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

2020 ◽  
Vol 30 (04) ◽  
pp. 2050015
Author(s):  
Vedran Novaković
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Data Layout ◽  
Arbitrary Length ◽  
Straightforward Method ◽  
Singular Value Decompositions ◽  
Matrix Formula ◽  
Improving Accuracy ◽  
Single Matrix ◽  
Svd Method

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.

scholarly journals A Star Sensor On-Orbit Calibration Method Based on Singular Value Decomposition

Sensors ◽  
2019 ◽  
Vol 19 (15) ◽  
pp. 3301 ◽  
Author(s):  
Liang Wu ◽  
Qian Xu ◽  
Janne Heikkilä ◽  
Zijun Zhao ◽  
Liwei Liu ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Singular Values ◽  
Calibration Method ◽  
Angular Distance ◽  
Singular Value ◽  
Optical Parameters ◽  
Star Sensor ◽  
Calibration Methods ◽  
Value Decomposition ◽  
Svd Method

The navigation accuracy of a star sensor depends on the estimation accuracy of its optical parameters, and so, the parameters should be updated in real time to obtain the best performance. Current on-orbit calibration methods for star sensors mainly rely on the angular distance between stars, and few studies have been devoted to seeking new calibration references. In this paper, an on-orbit calibration method using singular values as the calibration reference is introduced and studied. Firstly, the camera model of the star sensor is presented. Then, on the basis of the invariance of the singular values under coordinate transformation, an on-orbit calibration method based on the singular-value decomposition (SVD) method is proposed. By means of observability analysis, an optimal model of the star combinations for calibration is explored. According to the physical interpretation of the singular-value decomposition of the star vector matrix, the singular-value selection for calibration is discussed. Finally, to demonstrate the performance of the SVD method, simulation calibrations are conducted by both the SVD method and the conventional angular distance-based method. The results show that the accuracy and convergence speed of both methods are similar; however, the computational cost of the SVD method is heavily reduced. Furthermore, a field experiment is conducted to verify the feasibility of the SVD method. Therefore, the SVD method performs well in the calibration of star sensors, and in particular, it is suitable for star sensors with limited computing resources.

A QUINTET SINGULAR VALUE DECOMPOSITION THROUGH EMPIRICAL MODE DECOMPOSITIONS

2014 ◽  
Vol 06 (02n03) ◽  
pp. 1450010
Author(s):  
MIN-SUNG KOH
Keyword(s):  
Singular Value Decomposition ◽  
Speech Enhancement ◽  
Singular Values ◽  
Singular Value ◽  
Diagonal Matrix ◽  
Low Rank ◽  
Orthogonal Matrices ◽  
Low Rank Approximations ◽  
Single Matrix ◽  
Value Decomposition

A particular quintet singular valued decomposition (Quintet-SVD) is introduced in this paper via empirical mode decompositions (EMDs). The Quintet-SVD results in four specific orthogonal matrices with a diagonal matrix of singular values. Furthermore, this paper shows relationships between the Quintet-SVD and traditional SVD, generalized low rank approximations of matrices (GLRAM) of one single matrix, and EMDs. One application of the Quintet-SVD for speech enhancement is shown and compared with an application of traditional SVD.

Feature frequency extraction based on singular value decomposition and its application on rotor faults diagnosis

2019 ◽  
Vol 25 (6) ◽  
pp. 1246-1262 ◽  
Author(s):  
Zhen Li ◽  
Weiguang Li ◽  
Xuezhi Zhao
Keyword(s):  
Singular Value Decomposition ◽  
Singular Values ◽  
Frequency Component ◽  
Singular Value ◽  
Frequency Separation ◽  
Rotor Faults ◽  
Frequency Components ◽  
Value Decomposition ◽  
Svd Method ◽  
Feature Frequency

The selection of effective singular values using the singular value decomposition (SVD) method has always been a hot topic. In this paper, we found that there was a special relationship between effective singular values and feature frequency components. Theoretical derivations illustrated that each frequency component produced two adjacent nonzero singular values with one ranking another closely. Size of singular values was directly proportional to amplitude of feature frequency. The number of singular values was only related to the number of feature frequency components. For these discoveries, a novel feature frequency separation method based on SVD was proposed, through which axis orbits of large rotating machines were readily purified. The results show that the algorithm was very accurate in feature frequency extraction.

