中文摘要：

流形正则化低秩矩阵分解（Manifold Regularized Low-rank Matrix Factorization,MRLMF）算法是一种最近提出的能考虑样本间流形结构的矩阵分解算法.MRLMF采用Laplacian图来表示样本的流形结构,但是,最近研究表明,由于Laplacian图的零空间中的测地线函数为常数,使得其往往不能较好的保持样本间的局部拓扑结构.为了解决这一问题,提出一种Hessian正则化的低秩矩阵分解算法（Hessian Regularized Low-rank Matrix Factorization,HRLMF）.HRLMF利用二阶Hessian能来保持样本的局部流形结构,而Hessian能可以使测地线函数随距离变化,从而使得其保持样本局部拓扑结构的能力更强.此外,也给出了一种求解HRLMF的高效算法.在实际数据库上的实验表明,MRLMF算法比现有的算法有着更好的性能.

英文摘要：

Manifold regularized low-rank matrix factorization （ MRLMF ）, which can consider the manifold structure of samples, is a recently proposed matrix factorization algorithm. Laplacian graph is used in MRLMF to learn the manifold structure. However, recent studies show that the geodesic function in null space of Laplacian graph is constant, and then the Laplacian embedding usually cannot properly preserve local manifold structure of samples. To address the problem of MRLMF, Hessian regularized low-rank matrix fac- torization （HRLMF） is proposed in this paper. The second-order Hessian energy, the geodesic function of which varies with dis- tance, is used in HRLMF to preserve local manifold structures. Then HRLMF can preserve local topology structure much better than MRLMF. Besides, an efficient algorithm for solving HRLMF is also proposed in this paper. Experiment results on some real data sets demonstrate that MRLMF can achieve much better performances than state-of-the-art algorithms.