Foundations of Data Science

Exercise 12.40 We are given the probability distribution for two random vectors x and
y and we wish to stretch space to maximize the expected distance between them. Thus,
d∑
we will multiply each coordinate by some quantity a i . We restrict a 2 i = d. Thus, if we
increase some coordinate by a i > 1, some other coordinate must shrink. Given random
vectors x = (x 1 , x 2 , . . . , x d ) and y = (y 1 , y 2 , . . . , y d ) how should we select a i to maximize
E ( |x − y| 2) ? The a i stretch different coordinates. Assume
{ 0
1
y i =
2
1
1
2
and that x i has some arbitrary distribution.
E ( |x − y| 2) ∑
= E d [
a
2
i (x i − y i ) 2] ∑
= d a 2 i E (x 2 i − 2x i y i + yi 2 )
i=1
i=1
∑
= d a 2 i E ( )
x 2 i − x i + 1 2
i=1
Since E (x 2 i ) = E (x i ) we get . Thus, weighting the coordinates has no effect assuming
d∑
a 2 i = 1. Why is this? Since E (y i ) = 1.
2
i=1
E ( |x − y| 2) is
{
independent of the value of x i hence its distribution.
0
3
What if y i =
4
1 and E (y
1 i ) = 1. Then
4
4
E ( |x − y| 2) ∑
= d ∑
a 2 i E (x 2 i − 2x i y i + yi 2 ) = d a 2 i E ( x i − 1x )
2 i + 1 4
i=1
i=1
∑
= d (
.
1 E (x 2 i) + 4)
1
a 2 i
i=1
To maximize put all weight on the coordinate of x with highest probability of one. What
if we used 1-norm instead of the two norm?
i=1
E (|x − y|) = E
d∑
a i |x i − y i | =
i=1
d∑
a i E |x i − y i | =
i=1
d∑
a i b i
i=1
where b i = E (x i − y i ). If
d∑
i=1
a 2 i = 1, then to maximize let a i = b i
b
. Taking the dot product
of a and b is maximized when both are in the same direction.
Exercise 12.41 Maximize x+y subject to the constraint that x 2 + y 2 = 1.
Exercise 12.42 Draw a tree with 10 vertices and label each vertex with a unique integer
from 1 to 10. Construct the Prfer sequence for the tree. Given the Prfer sequence recreate
the tree.
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