title: A Novel Riemannian Metric Based on Riemannian Structure and Scaling Information for Fixed Low-Rank Matrix Completion.
abstract: Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.
authors.forename: Shasha
authors.initials: S
authors.lastname: Mao
authors.forename: Lin
authors.initials: L
authors.lastname: Xiong
authors.forename: Licheng
authors.initials: L
authors.lastname: Jiao
authors.forename: Tian
authors.initials: T
authors.lastname: Feng
authors.forename: Sai-Kit
authors.initials: SK
authors.lastname: Yeung