Fuzzy order-equivalence for similarity measures

Similarity measures constitute a central component of machine learning and retrieval systems, and the choice of an appropriate measure is of major importance. In this paper, we consider this issue from the point of view of the order induced by the measures when comparing a set of objects to a given reference, i.e. the ranking from the most similar object to the least similar one. We introduce the notion of fuzzy order- equivalence, based on degrees that quantify the extent to which the induced orders differ. We define these degrees using the generalized Kendall's rank correlation, taking into account the number of order permutations as well as their positions. We then present an automatic and hierarchical classification of usual similarity measures that makes it possible to indicate, for a given number of tolerated variations, the measures that will yield rankings without significant changes; it thus provides a guideline for set data similarity measure selection.