A Riemannian rank-adaptive method for low-rank optimization

Abstract This paper presents an algorithm that solves optimization problems on a matrix manifold M ⊆ R m × n with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted low-rank approximation problem is used to illustrate the efficiency and effectiveness of the algorithm.