This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.

1. Characterizing the Distortion of
Some Simple Euclidean
Embeddings
Jonathan Lenchner*, Krzysztof P. Onak*, Don Sheehy**, and Liu Yang*
!
!
*IBM
**University of Connecticut

2. Finite Euclidean Metrics
Finite Point Sets in Euclidean Space
Treat the point set P as a
finite metric space (P,d).
d(p, q) := kp qk =
q
(px qx)2 + (py qy)2

3. Embeddings and Distortion
P

4. Embeddings and Distortion
P Q

5. Embeddings and Distortion
P Q
Let ⇧ : P ! Q be a bijection.
⇧ is t-Lipschitz if d(p, q) < t d(⇧(p), ⇧(q))
for all p, q 2 P.

6. Embeddings and Distortion
P Q
Let ⇧ : P ! Q be a bijection.
⇧ is t-Lipschitz if d(p, q) < t d(⇧(p), ⇧(q))
for all p, q 2 P.
The distortion of ⇧ is the min t such that
⇧ and ⇧ 1
are t-Lipschitz.

7. Embeddings and Distortion
P Q
Let ⇧ : P ! Q be a bijection.
⇧ is t-Lipschitz if d(p, q) < t d(⇧(p), ⇧(q))
for all p, q 2 P.
The distortion of ⇧ is the min t such that
⇧ and ⇧ 1
are t-Lipschitz.
Dist(⇧) := maxp,q2P max( d(p,q)
d(⇧(p),⇧(q)) , d(⇧(p),⇧(q))
d(p,q) )

8. Lower Bounds [Badiou et al.]
Embedding n evenly spaced points on a circle
into a line requires ⌦(
p
n) distortion.

9. Lower Bounds [Badiou et al.]
Embedding n evenly spaced points on a circle
into a line requires ⌦(
p
n) distortion.

10. Lower Bounds [Badiou et al.]
Proof Idea:
Lipschitz Extensions
Borsuk-Ulam Theorem
Embedding n evenly spaced points on a circle
into a line requires ⌦(
p
n) distortion.

11. Lower Bounds [Badiou et al.]
Proof Idea:
Lipschitz Extensions
Borsuk-Ulam Theorem
Embedding n evenly spaced points on a circle
into a line requires ⌦(
p
n) distortion.
⌦(n1/4
) distortion for
embedding a sphere
into a plane.

12. Embedding in Pairs of Lines

13. Embedding in Pairs of Lines
Gap 1
2n

14. Embedding in Pairs of Lines
Between lines  1
2
p
n
Gap 1
2n

15. Embedding in Pairs of Lines
Between lines  1
2
p
n
Gap 1
2n
Resulting distortion is ⇥(
p
n)

16. Embedding in Pairs of Lines
Between lines  1
2
p
n
Gap 1
2n
Resulting distortion is ⇥(
p
n)
Distortion is ⇥(n1/4
)
for embedding
sphere to two planes.

17. Embedding in Triples of Lines

18. Embedding in Triples of Lines

19. Embedding in Triples of Lines

20. Embedding in Triples of Lines
Distortion is constant.

21. Embedding in Triples of Lines
Distortion is constant.
Similarly, embedding points a on sphere to 4 planes can also be done
with constant distortion.
!
Open: What about embedding into 3 planes?

22. One Point Off the Line/Plane
q
p
a
b

23. One Point Off the Line/Plane
p
n
q
p
a
b

24. One Point Off the Line/Plane
p
n
Distortion O(n1/4
) is possible.
q
p
a
b

25. One Point Off the Line/Plane
Lemma
Consider a collection of n points on a line, each point one unit from the next,
together with one additional point at height
p
n above the center point of the
points on the line. Then any embedding of these points into a line has distortion
⌦(n1/4
)
p
n
Distortion O(n1/4
) is possible.
q
p
a
b

26. One Point Off the Line/Plane
Lemma
Consider a collection of n points on a line, each point one unit from the next,
together with one additional point at height
p
n above the center point of the
points on the line. Then any embedding of these points into a line has distortion
⌦(n1/4
)
p
n
Distortion O(n1/4
) is possible.
Proof Idea:
If q is not on one end after the embedding, then it forces two adjacent points
to be stretched. Otherwise, the n/4 nearest points on the line to q must be
from the middle half. It follows that a or b separates a pair of adjacent
points from the middle half.
q
p
a
b

27. One Point Off the Line/Plane
p
n
q
p
a
b
q
Case 1: If q is not on the end, it separates c,d (previously adjacent pts).
c
d

28. One Point Off the Line/Plane
p
n
q
p
a
b
middle half
q
Case 2: If q is on the end, either a or b separates c, d (previously
adjacent pts from middle half).
c
d
a
The n/4 points closest to q must all be from the middle half.
Otherwise, dist(q,p) is stretched

29. Open Questions

30. Open Questions
Lower bounds for embedding into pairs of lines/planes.

31. Open Questions
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.

32. Open Questions
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.
Bounds on embedding points on a sphere into 3 planes.

33. Open Questions
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.
Bounds on embedding points on a sphere into 3 planes.
Upper bounds for one point off the line/plane.

34. Open Questions
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.
Bounds on embedding points on a sphere into 3 planes.
Upper bounds for one point off the line/plane.
Lower bounds for one point off a hyperplane in d>3.

35. Open Questions
Thanks.
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.
Bounds on embedding points on a sphere into 3 planes.
Upper bounds for one point off the line/plane.
Lower bounds for one point off a hyperplane in d>3.