Show that the second-order polynomial kernel (i.e., xi , xj = ???? x|i xj + 1 2) corresponds to the following mapping function hx

Show that the second-order polynomial kernel (i.e., ¹xi , xj º =

????

x|i xj + 1

2) corresponds to the following mapping function h¹xº from Rd to Rd¹d+1º:

Then, consider the mapping function for a third-order polynomial kernel and a general pth order polynomial kernel.

x = x1 X2 Px -> x 2x1.x2 2xd-1xd 2x1 2xd