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Wasserstein Distances
Wasserstein Distances
5 selected
Difficulty 5-7
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Intermediate
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2 intermediate
3 advanced
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Question 1 of 5
120s
intermediate (5/10)
compute
For two empirical distributions on
R
,
P
n
=
n
1
∑
δ
x
i
and
Q
n
=
n
1
∑
δ
y
i
with
x
i
,
y
i
sorted, what closed form does
W
1
(
P
n
,
Q
n
)
take?
Hide and think first
A.
Mean absolute difference between sorted samples,
n
1
∑
i
=
1
n
∣
x
(
i
)
−
y
(
i
)
∣
, the optimal-transport cost in 1D with sorted matching.
B.
Average pairwise distance,
n
2
1
∑
i
,
j
∣
x
i
−
y
j
∣
, taken over all sample pairs without sorting.
C.
Sum of squared differences between sorted samples,
n
1
∑
i
(
x
(
i
)
−
y
(
i
)
)
2
, the empirical
L
2
distance of order statistics.
D.
Maximum absolute difference between empirical CDFs,
max
t
∣
F
P
n
(
t
)
−
F
Q
n
(
t
)
∣
, also called the Kolmogorov-Smirnov distance.
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