Metric Embedding with Relaxed Guarantees

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Metric Embedding with Relaxed Guarantees
Ofer Neiman Ittai Abraham Yair Bartal
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Embedding metric spaces

• Representation of the metric in a simple and
  structured space.

• Common target spaces lp, trees (ultra-metrics).

• The price of simplicity distortion, which is the
  multiplicative amount by which distances can
  change.

• Goal find low distortion embeddings.

• A tool for approximation algorithms

• Useful for many practical applications.

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Metric embedding

• Let X,Y be metric spaces with metrics dx, dy
  respectively.
• f X?Y is an embedding of X into Y.

• The distortion of f is the minimal a such
  that for some c

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Basic results

• Every metric space on n points can be embedded
  into Euclidean space with distortion O(log n) and
  dimension O(log2n). Bourgain/LLR

• Every metric space on n points can be embedded
  into a tree metric with distortion
  
• Bartal/BLMN/RR.

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problem

• The lower bounds on the distortion and the
  dimension are high, and grow with n.
• In some cases, weaker guarantees are acceptable..

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Some Alternative Schemes

• Probabilistic embedding considering the expected
  distortion. Bartal, FRT
• Ramsey theorems embedding a large subspace of
  the original metric. BFM, BLMN
• Partial embedding embedding all but a fraction
  of the distances. KSW, ABCDGKNS

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Motivation

• Estimating latencies (round-trip time) in the
  internet.
• - the distance matrix is almost a metric.
• - embedding heuristics yield surprisingly
  good results...
• NgZhang02, ST03, DCKM04
• Practical network embedding requires
• - Small number of dimensions.
• - No centralized co-ordination.
• - Linear number of distances measurement.
• Finding nearest copy of a file, service from
  some server, ect.

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(1-e) partial embedding

• X, Y are metric spaces.
• f X?Y has (1-e) partial distortion at
• most a if there exists a set of pairs Ge such
  that
• For all pairs (u,v)?Ge.

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Scaling Embedding

• A stronger requirement is a map that will be good
  for all e simultaneously.
• Definition an embedding f has scaling
  distortion D(e) if for any egt0, it is an (1-e)
  partial embedding with distortion D(e).

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Scaling Average Distortion

• Thm every metric space has scaling
  probabilistic embedding with distortion
  O(log(1/e)) into trees.
• Thm every metric space has scaling embedding
  with distortion O(log(1/e)) and dimension O(log
  n) into Euclidean space.
• implies constant average distortion!
• Applications weighted average problems
• sparsest cut, quadratic assignment,
  linear arrangement, ect.

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Previous work

• Any ?-doubling metric space X can be embedded
  into l2 with (1-e) partial distortion
  KSW.

• Definition a metric space X is called
  ?-doubling if for any rgt0, any ball of radius r
  can be covered by ? balls of radius r/2.

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Our Results

• Partial embedding into l2 with distortion and
  dimension O(log(1/e)).

• General theorem converting classical lp
  embeddings into the partial model.

• Distortion Dimension dont depend on the size
  of X!

• Partial embedding into trees with distortion
  .

• Tight lower bounds.

Appeared in FOCS05 together with CDGKS
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Embedding into lp

• Thm Any subset-closed family of metric spaces X
  , that has
• for any X?X on n points, an
  embedding fX? lp with
• - distortion a(n).
• - dimension ß(n).
• f can be converted into (1-e) partial
  embedding of X with
• - distortion
• - dimension

In practice..
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Main Results

• (1-e) partial embedding of any metric space into
  lp with distortion
• and dimension
  Bourgain,Matousek,Bartal
• (1-e) partial embedding of any negative type
  metric (l1 metrics) into l2 with distortion
  and dimension
• ARV, ALN
• (1-e) partial embedding of any doubling metric
  into lp with distortion
  and dimension KLMN
• (1-e) partial embedding of any tree metric into
  l2 with distortion
• and dimension
  Matousek

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Definitions

• Let re(u) be the minimal radius such that
  B(u,re(u)) en.
• A pair (u,v), w.l.o.g re(u) re(v)
• has short distance if d(u,v) lt re(u)
• has medium distance if re(u) d(u,v) lt
  4re(u).
• has long distance if 4re(u) d(u,v).

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Close Distances

• (u,v) is a short pair.
• Short pairs are ignored - at most en2.

re(w)
w
re(u)
u
re(v)
v
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Beacon Based Embedding

• Randomly choose beacons B.
• Each point attached to nearest beacon.

