Tensor¶

Calculation¶

Kronecker product¶

If \(A\) is an \(m \times n\) matrix and \(B\) is a \(p \times q\) matrix, then the Kronecker product \(A \otimes B\) is the \(pm \times qn\) block matrix:

more explicitly:

Example

vector case (row or column form):

\[\begin{align} a\otimes b &= (\begin{array}{ccc} a_1 & \dots & a_n \end{array}) \otimes (\begin{array}{ccc} b_1 & \dots & b_m \end{array}) \\ &= (\begin{array}{ccccccc} a_1 b_1 & \dots & a_1 b_m & \dots & a_n b_1 & \dots & a_n b_m \end{array}) \end{align}\]

and

\[\begin{align} a\otimes b &= (\begin{array}{ccc} a_1 & \dots & a_n \end{array}) \otimes \left(\begin{array}{c} b_1 \\ \dots \\ b_m \end{array}\right) \\ &= \left( \begin{array}{ccccccc} a_1 b_1 & \dots & a_n b_1 \\ \vdots & \ddots & \vdots \\ a_1 b_m & \dots & a_n b_m \end{array} \right) \end{align}\]

Khatri-Rao product¶

In mathematics, the Khatri–Rao product is defined as:

Example

if A and B both are 2 × 2 partitioned matrices e.g.:

we obtain:

Hadamard product¶

Only for matrices of equal size. Also known as the element-wise product, entrywise product, or Schur product.

\[ (A\odot B)_{ij} = A_{ij} B_{ij} \]

Example

n-mode tensor-matrix product¶

The n-mode (matrix) product of a tensor \(\mathcal A \in \mathbb{R}^{I_1 \times I_2 \times \ldots \times I_N}\) with a matrix \(\mathbf U \in \mathbb{R}^{J \times I_n}\) is denoted by \(\mathcal{A} \times_n \mathbf U\) and is of size \(I_1 \times \ldots I_{n-1} \times J \times I_{n+1} \times \ldots \times N\). Elementwise, we have

\[\left(\mathcal A \times_n \mathbf U \right)_{i_1\ldots i_{n-1}\ j\ i_{n+1} \ \ldots \ i_N} = \sum_{i_n=1}^{I_n} a_{i_1i_2\ldots i_N }\ u_{j i_n}.\]

For multiple n-mode product the order is irrelevant:

\[ \text{for } n\neq m: A \times_m U \times_n V = A \times_n V \times_m U \]

For multiple n-mode product with the same n the order is relevant:

\[ A \times_n U \times_n V = A \times_n (VU) \]

最后更新: March 8, 2023