1. Cone Depth and the
Center Vertex Theorem
✦
Gary Miller
✦
Todd Phillips
✦
Don Sheehy

2. Let P be n points in Rd Center Points
A center point is a point
c (not necessarily in P)
such that every closed
half space containing c
contains at least n/d+1
points of P.
Center points always
exist. [Danzer et al, ’63]

3. Let P be n points in Rd Center Points
A center point is a point
c (not necessarily in P)
such that every closed
half space containing c
contains at least n/d+1
points of P.
Center points always
exist. [Danzer et al, ’63]

4. Let P be n points in Rd Center Points
A center point is a point
c (not necessarily in P)
such that every closed
half space containing c
contains at least n/d+1
points of P.
Center points always
exist. [Danzer et al, ’63]

5. Let P be n points in Rd Center Points
A center point is a point
c (not necessarily in P)
such that every closed
half space containing c
contains at least n/d+1
points of P.
Center points always
exist. [Danzer et al, ’63]

6. Let P be n points in Rd Center Points
A center point is a point
c (not necessarily in P)
such that every closed
half space containing c
contains at least n/d+1
points of P.
Center points always
exist. [Danzer et al, ’63]

7. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

8. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

9. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

10. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

11. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

12. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

13. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

14. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

15. Tukey Depth
The Tukey Depth of x is the
minimum number of points in
any half space containing x.
TD(x) = min||v||=1 |{p in P | (p-x)v > 0}|

16. Other notions of
statistical depth.
✦
Simplicial depth
✦
Convex hull peeling
✦
Regression Depth
✦ k-order α-hulls
✦
Travel Depth
✦
...... many others

17. Can we pick a center
from P?
When points are in
convex position, the
Tukey depth of every
p in P is 1.

18. Can we pick a center
from P?
When points are in
convex position, the
Tukey depth of every
p in P is 1.

19. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

20. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

21. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

22. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

23. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

24. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||

25. Cone Depth
Intuition: Narrow the field
of view.
TD(x) = min |{p ∈ P | (p-x)v > 0}|
||v||=1
(p-x)v
CD(x) = min |{p ∈ P | > c}|
||v||=1 ||p-x||
For this talk: cones
have half-angle 45o

26. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

27. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

28. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

29. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

30. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

31. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

32. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

33. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

34. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

35. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

36. The Center Vertex
Theorem
a center vertex is a
point p ∈ P such that
CD(p) ≥ n/d+1.
Thm: For all P ⊂ Rd,
there exists a center
vertex.
Pf: Pick p ∈ P closest
to a center point.

37. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

38. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

39. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

40. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

41. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

42. In Rd, the idea is the same. Beyond the plane
Pick the right hyperplane
through the center point, c.
Show that the bounded
part of the cone is empty.
The “right” hyperplane is
the one that intersects the
cone at a hyperellipsoid
centered at c.

43. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

44. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

45. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

46. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

47. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

48. Average Cone Depth
Let pk be the k-th nearest
point to c.
CD(pk) ≥ (n/d+1) - (k-1)
So, p ,...,p
1 have depth
n/2(d+1)
at least n/2(d+1).
Thus, the average depth
is at least
n/4(d+1)2 = O(n).

49. Some open questions.
✦
Is 45o the largest cone
half-angle for which a
center vertex always
exists?
✦
How fast can we compute
the cone depth of a point
in space?
✦
How fast can we find a
center vertex
deterministically.

50. Thanks.