To Örebro University

Rank data, in which each row is a complete or partial ranking of available items (columns), is ubiquitous. Among others, itcan be used to represent preferences of users, levels of gene expression, and outcomes of sports events. It can have many types ofpatterns, among which consistent rankings of a subset of the items in multiple rows, and multiple rows that rank the same subset of theitems highly. In this article, we show that the problems of finding such patterns can be formulated within a single generic framework thatis based on the concept of semiring matrix factorisation. In this framework, we employ the max-product semiring rather than theplus-product semiring common in traditional linear algebra. We apply this semiring matrix factorisation framework on two tasks: sparserank matrix factorisation and rank matrix tiling. Experiments on both synthetic and real world datasets show that the framework iscapable of discovering different types of structure as well as obtaining high quality solutions.