Ranking with hamiltonian dynamics

Abstract

In data science, a ranking or linear ordering problem is to place items into a linear order based on comparison data. In this article, we consider a Hamiltonian system, in which particles, representing the items to be ordered, exert forces on each other as determined by the comparison data. An ordering of the items can be derived from the relative positions of the particles in any state of the system. The Hamiltonian is designed so that states with low potential energy yield orderings that agree strongly with the comparison data. Although the dynamics are related to the Toda lattice, no usable ranking information seems to be available from that variation of the Hamiltonian. Instead, a Toda-like Hamiltonian plus a confinement potential yields better results. Several algorithms based on Hamiltonian trajectories, minimization of a ranking potential, and a Hamiltonian Markov chain are compared to the widely used RankBoost algorithm. They all perform relatively well on synthetic test data and on problems from a library of test cases, but none is clearly the best in all circumstances. The trajectory-based methods and Markov chain show interesting dynamics. Furthermore, since these methods attach particle positions to the items as well as an ordering, the distances between the particles indicate how rankable the items are.