Objective Functions for Topography: A Comparison of Optimal Maps

Topographic mappings are important in several contexts, including data visualization, connectionist representation, and cortical structure. Many different ways of quantifying the degree of topography of a mapping have been proposed. In order to investigate the consequences of the varying assumptions that these diierent approaches embody, we have optimized the mapping with respect to a number of different measures for a very simple problem the mapping from a square to a line. The principal results are that (1) different objective functions can produce very different maps, (2) only a small number of these functions produce mappings which match common intuitions as to what a topographic mapping "should" actually look like for this problem, (3) the objective functions can be put into certain broad categories based on the overall form of the maps, and (4) certain categories of objective functions may be more appropriate for particular types of problem than other categories.