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Exercise 1: Feature Map Let

`x,yinR^(2)`

, i.e.

`x=(x_(1),x_(2))`

and

`y=(y_(1),y_(2))`

. Define polynomial kernel

`K:R^(2)\times R^(2)->R`

by

`K(x,y)=(x^(TT)y)^(2)`

. Find the feature map

`\phi :R^(2)->R^(D)`

. Please specify the dimension

`D`

. Consider the same kernel function

`K(x,y)=(x^(TT)y)^(2)`

which is defined on

`R^(3)\times R^(3)`

in this part, i.e.

`x,yinR^(3)`

. What is the feature map

`\phi :R^(3)->R^(D)`

? What is the dimension

`D`

? In general, suppose that

`x,yinR^(d)`

and polynomial kernel is defined as

`K(x,y)=(x^(TT)y)^(2)`

. Without writing the feature map

`\phi :R^(d)->R^(D)`

explicitly, write down the dimension

`D`

as a function of input dimension

`d`

.