I am trying to follow some lecture notes about computing a kernel from the inner product of two features maps. I don't understand how this equation:
$<\phi(x),\phi(z)>=1+\sum_{i=1}^{d}x_{i}z_{i} +\sum_{i,j\in\{1....d\}}x_{i}x_{j}z_{i}z_{j} +\sum_{i,j,k\in\{1,...,d\}}x_{i}x_{j}x_{k}z_{i}z_{j}z_{k}$
is equal to this:
$=1 + \sum_{i=1}^{d}x_{i}z_{i}+\left(\sum_{i=1}^{d}x_{i}z_{i}\right)^{2}+\left(\sum_{i=1}^{d}x_{i}z_{i}\right)^{3}$
I understand the first two parts of the sum, but not the last two. Can someone explain how:
$\sum_{i,j,k\in\{1,...,d\}}x_{i}x_{j}x_{k}z_{i}z_{j}z_{k} = \left(\sum_{i=1}^{d}x_{i}z_{i}\right)^{3}$
This equation appears on page 5 of the lecture notes located here [Stanford notes].
