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tensor inversion and its application to the tensor equations with Einstein product

LINEAR & MULTILINEAR ALGEBRA 2019年第4期67卷 843-870页

作者： Liang, Mao-lin Zheng, Bing Zhao, Rui-juan Lanzhou Univ Sch Math & Stat Lanzhou 730000 Gansu Peoples R China Tianshui Normal Univ Sch Math & Stat Tianshui Peoples R China

Recently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542-570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University;2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.

关键词： Nung-Sing Sze Elementary tensor transformations tensor rank tensor determinant tensor inversion tensor equations

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Numeric tensor framework: Exploiting and extending Einstein notation

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JOURNAL OF COMPUTATIONAL SCIENCE 2016年 16卷 128-139页

作者： Harrison, Adam P. Joseph, Dileepan Univ Alberta Dept Elect & Comp Engn Edmonton AB Canada

The numeric tensor (NT) framework addresses and unifies a growing body of work on high-dimensional algebra and software for technical computing. Its NT algebra exploits and extends Einstein notation, offering unmatched capabilities, including N-dimensional operators, associativity, commutativity, entrywise products, and linear invertibility. High-performance C++ and MATLAB NT software allows practitioners to directly program with NT algebra. The advantages of NT algebra are highlighted using the example of canonical-polyadic (CP) tensor decomposition. Corresponding dense benchmarks demonstrate that the NT software matches or surpasses leading competitors, i.e., the MATLAB tensor Toolbox, NumPy, and Blitz++, while supporting a more general set of arithmetic operations. (C) 2016 Elsevier B.V. All rights reserved.

关键词： tensor algebra tensor computations tensor inversion C plus plus classes MATLAB classes

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tensor Quantization: High-Dimensional Data Compression

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2022年第8期32卷 5566-5580页

作者： Chang, Shih Yu Wu, Hsiao-Chun San Jose State Univ Dept Appl Data Sci San Jose CA 95192 USA Louisiana State Univ Sch Elect Engn & Comp Sci Baton Rouge LA 70803 USA

Quantization is an important technique to transform the input sample values from a large set (or a continuous range) into the output sample values in a small set (or a finite set). It has been applied broadly for lossy-data compression, pattern recognition, probability density estimation, and clustering. Vector quantization (VQ) is a prevalent image-compression technique, which treats image matrices as stretched vectors and then finds the representative stretched vectors accordingly for a given image data set. One can use tensor data representation to directly characterize the original two-dimensional image data rather than stretch the image matrix into a long vector so as to destroy the original two-dimensional data structure. In this work, we propose a new tensor quantization (TQ) framework which does not need to reduce the dimensionality of the original image data and destroy the original two-dimensional spatial relationship among data;these two serious drawbacks of vector quantization are well known. We first present tensor calculus and then propose a new parallel tensor-inversion algorithm for TQ thereupon. We also establish the pertinent theoretical proof to justify that our proposed new TQ approach is superior to the existing VQ approach especially as the image dimension becomes large. Finally, numerical experiments to evaluate the image-compression performances of VQ and TQ are demonstrated and their corresponding computational-complexities are also compared.

关键词： tensors Quantization (signal) Image coding Signal processing algorithms Matrix decomposition Transforms Speech recognition Vector quantization (VQ) tensor inversion tensor least squares (TLS) image compression tensor quantization (TQ)

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Multidimensional NMR inversion without Kronecker products: Multilinear inversion

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JOURNAL OF MAGNETIC RESONANCE 2016年 269卷 24-35页

作者： Medellin, David Ravi, Vivek R. Torres-Verdin, Carlos Univ Texas Austin Dept Petr & Geosyst Engn 200 E Dean KeetonMail Stop C0300 Austin TX 78712 USA

Multidimensional NMR inversion using Kronecker products poses several challenges. First, kernel compression is only possible when the kernel matrices are separable, and in recent years, there has been an increasing interest in NMR sequences with non-separable kernels. Second, in three or more dimensions, the singular value decomposition is not unique;therefore kernel compression is not welldefined for higher dimensions. Without kernel compression, the Kronecker product yields matrices that require large amounts of memory, making the inversion intractable for personal computers. Finally, incorporating arbitrary regularization terms is not possible using the Lawson-Hanson (LH) or the Butler Reeds-Dawson (BRD) algorithms. We develop a minimization-based inversion method that circumvents the above problems by using multilinear forms to perform multidimensional NMR inversion without using kernel compression or Kronecker products. The new method is memory efficient, requiring less than 0.1% of the memory required by the LH or BRD methods. It can also be extended to arbitrary dimensions and adapted to include non-separable kernels, linear constraints, and arbitrary regularization terms. Additionally, it is easy to implement because only a cost function and its first derivative are required to perform the inversion. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Multidimensional NMR inversion Two-dimensional NMR inversion Three-dimensional NMR inversion Multilinear inversion Bilinear inversion tensor inversion Multidimensional inverse Laplace

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Scale model seismicity : a detailed study of deformation localisation from laboratory acoustic emission data

Scale model seismicity : a detailed study of deformation loc...

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作者： Graham, Caroline C University of Edinburgh

学位级别：doctor

Acoustic emissions (AE) can provide information relating to the internal state of a deforming rock sample during laboratory testing and have been utilised to quantify damage progression for time-dependent failure modeling. However, the underlying physical mechanisms that produce AE in different materials and their evolution during the process of damage localisation are not fully understood, particularly in porous media. In order to investigate the sources of laboratory acoustic emissions, a moment tensor inversion was applied to data from triaxial compression experiments on Aue granite and Clashach sandstone. The moment tensor inversion was verified for granite, by comparison with results obtained using a more simplistic source analysis technique. In the non-porous Aue granite, AE sources exhibited a predominantly tensile behaviour in the early stages of AE activity. However, shear sources become dominant in the vicinity of the peak stress. In contrast, during deformation of the Clashach sandstone, which has a significant pre-existing porosity, AE sources are dominated by a collapse signature and generally involve a notable shear component. AE that have a predominantly shear mechanism are also a major contributor to the microscale deformation imaged by the technique, and dominate during shear localisation. A combination of correlation analysis and source analysis was used to elucidate the temporal and spatial evolution of the AE source mechanisms involved in the localisation process, as well as during a temporary hiatus in the progression to failure. The results support the concept that the cascade to failure requires the simultaneous involvement of a range of micromechanical behaviours to maintain the progression of localised damage, and eventual formation of a fault. Localisation of collapse mechanisms was not observed until the final approach to failure. Finally, AE sources produced during brittle deformation of the Clashach sandstone were characterised in detail

关键词： 550 Acoustic emissions deforming rock tensor inversion shear mechanism deformation

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Further results on the Drazin inverse of even-order tensors

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2020年第5期27卷

作者： Behera, Ratikanta Nandi, Ashish Kumar Sahoo, Jajati Keshari Indian Inst Sci Educ & Res Kolkata Dept Math & Stat Mohanpur 741246 India Univ Cent Florida Dept Math Orlando FL 32826 USA BITS Pilani Dept Math KK Birla Goa Campus Sancoale Goa India

The notion of the Drazin inverse of an even-order tensors with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by establishing a few characterizations of the Drazin inverse and W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution of multilinear systems by using the Drazin inverse and iterative (higher order Gauss-Seidel) method of tensors. Besides these, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.

关键词： W-weighted Drazin inverse Drazin inverse Einstein product multilinear system tensor inversion

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