A Riemannian trust‐region method for low‐rank tensor completion. Issue 6 (16th May 2018)

• Record Type: Journal Article
• Authors: Heidel, Gennadij; Schulz, Volker
• Other Names: Benner Peter guestEditor.; Faßbender Heike guestEditor.; Grasedyck Lars guestEditor.; Kressner Daniel guestEditor.; Meini Beatrice guestEditor.; Simoncini Valeria guestEditor.
• Is Part Of: Numerical linear algebra with applications. Volume 25:Issue 6(2018)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2018-05-16
• Subjects: low‐rank tensors -- multilinear rank -- Riemannian Hessian -- Riemannian optimization -- trust‐region methods -- Tucker decomposition; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2175
• Languages: English
• ISSNs: 1070-5325

Abstract

Summary: The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low‐rank constraint. This may be formulated as a least‐squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure; thus, methods of Riemannian optimization are applicable. In our work, we derive the Riemannian Hessian of an objective function on the low‐rank tensor manifolds using the Weingarten map, a concept from differential geometry. We discuss the convergence properties of Riemannian trust‐region methods based on the exact Hessian and standard approximations, both theoretically and numerically. We compare our approach with Riemannian tensor completion methods from recent literature, both in terms of convergence behavior and computational complexity. Our examples include the completion of randomly generated data with and without noise and the recovery of multilinear data from survey statistics.