Clustering Rankings in the Fourier Domain

It is the purpose of this paper to introduce a novel approach to clustering rank data on a set of possibly large cardinality n ∈ N*, relying upon Fourier representation of functions defined on the symmetric group Sn. In the present setup, covering a wide variety of practical situations, rank data are viewed as distributions on Sn. Cluster analysis aims at segmenting data into homogeneous subgroups, hopefully very dissimilar in a certain sense. Whereas considering dissimilarity measures/distances between distributions on the non commutative group Sn, in a coordinate manner by viewing it as embedded in the set [0, 1]n! for instance, hardly yields interpretable results and leads to face obvious computational issues, evaluating the closeness of groups of permutations in the Fourier domain may be much easier in contrast. Indeed, in a wide variety of situations, a few well-chosen Fourier (matrix) coefficients may permit to approximate efficiently two distributions on Sn as well as their degree of dissimilarity, while describing global properties in an interpretable fashion. Following in the footsteps of recent advances in automatic feature selection in the context of unsupervised learning, we propose to cast the task of clustering rankings in terms of optimization of a criterion that can be expressed in the Fourier domain in a simple manner. The effectiveness of the method proposed is illustrated by numerical experiments based on artificial and real data.