Question

In this problem, we derive the kernel K-means algorithm.

a) We start by getting some intuition about kernels. A kernel is a function

K(x,y) = ⟨ϕ(x),ϕ(y)⟩ where ϕ(.)
is a transformation to a high-dimensional space.
Consider the kernel Kl(x,y) = (xTy)^l, l ∈ {1,2,3,…}. For l=1, this is the linear kernel xTy corresponding to the identity transformation ϕ1(x) = x. Show that for l=2, if x=(x₁, x₂, …, xn)ᵀ, the kernel corresponds to the transformation ϕ2(x) = [x₁ rho₁(x) x₂ ϕ₁(x) ⋮ xn ϕ₁(x)]. This is a transformation from an n-dimensional to an n^2-dimensional space. What is the dimension of ϕl(x), the transformation associated with Kl(x,y)?

Answer 1

For
`\( l = 2 \)`, the kernel
`\( K_2(x, y) = (x^Ty)^2 \)` corresponds to the transformation
`\( \phi_2(x) = [x_1, \rho_1(x), x_2, \rho_1(x), \ldots, x_n, \rho_1(x)] \),` where
`\( \rho_1(x) \)` is the linear transformation
`\( x^T \)`. This transformation maps from an
`\( n \)`-dimensional space to an
`\( n^2 \)` -dimensional space. The dimension of
`\( \phi_2(x) \)` is
`\( n^2 \)`.

Step-by-step explanation:

The kernel
`\( K_2(x, y) = (x^Ty)^2 \)` can be expanded as
`\( K_2(x, y) = (x_1y_1 + x_2y_2 + \ldots + x_ny_n)^2 \)`. When we multiply this out, we get terms like
`\( x_i^2 \)`,
`\( y_i^2 \)`, and
`\( x_iy_i \)`, which are elements of the transformation
`\( \phi_2(x) \).` Therefore,
`\( \phi_2(x) \)` has
`\( n \)` original dimensions
`\( x_i \)` and
`\( n \)` additional dimensions
`\( \rho_1(x) = x^T \)`, resulting in a total of
`\( n + n^2 = n^2 \)`dimensions.

The transformation
`\( \phi_2(x) \)` is a mapping to a higher-dimensional space, and each element of
`\( \phi_2(x) \)` corresponds to a combination of the original features
`\( x_i \)` and the linear transformation
`\( x^T \)`. This transformation allows the algorithm to capture non-linear relationships between data points by implicitly mapping them to a higher-dimensional space. The increase in dimensionality facilitates the separation of data points in a way that might not be achievable in the original space, enhancing the effectiveness of the kernelized algorithm.