Quantifying neighbourhood preservation in topographic mappings

Mappings that preserve neighbourhood relationships are important in many contexts, from neurobiology to multivariate data analysis. It is important to be clear about precisely what is meant by preserving neighbourhoods. At least three issues have to be addressed: how neighbourhoods are defined, how a perfectly neighbourhood preserving mapping is defined, and how an objective function for measuring discrepancies from perfect neighbourhood preservation is defined. We review several standard methods, and using a simple example mapping problem show that the different assumptions of each lead to non-trivially different answers. We also introduce a particular measure for topographic distortion, which has the form of a quadratic assignment problem. Many previous methods are closely related to this measure, which thus serves to unify disparate approaches.

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