A Note on Topology Preservation in Classification, and the Construction of a Universal Neuron Grid
This work provides theoretical insights into neural network topology for researchers in machine learning, but it appears incremental as it builds on existing theorems without introducing new methods or data.
The paper addresses the problem of topology preservation in classification by showing that any neuron grid can preserve the qualitative structure of a data space, based on K. Menger's theorems, and constructs a universal neuron grid in three dimensions.
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.