Low-CP-rank tensor completion via practical regularization

Jiahua Jiang, Fatoumata Sanogo, Carmeliza Navasca

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Abstract

Dimension reduction is analytical methods for reconstructing high-order tensors that the intrinsic rank of these tensor data is relatively much smaller than the dimension of the ambient measurement space. Typically, this is the case for most real world datasets in signals, images and machine learning. The CANDECOMP/PARAFAC (CP, aka Canonical Polyadic) tensor completion is a widely used approach to find a low-rank approximation for a given tensor. In the tensor model (Sanogo and Navasca in 2018 52nd Asilomar conference on signals, systems, and computers, pp 845–849, https://doi.org/10.1109/ACSSC.2018.8645405, 2018), a sparse regularization minimization problem via ℓ1 norm was formulated with an appropriate choice of the regularization parameter. The choice of the regularization parameter is important in the approximation accuracy. Due to the emergence of the massive data, one is faced with an onerous computational burden for computing the regularization parameter via classical approaches (Gazzola and Sabaté Landman in GAMM-Mitteilungen 43:e202000017, 2020) such as the weighted generalized cross validation (WGCV) (Chung et al. in Electr Trans Numer Anal 28:2008, 2008), the unbiased predictive risk estimator (Stein in Ann Stat 9:1135–1151, 1981; Vogel in Computational methods for inverse problems, 2002), and the discrepancy principle (Morozov in Doklady Akademii Nauk, Russian Academy of Sciences, pp 510–512, 1966). In order to improve the efficiency of choosing the regularization parameter and leverage the accuracy of the CP tensor, we propose a new algorithm for tensor completion by embedding the flexible hybrid method (Gazzola in Flexible krylov methods for lp regularization) into the framework of the CP tensor. The main benefits of this method include incorporating the regularization automatically and efficiently as well as improving accuracy in the reconstruction and algorithmic robustness. Numerical examples from image reconstruction and model order reduction demonstrate the efficacy of the proposed algorithm.
Original languageEnglish
Article number18
Number of pages20
JournalJournal of Scientific Computing
Volume91
Issue number1
Early online date1 Mar 2022
DOIs
Publication statusPublished - Apr 2022

Keywords

  • Tensor
  • Tensor completion
  • Model order reduction
  • Regularization
  • Hybrid projection methods

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