Non-linear Classificaton

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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 $R^{d}$ .

If the principle axis of the plane are aligned with the coorindate axis then the polynomial is without the cross-terms where $Φ : R^{d} \to R^{2 d}$ ,

\begin{array}{r} Φ (X_{i}) = [\begin{array}{c} X_{i}^{2} \\ X_{i} \end{array}] \end{array}

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

\begin{array}{r} Φ (X_{i}) = [\begin{array}{c} X_{i}^{2} \\ X_{i} X_{j \neq i} \\ X_{i} \end{array}] \end{array}