Ieee Transactions on Signal Processing

|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 distortions due to the vector quantization are simple to compute and to compare between diierent maps, topography of a map is diicult to deene 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 visualiza-tion, and in numerous other applications. In this paper we review some conceptual aspects of deenitions of topography, and some recently proposed measures to quantify topography. We apply the measures rst to neural maps trained on synthetic data sets, and check the measures for properties like reproducability, 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 not a perfect , but an adequate job, e.g. in selecting a topographically optimal output space dimension.