Permutation Classifiers

We consider permutations of a given set of n different symbols. We are given two unordered training sets, T1 and T2, of such permutations that are each assumed to contain examples of permutations of the corresponding type, t1 and t2. Our goal is to train a classifier, C(q), by computing a statistical model from T1 and T2, which, when given a candidate permutation, q, decides whether q is of type t1 or t2. We discuss two versions of this problem. The ranking version focuses on the order of the symbols. Our Separation Average Distance Matrix (SADiM) solution expands on previously proposed ranking aggregation formulations. The grouping version focuses on contiguity of symbols and hierarchical grouping. We propose and compare two solutions: (1) The Population Augmentation Ratio (PAR) solution computes a PQ-tree for each training set and uses a novel measure of distance between these and q that is based on ratios of population counts (i.e., of numbers of permutations explained by specific PQ-trees). (2) The Difference of Positions (DoP) solution is computationally less expensive than PAR and is independent of the absolute population counts. Although DoP does not have the simple statistical grounding of PAR, our experiments show that it is consistently effective.