A Note on Topology Preservation in Classification, and the Construction of a Universal Neuron Grid

It will be shown that according to theorems of K. Menger, every neuron grid if identified with a curve is able to preserve the adopted qualitative structure of a data space. Furthermore, if this identification is made, the neuron grid structure can always be mapped to a subset of a universal neuron grid which is constructable in three space dimensions. Conclusions will be drawn for established neuron grid types as well as neural fields. 1 Mathematical Preliminaries Topology is one of the basic branches of mathematics. It is sometimes also referred to as qualitative geometry, in a way that it deals with the qualitative properties and structure of geometrical objects. The geometrical objects of interest in this paper are vector spaces, manifolds, and curves. These form the basis of the presented mathematical treatment of classification with neuron grids. Consequently, the paper has to begin with some mathematical preliminaries. 1.1 Manifolds In the following, a n-dimensional vector space will be identified with a subspace of IRn. It is assumed that IRn is equipped with a topology which in turn is induced by a metric. Mappings between subspaces of IRn are called homeomorphic or topology preserving if they are one-to-one and continuous in both directions. It is known from the theorem of dimension invariance (Brouwer, 1911) that mappings between non-empty open sets U ⊂ IRm and V ⊂ IRn for m 6= n are never homeomorphic. In the light of Brouwer’s theorem it is the open sets that fix the topological dimension of a subset of IRn. As the IRn is introduced as a metric space, the open sets are given by open balls that formalize the concept of distance between points of this metric space. Homeomorphic mappings preserve the neighborhood relationship between points of IRn. Amongst the huge variety of subsets of the metric space IRn, the n-dimensional manifolds (or n-manifolds) have turned out to be of interest as these describe solution spaces of equations or geometrical entities. Manifolds are parameterized geometrical objects, parameters could describe e.g. the coordinates of a physical space. As n-manifolds are locally Euclidean of dimension n they are subject to Brouwer’s theorem and therefore of fixed topological 1 ar X iv :1 30 8. 16 03 v2 [ cs .N E ] 8 A ug 2 01 3 dimension. As a consequence, dimension reducing mappings between manifolds will not be able to transfer mutual topological structures. A vivid example of the dimension conflict of two manifolds of different dimension is given by a surjective and continuous mapping of [0,1] onto [0,1]× [0,1] which is also called a ’Peano curve’ (Peano, 1890). The convoluted structure of a Peano curve does not reflect the neighborhood relationship of elements of the underlying manifold ⊂ IR2, not even locally: There is no topology preserving mapping of [0,1] onto [0,1]× [0,1]. An illustration of a Peano curve to 4th iteration is given in figure 1. Figure 1: A Peano curve (source: Wikipedia) 1.2 Curves The Peano curve introduced in the preceding section is based on the conventional definition of a compact curve as a continuous mapping of [0,1]. The following definition according to (Menger, 1968) renders the definition of a compact curve more precisely. Beforehand, a definition of a continuum is required. Definition A compact, connected set ⊂ IRn having more than one element is a continuum. A set ⊂ IRn which contains no continuum is called discontinuous. Definition A continuum K as a subset of a metric space is called a curve if every point of K is contained in arbitrary small neighborhoods having discontinuous intersects with K. From Menger’s definition of a curve the following theorem results: Theorem[Menger] Every curve defined in a metric space is homeomorphic to a curve defined in IR3 . The proof of this theorem is left here. The interested reader is referred to Menger’s textbook (Menger, 1968). The theorem states that with regard to topological aspects, the transition from curves defined in three-dimensional Euclidean space to curves de-