Foundations of Data Science

and hence
dist 2 (X 1 , X 2 ) = ‖X 1 ‖ 2 F − ∥ ∥
∥X 2 X2 T X 1 2
. F
Intuitively, the distance between X 1 and X 2 is the Frobenius norm of the component of
X 1 not in the space spanned by the columns of X 2 .
If X 1 and X 2 are 1-dimensional unit length vectors, dist 2 (X 1 , X 2 ) is the sin squared
of the angle between the spaces.
Example: Consider two subspaces in four dimensions
⎛ ⎞
√1
⎛
2
0
1
0 √3 ⎟
X 1 = ⎜
⎝
1 √
2
1 √3
0
1 √3
⎟
⎠
Here
⎛ ⎞
1 ⎛
√ 2
0
dist 2 0 √3 1
(X 1 , X 2 ) =
⎜ 1
⎝ √ √3 1 ⎟
2
⎠ − ⎜
⎝
∥
1
0 √3
⎛ ⎞
2
0 0
0 0
=
⎜
⎝ √1
√3 1
⎟
2
⎠
= 7 6
∥
∥
0
1 √3
F
1 0
0 1
0 0
0 0
X 2 =
⎞
⎟
⎠
⎜
⎝
1 0
0 1
0 0
0 0
⎞
⎟
⎠
( 1 0 0 0
0 1 0 0
⎛
)
⎜
⎝
√1
2
0
1
0 √3
√1
√3 1
2
0
1 √3
In essence, we projected each column vector of X 1 onto X 2 and computed the Frobenius
norm of X 1 minus the projection. The Frobenius norm of each column is the sin squared
of the angle between the original column of X 1 and the space spanned by the columns of
X 2 .
12.8 Generating Functions
∑
A sequence a 0 , a 1 , . . ., can be represented by a generating function g(x) = ∞ a i x i . The
advantage of the generating function is that it captures the entire sequence in a closed
form that can be manipulated as an entity. For example, if g(x) is the generating function
for the sequence a 0 , a 1 , . . ., then x d g(x) is the generating function for the sequence
dx
0, a 1 , 2a 2, 3a 3 , . . . and x 2 g ′′ (x) + xg ′ (x) is the generating function for the sequence for
0, a 1 , 4a 2 , 9a 3 , . . .
Example: The generating function for the sequence 1, 1, . . . is ∞ ∑
x i = 1 . The gener-
1−x
ating function for the sequence 0, 1, 2, 3, . . . is
418
i=0
⎞
⎟
⎠
∥
i=0
2
F