Neural maps and topographic vector quantization

Neural maps combine the representation of data by codebook vectors, like a vector quantizer, with the property of topography, like a continuous function. While the quantization error is simple to compute and to compare between different maps, topography of a map is difficult to define and to quantify. Yet, topography of a neural map is an advantageous property, e.g. in the presence of noise in a transmission channel, in data visualization, and in numerous other applications. In this article we review some conceptual aspects of definitions of topography, and some recently proposed measures to quantify topography. We apply the measures first to neural maps trained on synthetic data sets, and check the measures for properties like reproducibility, scalability, systematic dependence of the value of the measure on the topology of the map, etc. We then test the measures on maps generated for four real-world data sets, a chaotic time series, speech data, and two sets of image data. The measures are found to do an imperfect, but an adequate job in selecting a topographically optimal output space dimension, while they consistently single out particular maps as non-topographic.

Checking the projection display of multivariate data with colored graphs

Projection methods such as principal component analysis (PCA), nonlinear mapping (NLM), and the self-organizing map (SOM) are valuable algorithms for visualizing multidimensional data in a two-dimensional plane. Unfortunately, the reduction of the dimensionality involves distortions. In an attempt to graphically localize the distortions of the projected data, we suggest superposing colored graphs onto the 2D plots. The color of the edges of these graphs encodes the original high-dimensional distances between the connected points. The method is applied to a cluster analysis of 37 biologically active compounds and 471 molecules represented by a structural 3D descriptor.