Maps for the Visualization of high-dimensional Data Spaces

TLDR: The U-Matrix, as defined here, is a display of the density relationships in the data space using Pareto Density Estimation and can be used for a non-ambiguous display of a non planar neuron space.

Abstract: -The U-Matrix is a canonical tool for the display of distance structures in data space using emergent SOM (ESOM). The U-Matrix defined originally for planar map spaces is extended in this work to toroid neuron spaces. Embedding the neuron space in a finite but borderless space, such as a torus, avoids border effects of planar spaces. A planar display of a toroid map space disrupts, however, coherent U-Matrix structures. Tiling multiple instances of the U-Matrix solves this problem at the cost of multiple images of data points. The P-Matrix, as defined here, is a display of the density relationships in the data space using Pareto Density Estimation. While the P-Matrix is useful for clustering, it can also be used for a non-ambiguous display of a non planar neuron space. Centering the display for high density regions and removing ambiguous images of data points leads to U-Maps and P-Maps. U-Maps depict the distance structure of a data space as a borderless three dimensional landscape whose floor space is ordered according to the topology preserving features of ESOM. P-Maps display the density structures. Both maps are specially suited for data mining and knowledge discovery.