Question

A kernel is an efficient way to write out an inner product between two feature vectors computed from a pair of input vectors as follows:

K(x, y) = φ(x)Tφ(y).

Assume that both inputs are 2-dimensional and write out the explicit mapping

φ that mimics the kernel value for a 3rd-order polynomial kernel as follows:

K(x, y) = (xTy + 1)^3.

Answer 1

`\[ \phi(x) = \begin{bmatrix} x_1^2 \\ √(2)x_1x_2 \\ x_2^2 \\ √(2)x_1 \\ √(2)x_2 \\ 1 \end{bmatrix} \]`

Step-by-step explanation:

The 3rd-order polynomial kernel can be expressed as:

`\[ K(x, y) = (x^T y + 1)^3 \].`

To find the explicit mapping \(\phi\), we can use the binomial expansion:

`\[ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \].`

For our kernel,
`\(a = 1\)` and
`\(b = x^T y\)`. Expanding, we get:

`\[ K(x, y) = (1 + x^T y)^3 = 1 + 3(x^T y) + 3(x^T y)^2 + (x^T y)^3 \].`

Now, express this in terms of the feature vector
`\(\phi(x)\)`:

`\[ \phi(x)^T \phi(y) = 1 + √(3)x_1y_1 + 3x_1x_2y_1y_2 + √(3)x_2y_2 + (x^T y)^3 \].`

Comparing coefficients, we find
`\(\phi(x)\)`:

`\[ \phi(x) = \begin{bmatrix} x_1^2 \\ √(2)x_1x_2 \\ x_2^2 \\ √(2)x_1 \\ √(2)x_2 \\ 1 \end{bmatrix} \].`

This mapping
`\(\phi\)` allows us to compute the inner product efficiently as
`\(K(x, y) = \phi(x)^T \phi(y)\).`

Answer 2

A kernel is a function that allows us to compute the inner product between two feature vectors. The explicit mapping for a 3rd-order polynomial kernel can be written as: φ(x) = (xTy + 1)^3.

Step-by-step explanation:

A kernel is a function that allows us to compute the inner product between two feature vectors. The explicit mapping φ for a 3rd-order polynomial kernel can be written as: φ(x) = (xTy + 1)3

This means that the mapping φ takes each 2-dimensional input vector x and y, computes their inner product xTy, adds 1, and then raises the result to the power of 3.