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Characterizing the Distortion of Some Simple Euclidean Embeddings Jonathan Lenchner*, Krzysztof P. Onak*, Don Sheehy**, and Liu Yang* *IBM **University of Connecticut
Characterizing the Distortion of Some Simple Euclidean
EmbeddingsJonathan Lenchner*, Krzysztof P. Onak*, Don Sheehy**, and Liu Yang* !!*IBM **University of Connecticut
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
Embeddings and Distortion
P
Embeddings and Distortion
P Q
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 .
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 ⇧
�1are t-Lipschitz.
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 ⇧
�1are t-Lipschitz.
Dist(⇧) := maxp,q2P max(
d(p,q)d(⇧(p),⇧(q)) ,
d(⇧(p),⇧(q))d(p,q) )
Lower Bounds [Badiou et al.]
Embedding n evenly spaced points on a circle
into a line requires ⌦(
pn) distortion.
Lower Bounds [Badiou et al.]
Embedding n evenly spaced points on a circle
into a line requires ⌦(
pn) distortion.
Lower Bounds [Badiou et al.]
Proof Idea: Lipschitz Extensions Borsuk-Ulam Theorem
Embedding n evenly spaced points on a circle
into a line requires ⌦(
pn) distortion.
Lower Bounds [Badiou et al.]
Proof Idea: Lipschitz Extensions Borsuk-Ulam Theorem
Embedding n evenly spaced points on a circle
into a line requires ⌦(
pn) distortion.
⌦(n1/4) distortion for
embedding a sphere
into a plane.
Embedding in Pairs of Lines
Embedding in Pairs of Lines
Gap � 12n
Embedding in Pairs of Lines
Between lines 12pn
Gap � 12n
Embedding in Pairs of Lines
Between lines 12pn
Gap � 12n
Resulting distortion is ⇥(
pn)
Embedding in Pairs of Lines
Between lines 12pn
Gap � 12n
Resulting distortion is ⇥(
pn)
Distortion is ⇥(n1/4)
for embedding
sphere to two planes.
Embedding in Triples of Lines
Embedding in Triples of Lines
Embedding in Triples of Lines
Embedding in Triples of Lines
Distortion is constant.
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?
One Point Off the Line/Plane
q
pa
b
One Point Off the Line/Plane
pn
q
pa
b
One Point Off the Line/Plane
pn
Distortion O(n1/4) is possible.
q
pa
b
One Point Off the Line/Plane
LemmaConsider a collection of n points on a line, each point one unit from the next,
together with one additional point at height
pn above the center point of the
points on the line. Then any embedding of these points into a line has distortion
⌦(n
1/4)
pn
Distortion O(n1/4) is possible.
q
pa
b
One Point Off the Line/Plane
LemmaConsider a collection of n points on a line, each point one unit from the next,
together with one additional point at height
pn above the center point of the
points on the line. Then any embedding of these points into a line has distortion
⌦(n
1/4)
pn
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
pa
b
One Point Off the Line/Plane
pn
q
pa
b
q
Case 1: If q is not on the end, it separates c,d (previously adjacent pts).
cd
One Point Off the Line/Plane
pn
q
pa
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).
cd
a
The n/4 points closest to q must all be from the middle half. Otherwise, dist(q,p) is stretched
Open Questions
Open Questions
Lower bounds for embedding into pairs of lines/planes.
Open Questions
Lower bounds for embedding into pairs of lines/planes.
Extensions to measures and expected distortion.
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.
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.
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.
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.
