张量分解和应用（1）_tamara g. kolda

注： 此文章为对Tamara G.Kolda和Brett W.Bader的<Tensor Decompositions and Applications>摘抄和分析。

介绍

注：定义不敢乱翻译。

The order of a tensor is the number of dimensions, also known as ways or modes.

A colon is used to indicate all elements of a mode.

Fibers are the higher-order analogue of matrix rows and columns.
A fiber is defined by fixing every index but one.

Slices are two-dimensional sections of a tensor, defined by fixing all but two indices.

The norm of a tensor $\mathcal{X}\in {\mathbb{R}}^{I_1\times I_2\times \cdots \times I_N}$ is the square root of the sum of the squares of all its elements.
$$||\mathcal{X}||=\sqrt{\sum _{i_1=1}^{I_1}\sum _{i_2=1}^{I_2}\cdots \sum _{i_N=1}^{I_N}{x}_{i_1{i}_2\cdots i_N}^2}$$

The inner product of two same-sized tensors $\mathcal{X},\mathcal{Y}\in {\mathbb{R}}^{I_1\times I_2\times \cdots \times I_N}$ is the sum of the products of their entries, i.e.,
$$<\mathcal{X},\mathcal{Y}>=\sum _{i_1=1}^{I_1}\sum _{i_2=1}^{I_2}\cdots \sum _{i_N=1}^{I_N}{x}_{i_1{i}_2\cdots i_N}{y}_{i_1{i}_2\cdots i_N}.$$

秩一张量

An N-way tensor $\mathcal{X}\in {\mathbb{R}}^{I_1\times I_2\times \cdots \times I_N}$ is rank one if it can be written as the outer product of N vectors, i.e.,
$$\mathcal{X}=a^{(1)}\circ a^{(2)}\circ \cdots \circ a^{(3)}.$$
The symbol “$\circ$” represents the vector outer product.

张量对称性

对角张量

矩阵化(将一个张量转化为矩阵)

Matricization, also known as unfolding or flattening, is the process of recording the elements of an N-way array into a matrix.

张量乘法(n模态积)

矩阵的Kronecker、Khatri-Rao、和Hadamard积