Low-rank tensor completion: a Riemannian manifold preconditioning approach

Hiroyuki Kasai, Bamdev Mishra
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:1012-1021, 2016.

Abstract

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms in batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v48-kasai16, title = {Low-rank tensor completion: a Riemannian manifold preconditioning approach}, author = {Kasai, Hiroyuki and Mishra, Bamdev}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {1012--1021}, year = {2016}, editor = {Balcan, Maria Florina and Weinberger, Kilian Q.}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/kasai16.pdf}, url = {https://proceedings.mlr.press/v48/kasai16.html}, abstract = {We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms in batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.} }
Endnote
%0 Conference Paper %T Low-rank tensor completion: a Riemannian manifold preconditioning approach %A Hiroyuki Kasai %A Bamdev Mishra %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-kasai16 %I PMLR %P 1012--1021 %U https://proceedings.mlr.press/v48/kasai16.html %V 48 %X We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms in batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.
RIS
TY - CPAPER TI - Low-rank tensor completion: a Riemannian manifold preconditioning approach AU - Hiroyuki Kasai AU - Bamdev Mishra BT - Proceedings of The 33rd International Conference on Machine Learning DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-kasai16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 48 SP - 1012 EP - 1021 L1 - http://proceedings.mlr.press/v48/kasai16.pdf UR - https://proceedings.mlr.press/v48/kasai16.html AB - We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms in batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets. ER -
APA
Kasai, H. & Mishra, B.. (2016). Low-rank tensor completion: a Riemannian manifold preconditioning approach. Proceedings of The 33rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 48:1012-1021 Available from https://proceedings.mlr.press/v48/kasai16.html.

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