A New Modified SVD Method Applied in 3D Acoustic Temperature Field Reconstruction

2013 ◽  
Vol 732-733 ◽  
pp. 218-223 ◽  
Author(s):  
Wei Yang ◽  
Xiao Liang Zhu
Keyword(s):  
Temperature Field ◽  
Singular Value Decomposition ◽  
Singular Values ◽  
Singular Value ◽  
Temperature Fields ◽  
Time Data ◽  
Field Reconstruction ◽  
Value Decomposition ◽  
Svd Method ◽  
Temperature Field Reconstruction

In order to solve the ill-posed problem in 3D acoustic temperature field reconstruction, the paper proposed a new modified singular value decomposition (SVD) method.According to their reliability ,singular values were divided into three parts and got various degree of modification respectively. To verify the performance of the new algorithm based on the modified SVD method,two model temperature fields were reconstructed when the signal-to-noise ratio (SNR) of sound flight-time data was 50dB , 40dB and 30dB respectively.And the results were compared with those based on routine Truncated singular value decomposition (TSVD) and Tikhonov methods. Simulation results show that the new algorithm has higher precision, better anti-noise ability than the routine methods and it is more suitable for the complex temperature fields reconstruction, thus it is expected to be used for temperature field reconstruction on-line.

scholarly journals Random Toeplitz matrices: The condition number under high stochastic dependence

2021 ◽  
Author(s):  
Paulo Manrique-Mirón
Keyword(s):  
Generating Function ◽  
Condition Number ◽  
Toeplitz Matrix ◽  
Singular Values ◽  
Moment Generating Function ◽  
Circulant Matrix ◽  
Singular Value ◽  
The Matrix ◽  
Matrix Formula ◽  
The Moment

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

scholarly journals Distributed Singular Value Decomposition Method for Fast Data Processing in Recommendation Systems

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2284
Author(s):  
Krzysztof Przystupa ◽  
Mykola Beshley ◽  
Olena Hordiichuk-Bublivska ◽  
Marian Kyryk ◽  
Halyna Beshley ◽  
...  
Keyword(s):  
Distributed Systems ◽  
Singular Value Decomposition ◽  
Data Processing ◽  
Message Passing ◽  
Message Passing Interface ◽  
Recommendation Systems ◽  
Singular Value ◽  
Singular Value Decomposition Method ◽  
Value Decomposition ◽  
Svd Method

The problem of analyzing a big amount of user data to determine their preferences and, based on these data, to provide recommendations on new products is important. Depending on the correctness and timeliness of the recommendations, significant profits or losses can be obtained. The task of analyzing data on users of services of companies is carried out in special recommendation systems. However, with a large number of users, the data for processing become very big, which causes complexity in the work of recommendation systems. For efficient data analysis in commercial systems, the Singular Value Decomposition (SVD) method can perform intelligent analysis of information. With a large amount of processed information we proposed to use distributed systems. This approach allows reducing time of data processing and recommendations to users. For the experimental study, we implemented the distributed SVD method using Message Passing Interface, Hadoop and Spark technologies and obtained the results of reducing the time of data processing when using distributed systems compared to non-distributed ones.

scholarly journals Asymptotic Behavior of Singular and Entropy Numbers for Some Riemann–Liouville Type Operators

2001 ◽  
Vol 8 (2) ◽  
pp. 323-332
Author(s):  
A. Meskhi
Keyword(s):  
Asymptotic Behavior ◽  
Integral Operators ◽  
Integral Transforms ◽  
Singular Value ◽  
Entropy Numbers ◽  
Liouville Type ◽  
Singular Value Decompositions

Abstract The asymptotic behavior of the singular and entropy numbers is established for the Erdelyi–Köber and Hadamard integral operators (see, e.g., [Samko, Kilbas and Marichev, Integrals and derivatives. Theoryand Applications, Gordon and Breach Science Publishers, 1993]) acting in weighted L 2 spaces. In some cases singular value decompositions are obtained as well for these integral transforms.

scholarly journals Exact relation between singular value and eigenvalue statistics

2016 ◽  
Vol 05 (04) ◽  
pp. 1650015 ◽  
Author(s):  
Mario Kieburg ◽  
Holger Kösters
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Specific Class ◽  
Rotation Invariant ◽  
Exact Relation ◽  
Finite Matrix ◽  
Eigenvalue Statistics ◽  
Probability Densities ◽  
Analytical Relations ◽  
Matrix Spaces

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

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