arXiv subjects

Zhengbang Cao

Publications and source records attributed to Zhengbang Cao.

6 recordsLinked to original sources

arXiv2023

An L-DEIM Induced High Order Tensor Interpolatory Decomposition

This paper derives the CUR-type factorization for tensors in the Tucker format based on a new variant of the discrete empirical interpolation method known as L-DEIM. This novel sampling technique allows us to construct an efficient algorithm for computing the structure-preserving decomposition, which significantly reduces the computational cost. For large-scale datasets, we incorporate the random sampling technique with the L-DEIM procedure to further improve efficiency. Moreover, we propose randomized algorithms for computing a hybrid decomposition, which yield interpretable factorization and provide a smaller approximation error than the tensor CUR factorization. We provide comprehensive analysis of probabilistic errors associated with our proposed algorithms, and present numerical results that demonstrate the effectiveness of our methods.

math.NA↗

arXiv2023

Randomized GCUR decompositions

By exploiting the random sampling techniques, this paper derives an efficient randomized algorithm for computing a generalized CUR decomposition, which provides low-rank approximations of both matrices simultaneously in terms of some of their rows and columns. For large-scale data sets that are expensive to store and manipulate, a new variant of the discrete empirical interpolation method known as L-DEIM, which needs much lower cost and provides a significant acceleration in practice, is also combined with the random sampling approach to further improve the efficiency of our algorithm. Moreover, adopting the randomized algorithm to implement the truncation process of restricted singular value decomposition (RSVD), combined with the L-DEIM procedure, we propose a fast algorithm for computing an RSVD based CUR decomposition, which provides a coordinated low-rank approximation of the three matrices in a CUR-type format simultaneously and provides advantages over the standard CUR approximation for some applications. We establish detailed probabilistic error analysis for the algorithms and provide numerical results that show the promise of our approaches.

math.NA↗

arXiv2021

Incremental tensor regularized least squares with multiple right-hand sides

Solving linear discrete ill-posed problems for third order tensor equations based on a tensor t-product has attracted much attention. But when the data tensor is produced continuously, current algorithms are not time-saving. Here, we propose an incremental tensor regularized least squares (t-IRLS) algorithm with the t-product that incrementally computes the solution to the tensor regularized least squares (t-RLS) problem with multiple lateral slices on the right-hand side. More specifically, we update its solution by solving a t-RLS problem with a single lateral slice on the right-hand side whenever a new horizontal sample arrives, instead of solving the t-RLS problem from scratch. The t-IRLS algorithm is well suited for large data sets and real time operation. Numerical examples are presented to demonstrate the efficiency of our algorithm.

math.NA↗

arXiv2021

Perturbation analysis for t-product based tensor inverse, Moore-Penrose inverse and tensor system

This paper establishes some perturbation analysis for the tensor inverse, the tensor Moore-Penrose inverse and the tensor system based on the t-product. In the settings of structured perturbations, we generalize the Sherman-Morrison-Woodbury (SMW) formula to the t-product tensor scenarios. The SMW formula can be used to perform sensitivity analysis for a multilinear system of equations.

math.NA↗

arXiv2021

On some tensor inequalities based on the t-product

In this work, we investigate the tensor inequalities in the tensor t-product formalism. The inequalities involving tensor power are proved to hold similarly as standard matrix scenarios. We then focus on the tensor norm inequalities. The well-known arithmetic-geometric mean inequality, H{\" o}lder inequality, and Minkowski inequality are generalized to tensors. Furthermore, we obtain some t-eigenvalue inequalities.

math.NA↗

arXiv2021

CS decomposition and GSVD for tensors based on the T-product

This paper derives the CS decomposition for orthogonal tensors (T-CSD) and the generalized singular value decomposition for two tensors (T-GSVD) via the T-product. The structures of the two decompositions are analyzed in detail and are consistent with those for matrix cases. Then the corresponding algorithms are proposed respectively. Finally, T-GSVD can be used to give the explicit expression for the solution of tensor Tikhonov regularization. Numerical examples demonstrate the effectiveness of T-GSVD in solving image restoration problems.

math.NA↗