Contrastive Antichains in Hierarchies

Abstract

Concepts are often described in terms of positive integer-valued attributes that are organized in a hierarchy. For example, cities can be described in terms of how many places there are of various types (e.g. nightlife spots, residences, food venues), and these places are organized in a hierarchy (e.g. a Portuguese restaurant is a type of food venue). This hierarchy imposes particular constraints on the values of related attributes-e.g. there cannot be more Portuguese restaurants than food venues. Moreover, knowing that a city has many food venues makes it less surprising that it also has many Portuguese restaurants, and vice versa. In the present paper, we attempt to characterize such concepts in terms of so-called contrastive antichains: particular kinds of subsets of their attributes and their values. We address the question of when a contrastive antichain is interesting, in the sense that it concisely describes the unique aspects of the concept, and this while duly taking into account the known attribute dependencies implied by the hierarchy. Our approach is capable of accounting for previously identified contrastive antichains, making iterative mining possible. Besides the interestingness measure, we also present an algorithm that scales well in practice, and demonstrate the usefulness of the method in an extensive empirical results section. CCS CONCEPTS • Mathematics of computing → Information theory; Probabilistic representations; • Computing methodologies → Knowledge representation and reasoning; Learning in probabilistic graphical models; Unsupervised learning.