Riemannian Submanifold Tracking on Low-Rank Algebraic Variety

Abstract

Matrix recovery aims to learn a low-rank structure from high dimensional data, which arises in numerous learning applications. As a popular heuristic to matrix recovery, convex relaxation involves iterative calling of singular value decomposition (SVD). Riemannian optimization based method can alleviate such expensive cost in SVD for improved scalability, which however is usually degraded by the unknown rank. This paper proposes a novel algorithm RIST that exploits the algebraic variety of low-rank manifold for matrix recovery. Particularly, RIST utilizes an efficient scheme that automatically estimate the potential rank on the real algebraic variety and tracks the favorable Riemannian submanifold. Moreover, RIST utilizes the second-order geometric characterization and achieves provable superlinear convergence, which is superior to the linear convergence of most existing methods. Extensive comparison experiments demonstrate the accuracy and ef- ficiency of RIST algorithm.