Low-CP-Rank Tensor Completion via Practical Regularization

• Published in: Journal of scientific computing 2022-04, Vol.91 (1), p.18, Article 18
• Main Authors: Jiang, Jiahua, Sanogo, Fatoumata, Navasca, Carmeliza
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; Approximation; Computational Mathematics and Numerical Analysis; Computer science; Data science; Decomposition; Image reconstruction; Inverse problems; Linear algebra; Machine learning; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Methods; Missing data; Model reduction; Optimization techniques; Parameters; Regularization; Robustness (mathematics); Tensors; Theoretical
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01789-9

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 ℓ 1 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.