Support Vector Machines - The Auton Lab

Quadratic Dot
Products
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
Φ( a)
• Φ(
b)
=
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a
1
2a
2a
2a
:
2a
m
2
1
2
2
a
a
:
2
m
2a a
1
2a a
:
1
2a a
1
2a a
m
2a
a
:
:
2
1
1
2
2
3
3
m
a
⎞
⎛
⎟ ⎜
⎟ ⎜ 2b1
⎟ ⎜ 2b2
⎟ ⎜
⎟ ⎜ :
⎟ ⎜
⎟ ⎜
2bm
2
⎟ ⎜ b1
⎟ ⎜ 2
⎟ ⎜ b2
⎟ ⎜ :
⎟ ⎜
2
⎟ ⎜ bm
⎟
•
⎜ 2b1b
2
⎟
2b b
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
:
2b b
2b
1 3
1 m
2b
b
:
2 3
2b b
1 m
:
b
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
m−1
m
m−1
m
Copyright © 2001, 2003, Andrew W. Moore Support Vector Machines: Slide 49
1
+
m
∑
i=
1
+
m
∑
i=
1
m
+
2 a b i i
a 2
b i i
m
∑∑
i= 1 j=
i+
1
2
2a a b b
i
j
i
j
Quadratic Dot
Products
Φ( a)
• Φ(
b)
=
1+
2
m
∑
i=
1
a b +
i
i
m
∑
i=
1
a b
2 2
i i
+
m
m
∑∑
i= 1 j=
i+
1
2a a b b
i
j
i
j
Just out of casual, innocent, interest,
let’s look at another function of a and
b:
( a.
b + 1)
2
= ( a.
b)
+ 2a.
b + 1
2
2
m
m
⎛ ⎞
= ⎜∑
aibi
⎟ + 2∑
aibi
+ 1
⎝ i=
1 ⎠ i=
1
m
m
= ∑∑aibia
jb
j
+ 2∑
aibi
+ 1
i= 1 j=
1
m
m
i=
1
= 2
∑ ( a ) + 2∑∑
+ 2 ∑
ibi
aibi
a
jb
j
aibi
+ 1
i=
1
m
m
i= 1 j=
i+
1
m
i=
1
Copyright © 2001, 2003, Andrew W. Moore Support Vector Machines: Slide 50
25