the centervertex theorem (cccg)
TRANSCRIPT
The Centervertex Theorem for Wedge Depth
Gary Miller, Todd Phillips, and Don Sheehy
Carnegie Mellon University
Medians
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We want toget this back.
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wedge depth candistinguish between
these two points.
Tukey Depth
S ! Rd, |S| = n
Input (vertices)
Depth MeasuresD : R
d! Z
S ! Rd, |S| = n
Input (vertices)
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
Depth MeasuresD : R
d! Z
S ! Rd, |S| = n
Input (vertices)
D!(x) !n
d + 1
Centerpoints
D!(x) !n
d + 1
Centerpoints
Centerpoints always exist! [Rado ’47, Danzer et al ’63]
Wedge Depth
tr
α-Wedges
p
tr
α-Wedges
!(ptr) !!
2
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
The Centervertex Theorem
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
Expected Depth
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
Theorem:The expected 3!/2-wedge depth of a vertex of Sis at least n
2(d+1)2 .
Theorem:
Proof:Order the vertices (v1, . . . , vn) by increasing distanceto a centerpoint.
By the previous Lemma, D3!/2(vi) !n
d+1" i + 1.
The expected 3!/2-wedge depth of a vertex of Sis at least n
2(d+1)2 .
1
n
n!
i=1
D3!/2(vi) !1
n
n
d+1!
i=1
D3!/2(vi)
Ei[D3!/2(vi)] !1
n
n
d+1!
i=1
i !n
2(d + 1)2.
Algorithms
Algorithm 1: Pick a point at random.
With constant probability, the depth will be linear.( )
Algorithm 2: Find a centerpoint and then find the nearest vertex.
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(n) time in R2
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(n) time in R2
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(nlog d) time for approximate centerpoints
O(n) time in R2
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(nlog d) time for approximate centerpoints
Monte Carlo algorithms: Sublinear timeapproximate centerpoints w.h.p.
O(n) time in R2
Thank you.Questions?
OpenQuestions
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?
Open Questions