Low-CP-Rank Tensor Completion via Practical Regularization

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, 10.1109/ACSSC.2018.8645405, 2018), a sparse regularization minimization problem via $$\ell _1$$ 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.

A Customized Data Fusion Tensor Approach for Interval-Wise Missing Network Volume Imputation

“…In this tensor decomposition approaches, the latter term of equation ( 1) adds the regularization term. On this basis, inspired by Acar, et al [31] and Jiang, et al [17], we extra add customized mode averaging in the original regularization term. This adjustment is used to reduce the impact of diverse disturbances caused by different segment grades and different missing lengths and make full use of the unique characteristics of CL data as an auxiliary role in process of our interval-wise missing scenario.…”

“…The diverse disturbances from available data dimensions can significantly affect the imputation performance. Even if the addition of regularization items has been used to alleviate this impact [16][17][18], it is still limited by the non-convex problems in regularized parameter solving.…”

Streaming probabilistic tensor train decomposition

“…Effective numerical techniques, such as CANDECOMP/PARAFAC (CP) decomposition [10,11,12] and Tucker decomposition [13,14] are the most commonly used tensor decomposition approaches and have been proposed to compress full tensors and to obtain their low-rank representations. CP decomposition approximates a tensor by a sum of rank-one tensors, while Tucker decomposition decomposes a tensor into a core tensor and several factor matrices.…”