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.

在数据科学中，排名或线性排序问题是基于比较数据将项目置于线性顺序中。在本文中，我们考虑一个哈密顿系统，其中代表待订购项目的粒子相互之间施加力，这由比较数据确定。项目的排序可以从系统任何状态下粒子的相对位置得出。哈密​​顿量被设计成使得具有低势能产率排序的状态与比较数据强烈一致。尽管动力学与Toda晶格相关，但似乎无法从汉密尔顿学派的变化获得可用的排名信息。取而代之的是，像Toda那样的哈密顿量加上限制势能产生更好的结果。几种基于哈密顿轨迹的算法，可最大程度地降低排名潜力，并将汉密尔顿马尔可夫链与广泛使用的RankBoost算法进行比较。它们在综合测试数据和测试用例库中的问题上都表现相对较好，但是显然在所有情况下都不是最好的。基于轨迹的方法和马尔可夫链显示出有趣的动态。此外，由于这些方法将粒子的位置和顺序附加到项目上，因此粒子之间的距离表示项目的可分级性。