In order to facilitate the distinctions of scalars, matrices, and tensors, the following notations are used.
Therefore, to extract all matrices and tensors for the package model, we need first only to invert a single (3times3) matrix, and then, for each matrix or tensor function, we need to calculate (8) once.
Following the convention of multilinear algebra, we denote vectors, matrices, and tensors by lowercase boldface letters (e.g., M), uppercase boldface letters (e.g., M), and calligraphic letters (e.g., ℳ ), respectively, in this article.
In order to facilitate the distinction between scalars, matrices, and tensors, the following notation is used: Scalars are denoted as italic letters (a, b,..., A, B,..., α, β,...), column vectors as lower-case bold-face letters (a, b,...), matrices as bold-face capitals (A, B,...), and tensors are written as bold-face calligraphic letters.
To facilitate the distinction between scalars, vectors, matrices, and tensors, the following notation is used throughout the manuscript: scalars are represented by italic letters, vectors by lowercase bold-faced letters, matrices by uppercase bold-faced letters, and tensors as bold-faced calligraphic letters.
In the context of Big Data, matrix and tensor data recovery via an online rank minimization process [31] was recently proposed for scalable imputation of missing data.
We thank GH Golub for introducing us to matrix and tensor computations, and the American Institute of Mathematics in Palo Alto and Stanford University for hosting the 2004 Workshop on Tensor Decompositions and the 2006 Workshop on Algorithms for Modern Massive Data Sets, respectively, where some of this work was done.
We index matrix and tensor indices by using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $i,j,k=0,1\in \mathbb {Z}_{2}$\end{document } i, j, k = 0, 1 ∈ ℤ 2 and allow multiplication × in the ring of integers.
Scalars, column vectors, matrices, and high‐order tensors are denoted by lowercase, boldface lowercase, boldface uppercase, and calligraphic uppercase letters, e.g., a, a, A, and, respectively.
In addition, we incorporate forward-backward-averaging and find a similar link between the real-valued matrix-based and tensor-based subspace estimation.
The extension is based on an algebraic link between matrix-based and tensor-based subspace estimates via a Kronecker structured projection.
