Weighted differential topographic function: a refinement of topographic function

Topology preservation of Self-Organizing Maps (SOMs) is an advantageous property for correct clustering. Among several existing measures of topology violation, this paper studies the Topographic Function (TF) [1]. We find that this measuring method, demonstrated for low-dimensional data in [1], has a reliable foundation in its distance metric for the interpretation of the neighborhood relationship in the input space, for high-dimensional data. Based on the TF, we present a Differential Topographic Function (DTF) to reveal the topology violation more clearly and informatively. In addition, a Weighted Differential Topographic Function (WDTF) has been developed. For real world data, the DTF and WDTF unravel more details than the original TF, and help us estimate the topology preservation quality more accurately.