Low-rank tensor completion by Riemannian optimization

In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjug...

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Bibliographic Details
Published inBIT Numerical Mathematics Vol. 54; no. 2; pp. 447 - 468
Main Authors Kressner, Daniel, Steinlechner, Michael, Vandereycken, Bart
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.06.2014
Subjects
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Reconstruction
58C05
Tensors
65K05
Low-rank approximation
65F99
15A69
Tucker decomposition
High-dimensionality
Riemannian optimization
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Summary:In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjugate gradient method is developed. Paying particular attention to efficient implementation, our algorithm scales linearly in the size of the tensor. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries are unknown. We illustrate the use of the developed algorithm for the recovery of multidimensional images and for the approximation of multivariate functions.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-013-0455-z