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Some More Bad Points

• If d(u,B) gt re(u) then
• is bad.
• For each u?X
• With probability ½ at most 2en2 bad pairs.

re(w)
w
re(u)
u
re(v)
v
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Partial Embedding

• Use the embedding fB?lp.
• f has distortion guarantee of
  .
• The partial embedding is

h(u)
f (u)
f(b)
u attached to beacon b
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Upper Bound

• We assume for the pair (u,v)
• - Each point has a beacon in its ball.
• - Both u,v are outside each others ball.
  
• - The mapping f is a contraction.

re(v)
v
re(u)
u
bv
bu
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Lower Bound - Long Distances
d(u,v) 4maxre(u), re(v)
re(u)
re(v)
u
v
bv
bu
d(bu,bv) d(u,v)/2
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Medium Distances??

• There is a problem in this case

u,v are attached to the same beacon!!
re(u)
u
re(v)
v

• The additional coordinates h will guarantee
  enough contribution..

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Medium Distances
Pairs satisfying re(u) d(u,v) 4re(u)
w.l.o.g re(u) re(v)
h(u)-h(v)
re(u),0
re(u),0
re(u),0
re(v),0
re(v),0
re(v),0
With probability ¼ we get re(u)
re(u)
0
re(u)
In expectation ¼ of the coordinates will be
re(u).

• With probability lt e the pair (u,v) will be
  smaller than half its expectation.

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Medium Distances

• With probability ½, 2en2 medium pairs failed, but
  for the others

• End of proof!

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Coarse Partial Embedding

• Another version ignoring only the short
  distances (i.e., from each point to its nearest
  en neighbors). the dimension increases to
  O(log(n)ß(1/e)).

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Partial Embedding into Trees

• Thm every metric space has (1-e) partial
  embedding with distortion into a
  tree (ultra-metric).

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Ultra-metrics

• Metric on leaves of rooted labeled tree.
• 0 ?(D) ?(B) ?(A).
• d(x,y) ?(lca(x,y)).
• d(x,y) ?(D).
• d(x,w) ?(B).
• d(w,z) ?(A).

?(A)
?(C)
?(B)
?(D)
x
y
z
w
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Embedding into Ultra-metric

• Partition X into 2 sets X1, X2
• Create a root labeled ? diam(X).
• The children of the root are created recursively
  on X1, X2
• Using induction the number of distances we ignore
  is
• B bad distances for current level.

X
X1
X2
?
X1
X2
B eX1X2
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Where to Cut?

• Take a point u such that B(u,?/2) n/2.
• Let
• i1,,1/e
• Let SiAi1-Ai
• We need a slim shell
• only distances inside the shell are distorted
  by more than

Ai1
Ai
A1
u
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Where to Cut?
Ai

• Case 1 A1lt en.
• X1 u, X2 X\u

A1
u
?
X\u
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Where to Cut?

• Case 2
• Choose an i such that
• Let X1Ai½, X2 X\X1

X2
Ai1
Ai
A1
?
u
X1
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Finding Shell Si

• Assume by contradiction for all
• Si2 gt enAi
• Then by induction Ai eni2.
  which implies At n.
• End of proof!

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Lower Bounds

• General method to obtain partial lower bounds
  from known classical ones.
• Thm given a lower bound a for embedding a
  family X into a family Y i.e. for any n
  there is X?X on n points and any embedding of
  X requires distortion at least a(n).
• Then there is X?X for which any (1-e)
  partial embedding requires distortion

The family X must be nearly closed under
composition!
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Main corollaries

• distortion for partial
  embedding into lp.
• LLR, Mat
  

• distortion for partial
  embedding into trees.
• Bartal/BLMN/RR.

• distortion for
  probabilistic partial
• embedding into trees.
  Bartal

• distortion for
  partial embedding of
• doubling or l1 metrics
  into l2. NR

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General idea

• Choose X?X such that
• For each x?X create a metric Cx such that
• - Cx?X.
• -
• X contain many copies of X.
• Let f be a (1-e) partial embedding that
  ignores the set of edges I. By definition
  .

X
X
d
d
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Finding a copy of X

• T vertices intersecting less than
  edges in I.
• For each x?X, choose some
• vx?CxnT.
• For each pair (vx,vy) find t ?Cy such that

Cx
Cy
vx
vy
in T
in T
t
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Distortion of the Copy
f has distortion guarantees for both these
distances
d(t,vy) is negligible
vx
vy
t
Its distortion must be at least
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Thank you!!