The document summarizes research on the centervertex theorem for wedge depth. It states that for any set of points in d-dimensional space, there exists a point within the set whose wedge depth is at least (n/(d+1)), where n is the number of points. This means that for any wedge centered at that point, the number of points inside the wedge is greater than n/(d+1). The document provides the proof of the theorem and discusses algorithms for finding a centervertex with high wedge depth, including using centerpoints or approximate centerpoints. It concludes by posing open questions about computing centervertices and wedge depth efficiently.

1. The Centervertex Theorem
for Wedge Depth
Gary Miller, Todd Phillips, and Don Sheehy
Carnegie Mellon University

3. Medians

5. Warm up

6. Warm up in 1-D

7. Warm up in 1-D

8. Warm up in 1-D
1 2 3
Rank

9. Warm up in 1-D
1 2 3
Mean
Rank

10. Warm up in 1-D
Median
Mean
Rank

11. Warm up in 1-D
Median
In the set
Mean
Rank

12. Warm up in 1-D
Median
In the set
Robust
Mean
Rank

13. Warm up in 1-D
Median
In the set
Robust
∞
Mean
Rank

14. Warm up in 1-D
Median
In the set
Robust
∞
Mean ∞
Rank

15. Warm up in 1-D
Median
In the set
Robust
Mean
Rank

16. Warm up in 1-D
Median
In the set
Robust
Combinatorial
Mean
Rank

17. Warm up in 1-D
Median
In the set
Robust
Combinatorial
Mean
Rank

18. Warm up in 1-D
Median
In the set
Robust
Combinatorial
Mean
Rank

19. Warm up in 1-D
Median
In the set
Robust
Combinatorial
Mean
Rank
Depth

20. Warm up in 1-D
Median
In the set
Robust
Combinatorial
Mean
Rank
Depth

21. Warm up in 1-D
We want to
get this back.
Median
In the set
Robust
Combinatorial
Mean
Rank
Depth

22. Warm up in 1-D
wedge depth can
distinguish between
these two points.

24. Tukey Depth

26. Input (vertices)
d
S ⊂ R , |S| = n

27. Input (vertices)
d
S ⊂ R , |S| = n
Depth Measures
d
D:R →Z

28. Input (vertices)
d
S ⊂ R , |S| = n
Depth Measures
d
D:R →Z
Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}

39. Centerpoints
n
Dπ (x) ≥
d+1

40. Centerpoints
n
Dπ (x) ≥
d+1
Centerpoints always exist!
[Rado ’47, Danzer et al ’63]

42. Wedge Depth

43. α-Wedges
r
t

44. α-Wedges
p
α
r
t
∠(ptr) ≤
2

45. Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}

46. Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}

47. Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}
α-wedge Depth
Dα (x) = min {|W ∩ S| : W is an α-wedge with apex x}

48. Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}
α-wedge Depth
Dα (x) = min {|W ∩ S| : W is an α-wedge with apex x}

49. Tukey Depth
Dπ (x) = min {|H ∩ S| : H is a closed halfspace, and x ∈ H}
α-wedge Depth
Dα (x) = min {|W ∩ S| : W is an α-wedge with apex x}

56. The Centervertex
Theorem

57. The Centervertex Theorem
d
Given a set S ⊂ R , there exists a vertex
n
v ∈ S such that D3π/2 (v) ≥ d+1 .

58. The Centervertex Theorem
d
Given a set S ⊂ R , there exists a vertex
n
v ∈ S such that D3π/2 (v) ≥ d+1 .

59. The Centervertex Theorem
d
Given a set S ⊂ R , there exists a vertex
n
v ∈ S such that D3π/2 (v) ≥ d+1 .

60. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
v

61. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v

62. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v

63. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v

64. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v

65. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v

66. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T

67. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T
nd
|H ∩ S| ≤
d+1

68. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T
nd
|H ∩ S| ≤
d+1

69. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T
nd
|H ∩ S| ≤
d+1
|T ∩ S| = 0

70. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T
nd
|H ∩ S| ≤
d+1
|T ∩ S| = 0

71. n
For all 3π/2-wedges W with apex v, |W ∩ S| > d+1
nd
Equivalently, |S ∩ W | ≤ d+1 .
v
W ⊂H ∪T
nd
|H ∩ S| ≤
d+1
|T ∩ S| = 0

79. Expected Depth

80. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

81. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

82. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

83. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

84. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

85. Lemma:
Given a set S ⊂ Rd , a point x ∈ Rd and
a vertex s ∈ S, if s is the kth nearest vertex
to x then D3π/2 (s) ≥ Dπ (x) − k + 1.

87. Theorem:
The expected 3π/2-wedge depth of a vertex of S
n
is at least 2(d+1)2 .

88. Theorem:
The expected 3π/2-wedge depth of a vertex of S
n
is at least 2(d+1)2 .
Proof:
Order the vertices (v1 , . . . , vn ) by increasing distance
to a centerpoint. n
n d+1
1 1
D3π/2 (vi ) ≥ D3π/2 (vi )
n i=1
n i=1
n
By the previous Lemma, D3π/2 (vi ) ≥ d+1 − i + 1.
n
d+1
1 n
Ei [D3π/2 (vi )] ≥ i≥ .
n i=1
2(d + 1)2

90. Algorithms

91. Algorithm 1: Pick a point at random.
( With constant probability,
the depth will be linear. )

92. Algorithm 2: Find a centerpoint and
then find the nearest vertex.

93. Algorithm 2: Find a centerpoint and
then find the nearest vertex.
2
O(n) time in R

94. Algorithm 2: Find a centerpoint and
then find the nearest vertex.
2
O(n) time in R
O(nd−1 ) time for computing centerpoints (tukey medians)

95. Algorithm 2: Find a centerpoint and
then find the nearest vertex.
2
O(n) time in R
O(nd−1 ) time for computing centerpoints (tukey medians)
O(nlog d ) time for approximate centerpoints

96. Algorithm 2: Find a centerpoint and
then find the nearest vertex.
2
O(n) time in R
O(nd−1 ) time for computing centerpoints (tukey medians)
O(nlog d ) time for approximate centerpoints
Monte Carlo algorithms: Sublinear time
approximate centerpoints w.h.p.

98. Thank you.
Questions?

102. Open
Questions

103. Open Questions
How hard is it to compute a centervertex?
How hard is it to compute wedge depth?
How hard is it to test centervertices?
Is 270o tight?