Manifold Ranking using Hessian Energy

In recent years, learning on manifolds has attracted much attention in the academia community. The idea that the distribution of real-life data forms a low dimensional manifold embedded in the ambient space works quite well in practice, with applications such as ranking, dimensionality reduction, semi-supervised learning and clustering. This paper focuses on ranking on manifolds. Traditional manifold ranking methods try to learn a ranking function that varies smoothly along the data manifold by using a Laplacian regularizer. However, the Laplacian regularization suffers from the issue that the solution is biased towards constant functions. In this work, we propose using second-order Hessian energy as regularization for manifold ranking. Hessian energy overcomes the above issue by only penalizing accelerated variation of the ranking function along the geodesics of the data manifold. We also develop a manifold ranking framework for general graphs/hypergraphs for which we do not have an original feature space (i.e. the ambient space). We evaluate our ranking method on the COREL image dataset and a rich media dataset crawled from Last.fm. The experimental results indicate that our manifold ranking method is effective and outperforms traditional graph Laplacian based ranking method.