НАСЛЕДОВАНИЕ СИНГУЛЯРНЫХ ВЕКТОРОВ ПРИ ПОПОЛНЕНИИ МАТРИЦЫ СТОЛБЦОМ

Let the matrix A1 be obtained from the matrix A by adding a column to it on the right. The possibility of inheritance of singular numbers and the corresponding singular vectors when passing from matrix A to matrix A1 is investigated. The singular value decompositions of the matrix A are based on the scalar and vector properties of the square symmetric matrices ATA and AAT. The article deals with the singular value decomposition of the matrix A, which has more rows than columns, and the decomposition is based on the analysis of the ATA matrix.

Batched Computation of the Singular Value Decompositions of Order Two by the AVX-512 Vectorization

In this paper a vectorized algorithm for simultaneously computing up to eight singular value decompositions (SVDs, each of the form [Formula: see text]) of real or complex matrices of order two is proposed. The algorithm extends to a batch of matrices of an arbitrary length [Formula: see text], that arises, for example, in the annihilation part of the parallel Kogbetliantz algorithm for the SVD of matrices of order [Formula: see text]. The SVD method for a single matrix of order two is derived first. It scales, in most instances error-free, the input matrix [Formula: see text] such that the scaled singular values cannot overflow whenever the elements of [Formula: see text] are finite, and then computes the URV factorization of the scaled matrix, followed by the SVD of the non-negative upper-triangular middle factor. A vector-friendly data layout for the batch is then introduced, where the same-indexed elements of each of the input and the output matrices form vectors, and the algorithm’s steps over such vectors are described. The vectorized approach is shown to be about three times faster than processing each matrix in the batch separately, while slightly improving accuracy over the straightforward method for the [Formula: see text] SVD.

Fast dictionary learning for noise attenuation of multidimensional seismic data

SUMMARY The K-SVD algorithm has been successfully utilized for adaptively learning the sparse dictionary in 2-D seismic denoising. Because of the high computational cost of many singular value decompositions (SVDs) in the K-SVD algorithm, it is not applicable in practical situations, especially in 3-D or 5-D problems. In this paper, I extend the dictionary learning based denoising approach from 2-D to 3-D. To address the computational efficiency problem in K-SVD, I propose a fast dictionary learning approach based on the sequential generalized K-means (SGK) algorithm for denoising multidimensional seismic data. The SGK algorithm updates each dictionary atom by taking an arithmetic average of several training signals instead of calculating an SVD as used in K-SVD algorithm. I summarize the sparse dictionary learning algorithm using K-SVD, and introduce SGK algorithm together with its detailed mathematical implications. 3-D synthetic, 2-D and 3-D field data examples are used to demonstrate the performance of both K-SVD and SGK algorithms. It has been shown that SGK algorithm can significantly increase the computational efficiency while only slightly degrading the denoising performance.

A parallel hierarchical blocked adaptive cross approximation algorithm

This article presents a low-rank decomposition algorithm based on subsampling of matrix entries. The proposed algorithm first computes rank-revealing decompositions of submatrices with a blocked adaptive cross approximation (BACA) algorithm, and then applies a hierarchical merge operation via truncated singular value decompositions (H-BACA). The proposed algorithm significantly improves the convergence of the baseline ACA algorithm and achieves reduced computational complexity compared to the traditional decompositions such as rank-revealing QR. Numerical results demonstrate the efficiency, accuracy, and parallel scalability of the proposed algorithm.