-4exELEC 400M / EECE 571M Homework 3: Nonlinear Transform -8ex Homework is not handed in. Please solve for your own benefit.
ELEC 400M / EECE 571M
Homework 3: Nonlinear Transform
Homework is not handed in. Please solve for your own benefit.
• Exercise 3.11, solution attached
• Exercise 3.13, solution attached
1
• Exercise 3.11
z = Φ(x) = [1, x21, x
2
2]
T : decision boundary is w̃Tz = w̃0 + w̃1z1 + w̃2z2 = 0 in Z and
w̃Tz = w̃0 + w̃1x
2
1 + w̃2x
2
2 = 0 in X
(a) w̃1 > 0, w̃2 < 0: x22 |w̃1| − x 2 1 |w̃2| = − w̃0|w̃1w̃2| This is a hyperbola. -5 -4 -3 -2 -1 0 1 2 3 4 5 x 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x 2 tilde(w) 0 <0 tilde(w) 0 >0
(b) w̃1 > 0, w̃2 = 0: w̃1x
2
1 + w̃0 = 0 → x1 = ±
√
− w̃0
w̃1
→ two vertical lines in the (x2, x1)-plane (and w̃0 > 0)
(c) w̃1 > 0, w̃2 > 0, w̃0 < 0:
x22
|w̃1|
+
x21
|w̃2|
= − w̃0|w̃1w̃2| > 0
This is an ellipse.
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
2
(d) w̃1 > 0, w̃2 > 0, w̃0 > 0: |w̃0|+ |w̃1|x21 + |w̃2|x22 = 0
This equation has not solution x1, x2 ∈ R.
• Exercise 3.13
(a) z = [1, x1, x2, x
2
1]
T
2
(b) z = [1, x1, x2, x
2
1, x
2
2]
T
(c) z = [1, x1, x2, x
2
1, x
2
2]
T
(d) z = [1, x1, x2, x
2
1, x
2
2]
T
(e) z = [1, x1, x2, x
2
1, x1x2, x
2
2]
T
(f) z = [1, x1, x2]
T = x
3