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Learning from Data

Exercise 3.11

Answers

Feature transformation $Φ (x) = (1, x_{1}^{2}, x_{2}^{2})$ , a hyperplane in $Z$ can be expressed as

\tilde{w} z = {\tilde{w}}_{0} z_{0} + {\tilde{w}}_{1} z_{1} + {\tilde{w}}_{2} z_{2} = {\tilde{w}}_{0} + {\tilde{w}}_{1} x_{1}^{2} + {\tilde{w}}_{2} x_{2}^{2} = 0 .

(a): $| {\tilde{w}}_{1} | x_{1}^{2} - | {\tilde{w}}_{1} | x_{2}^{2} = c$ : Hyperbola
(b): ${\tilde{w}}_{0} + {\tilde{w}}_{1} x_{1}^{2} = 0$ : It can be two vertical lines parallel to axis of $x 1$ .
(c): Ellipse.
(d): No solution. For any $x_{1}$ and $x_{2}$ , $\tilde{w} z 0$ . So we can’t find $\tilde{w}$ that satisfies the condition.

niuers

2021-12-07 22:23

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