# Paper ID sheet UCL-INMA-2015.05

Title

## A Riemannian rank-adaptive method for low-rank optimization

Authors
Guifang Zhou, Wen Huang, Kyle Gallivan, Paul Van Dooren, P.-A. Absil
Abstract
This paper presents an algorithm that solves optimization problems on a matrix manifold $\mathcal{M} \subseteq \mathbb{R}^{m \times n}$ 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.
Key words
low-rank optimization; rank-constrained optimization; Riemannian manifold; fixed-rank manifold; low-rank approximation
Status
Accepted for publication in Neurocomputing, 2016