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

TL;DR

This paper demonstrates that neuron grids can preserve data structure and be mapped to a universal 3D neuron grid, with implications for neural fields and existing grid types.

Contribution

It introduces a topological framework showing neuron grids can preserve data structure and be embedded into a universal 3D neuron grid.

Findings

Neuron grids can preserve the qualitative structure of data spaces.

Neuron grids can be mapped to a universal 3D neuron grid.

Implications for existing neuron grid types and neural fields.

Abstract

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.