Rank Matrix Factorisation

We introduce the problem of rank matrix factorisation (RMF). That is, we consider the decomposition of a rank matrix, in which each row is a (partial or complete) ranking of all columns. Rank matrices naturally appear in many applications of interest, such as sports competitions. Summarising such a rank matrix by two smaller matrices, in which one contains partial rankings that can be interpreted as local patterns, is therefore an important problem. After introducing the general problem, we consider a specific instance called Sparse RMF, in which we enforce the rank profiles to be sparse, i.e., to contain many zeroes. We propose a greedy algorithm for this problem based on integer linear programming. Experiments on both synthetic and real data demonstrate the potential of rank matrix factorisation.