Parallel field ranking

Recently, ranking data with respect to the intrinsic geometric structure (manifold ranking) has received considerable attentions, with encouraging performance in many applications in pattern recognition, information retrieval and recommendation systems. Most of the existing manifold ranking methods focus on learning a ranking function that varies smoothly along the data manifold. However, beyond smoothness, a desirable ranking function should vary monotonically along the geodesics of the data manifold, such that the ranking order along the geodesics is preserved. In this article, we aim to learn a ranking function that varies linearly and therefore monotonically along the geodesics of the data manifold. Recent theoretical work shows that the gradient field of a linear function on the manifold has to be a parallel vector field. Therefore, we propose a novel ranking algorithm on the data manifolds, called Parallel Field Ranking. Specifically, we try to learn a ranking function and a vector field simultaneously. We require the vector field to be close to the gradient field of the ranking function, and the vector field to be as parallel as possible. Moreover, we require the value of the ranking function at the query point to be the highest, and then decrease linearly along the manifold. Experimental results on both synthetic data and real data demonstrate the effectiveness of our proposed algorithm.