Tensor ring (TR) decomposition has made remarkable achievements in numerous high-order data processing tasks. However, the current alternating least squares (ALS)- and singular value decomposition (SVD)-based algorith...
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Tensor ring (TR) decomposition has made remarkable achievements in numerous high-order data processing tasks. However, the current alternating least squares (ALS)- and singular value decomposition (SVD)-based algorithms for TR decomposition, i.e., TR-ALS and TR-SVD, especially the former, are computationally expensive, making them unfriendly for large-scale data processing. This paper adopts three strategies to propose a novel fast TR decomposition algorithm: (1) Use a more efficient lanczos bidiagonalization algorithm SVD to generate the TR core tensors. (2) Exploit the hierarchical strategy to generate the TR core tensors parallel. (3) Employ new reshaping and unfolding operations to reduce the dimensionality of the data to generate TR core tensors. By incorporating these three strategies, we propose the TR-Hlanczosalgorithm for fast TR decomposition. This algorithm seamlessly produces the TR core tensors through the lanczos bidiagonalization algorithm in a hierarchical manner. The effectiveness of the proposed TR-Hlanczosalgorithm is demonstrated through experimental results on both highly oscillatory functions and real-world datasets. For instance, when dealing with data of size 505, 5 , TR-Hlanczos is nearly 561 times and 18 times faster algorithms based on ALS and SVD, respectively.
When projection methods are employed to regularize linear discrete ill-posed problems, one implicitly assumes that the discrete Picard condition (DPC) is somehow inherited by the projected problems. In this paper we s...
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When projection methods are employed to regularize linear discrete ill-posed problems, one implicitly assumes that the discrete Picard condition (DPC) is somehow inherited by the projected problems. In this paper we show that, under some assumptions, the DPC holds for the projected uncorrupted systems computed by various Krylov subspace methods. By exploiting the inheritance of the DPC, some estimates on the behavior of the projected problems are also derived. Numerical examples are provided in order to illustrate the accuracy of the derived estimates.
Any real matrix A has associated with it the real symmetric matrix \[ B \equiv \left(\begin{array}{*{20}c} 0 \\ {A^T } \\ \end{array} \begin{array}{*{20}c} A \\ 0 \\ \end{array} \right) \] whose positive eigenva...
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Any real matrix A has associated with it the real symmetric matrix \[ B \equiv \left(\begin{array}{*{20}c} 0 \\ {A^T } \\ \end{array} \begin{array}{*{20}c} A \\ 0 \\ \end{array} \right) \] whose positive eigenvalues are the nonzero singular values of A. Using B and our lanczosalgorithms for computing eigenvalues and eigenvectors of very large real symmetric matrices, we obtain an algorithm for computing singular values and singular vectors of large sparse real matrices. This algorithm provides a means for computing the largest and the smallest or even all of the distinct singular values of many matrices
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