Normalized Forms for Two Common

| In this paper we demonstrate that two common metrics, symmetric set diierence, and Eu-clidian distance, have normalized forms which are nevertheless metrics. The rst of these jA4Bj=jABj is easily established and generalizes to measure spaces. The second applies to vectors in R n and is given by kX?Y k=(kXk+kY k). That this is a metric is more dii-cult to demonstrate and is true for Euclidian distance (the L2 norm) but for no other integral Minkowski metric. In addition to providing bounded distances when no a priori data bound exists, these forms are qualitatively diierent from their unnormalized counterparts , and are therefore also distinguished from simpler range companded constructions. Mixed forms are also deened which combine absolute and relative behavior, while remaining metrics. The result is a family of forms which resemble commonly used dissimilarity statistics but obey the triangle inequality.