Tensor completion using geodesics on Segre manifolds

• Published in: Numerical linear algebra with applications Vol. 29; no. 6
• Main Authors: Swijsen, Lars, Van der Veken, Joeri, Vannieuwenhoven, Nick
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.12.2022
• Subjects: Algorithms; canonical polyadic decomposition; Chemical compounds; conjugate gradient algorithm; Conjugate gradient method; Decomposition; Fluorescence; geodesics; Geodesy; Manifolds; Mathematical analysis; Missing data; Newton methods; Optimization; Parameterization; Recommender systems; Riemannian geometry; Riemannian optimization; Segre manifold; tensor rank decomposition; Tensors
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2446

We propose a Riemannian conjugate gradient algorithm for approximating incomplete tensors by canonical polyadic decompositions of low rank. Our main contribution consists of an explicit expression for an almost everywhere complete set of geodesics of the Segre manifold of rank‐1 tensors. These are leveraged in our Riemannian optimization algorithm over a geometrically convenient parametrization of rank‐r$$ r $$ tensors to move in the direction of a tangent vector over the r$$ r $$‐fold product of Segre manifolds. We apply our method to movie rating predictions in a recommender system for the MovieLens dataset, and for identifying pure fluorophores via fluorescent spectroscopy with missing data. In this last application, we can recover the tensor decomposition from only 6.5%$$ 6.5\% $$ of the data. In our numerical experiments, the proposed Riemannian conjugate gradient algorithm was competitive with a state‐of‐the‐art quasi‐Newton method with truncated conjugate gradient inner solves from Tensorlab in terms of accuracy and could reduce the computation time by half.