Non-linear Classificaton

Parabolic Transformations

This section is for kernel that follow parabolic functions (i.e., shapes that are conic or in general quadric).

Ellipsoid and Hyperboloid

The polynomial draws a ellipsoid or hyperboloid plane in \(\mathbb R^d\).

If the principle axis of the plane are aligned with the coorindate axis then the polynomial is without the cross-terms where \(\Phi: \mathbb R^d \rightarrow \mathbb R^{2d}\),

\[\begin{split} \Phi(X_i) = \begin{bmatrix} X_i^{2}\\ X_i \end{bmatrix} \end{split}\]

If the planes aren’t axis-aligned then (\(\Phi : \mathbb R^d \rightarrow \mathbb R^\frac{d^2 + 3d}{2}\)),

\[\begin{split} \Phi(X_i) = \begin{bmatrix} X_i^{2}\\ X_iX_{j\neq i}\\ X_i \end{bmatrix} \end{split}\]