Parallel matrix factorization for low-rank tensor completion

Yangyang Xu; Ruru Hao; Wotao Yin; Zhixun Su

doi:10.3934/ipi.2015.9.601

Article Contents

2015, Volume 9, Issue 2: 601-624. Doi: 10.3934/ipi.2015.9.601

This issue Previous Article Determining an obstacle by far-field data measured at a few spots Next Article

Parallel matrix factorization for low-rank tensor completion

1.
Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian
3.
Department of Mathematics, University of California, Los Angeles, CA 90095-1555

Received: November 30, 2013

Revised: September 30, 2014

Published: April 2015

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Abstract

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.
Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.

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Mathematics Subject Classification: Primary: 94A08, 94A12; Secondary: 90C90.

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