germany,
17-19 Aug. 2008, 20 pages.

**Abstract:**
This tutorial aims to give an introduction to neural computation with
Clifford Geometric Algebra (GA) rounded off with some application examples.
GA NNs (Clifford NNs) are unique in geometrically correct representation
and computation with multi-dimensional geometric data. Attendants
will learn to represent problems in GA, construct GA multi layer
perceptrons (MLPs), and interpret the results. Recommended for everyone
dealing with multi-dimensional data.

In the first half a general introduction to the basics of neural networks
of real numbers will be given. This includes types of learning problems
(from supervised to unsupervised), regression with linear weights, optimization,
layered NNs, and error propagation.

The second part treats the foundations of GA-neural computation. Already
number systems like complex numbers and quaternions have many
neural network applications. We explain the construction of basic, versor
(Clifford group) and spinor GA neurons, and GA MLPs. These have
advantages for processing, optimization, and interpretation of data as (geometric)
objects by GA NNs.

Finally an overview of practical applications of GA-neurons to geometric
data analysis will round off the tutorial. This will illustrate how
naturally and interpretatively GA-NNs can be used to represent and analyze
geometric objects and their relationships. Some explicit examples
will be introduced from diverse fields like feature recognition, biochemistry
and meteorology.

**Keywords: ** Clifford geometric algebra, multidimensional Neural Networks,
Self Organizing Maps, Clifford Neuron, Spinor Neuron, GA MLP,
geometric interpretation, molecular conformation, climate extremes.

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