Embedding Metric Spaces in
Their Intrinsic Dimension
Ittai Abraham , Yair Bartal*, Ofer Neiman
The Hebrew University
* also Caltech
Emebdding Metric Spaces
► Metric
spaces (X,dX), (Y,dY)
► Embedding is a function f : X→Y
► Distortion is the minimal α such that
dX(x,y)≤dY(f(x),f(y))≤α·dX(x,y)
Intrinsic Dimension
► Doubling
Constant : The minimal λ such any
ball of radius r>0, can be covered by λ balls
of radius r/2.
► Doubling Dimension : dim(X) = log2λ.
► The
problem: Relation between metric
dimension to intrinsic dimension.
Previous Results
► Given
a λ-doubling finite metric space (X,d) and
0<γ<1, it’s snow-flake version (X,dγ) can be
embedded into Lp with distortion and dimension
depending only on λ [Assouad 83].
► Conjecture
(Assouad) : This hold for γ=1.
► Disproved by Semmes.
►A


lower bound on distortion of  log n for L2,
with a matching upper bound [GKL 03].
Rephrasing the Question
► Is
there a low-distortion embedding for a finite
metric space in its intrinsic dimension?
Main result : Yes.
Main Results
► Any
finite metric space (X,d) embeds into Lp:
 With distortion O(log1+θn) and dimension
O(dim(X)/θ), for any θ>0.
 With constant average distortion and dimension
O(dim(X)log(dim(X))).
Additional Result
► Any
finite metric space (X,d) embeds into Lp:
distortion Olog n  log n  D  
~
and dimension O D dim  X  log log n  D .
► With
1 p
11 p
( For all D≤ (log n)/dim(X) ).
 In particular Õ(log2/3n) distortion and dimension into L2.
 Matches best known distortion result [KLMN 03] for
D=(log n)/dim(X) , with dimension O(log n log(dim(X))).
Distance Oracles
► Compact
data structure that approximately
answers distance queries.
► For general n-point metrics:
 [TZ 01] O(k) stretch with O(kn1/k) bits per label.
► For
a finite λ-doubling metric:
 O(1) average stretch with Õ(log λ) bits per label.
 O(k) stretch with Õ(λ1/k) bits per label.
Follows from
variation on “snowflake” embedding
(Assouad).
First Result
► Thm:
For any finite λ-doubling metric space
(X,d) on n points and any 0<θ<1 there exists
an embedding of (X,d) into Lp with
distortion O(log1+θn) and dimension O((log
λ)/θ) .
Probabilistic Partitions
►
►
►
►
►
P={S1,S2,…St} is a partition of X if
i  j : Si  S j  ,
 Si  X
i
P(x) is the cluster containing x.
P is Δ-bounded if diam(Si)≤Δ for all i.
A probabilistic partition P is a distribution over a set
of partitions.
A Δ-bounded P is η-padded if for all xєX :
PrP Bx,    Px  1 2
η-padded Partitions
►
►
►
►
►
The parameter η determines the quality of the
embedding.
[Bartal 96]: η=Ω(1/log n) for any metric space.
[CKR01+FRT03]: Improved partitions with η(x)=1/log(ρ(x,Δ)).
[GKL 03] : η=Ω(1/log λ) for λ-doubling metrics.
[KLMN 03]: Used to embed general + doubling metrics into
Lp : distortion O((log λ)1-1/p(log n)1/p), dimension O(log2n).
The local growth rate of x at radius r is:
  x, r  
B x,64r 
B x, r 64 
Uniform Local Padding Lemma
►
►
►
A local padding : padding probability for x is independent
of the partition outside B(x,Δ).
A uniform padding : padding parameter η(x) is equal for all
points in the same cluster.
There exists a Δ-bounded prob. partition with local uniform
padding parameter η(x) :
 η(x)>Ω(1/log λ)
 η(x)> Ω(1/log(ρ(x,Δ)))
C1
v1
η(v1) 
v2
C2
v3
η(v3) 
Plan:
►A
simpler result of:
 Distortion O(log n).
 Dimension O(loglog n·log λ).
► Obtaining
lower dimension of O(log λ).
► Brief overview of:
 Constant average distortion.
 Distortion-dimension tradeoff.
Embedding
into one dimension
►
For each scale iєZ, create uniformly padded local
probabilistic 8i-bounded partition Pi.
►
For each cluster choose σi(S)~Ber(½) i.i.d.
fi(x)=σi(Pi(x))·min{ηi-1(x)·d(x,X\Pi(x)), 8i}
f x    f i x 
Pi
i
►
Deterministic upper bound :
|f(x)-f(y)| ≤ O(log n·d(x,y)).
using
1
i



  Olog n

x

log

x
,
8
 i

i
i
x
d(x,X\Pi(x)
Lower Bound - Overview
► Create
a ri-net for all integers i.
► Define success event for a pair (u,v) in the ri-net,
d(u,v)≈8i : as having contribution >8i/4 , for many
coordinates.
► In every coordinate, a constant probability of
having contribution for a net pair (u,v).
► Use Lovasz Local Lemma.
► Show lower bound for other pairs.
Lower Bound – Other Pairs?
►
►
►
►
►
x,y some pair, d(x,y)≈8i. u,v the nearest in the ri-net to x,y.
Suppose that |f(u)-f(v)|>8i/4.
We want to choose the net such that
|f(u)-f(x)|<8i/16, choose ri= 8i/(16·log n).
Using the upper bound |f(u)-f(x)| ≤ log n·d(u,x) ≤ 8i/16
|f(x)-f(y)| ≥ |f(u)-f(v)|-|f(u)-f(x)|-|f(v)-f(y)| ≥ 8i/4-2·8i/16 = 8i/8.
8i/(16log n)
v
u
x
y
Lower
Bound:
v
 8i
u
► ri-net pair (u,v).
► It must be that
►
►
Can assume that 8i ≈d(u,v)/4.
Pi(u)≠Pi(v)
With probability ½ : d(u,X\Pi(u))≥ηi8i
With probability ¼ : σi(Pi(u))=1 and σi(Pi(v))=0
f i u   f i v   i1 u  i u   8i  0  8i
Lower Bound – Net Pairs
►
►
d(u,v)≈8i. Consider R 
If R<8i/2 :
 f u   f v 
j i
j
j
 With prob. 1/8 fi(u)-fi(v)≥ 8i.
►
If R≥ 8i/2 :
 With prob. 1/4 fi(u)=fi(v)=0.
►
In any case

j i
►
f j u   f j v   8i 2
The good event for pair in
scale i depend on higher
scales, but has constant
probability given any
outcome for them.
Oblivious to lower scales.
Lower scales do not matter
v
u

j i
ηi(u) 8i
f j u   f j v    8 j  8i 4
j i
Local Lemma
►
Lemma (Lovasz): Let A1,…An be “bad” events. G=(V,E) a
directed graph with vertices corresponding to events with
out-degree at most d. Let c:V→N be “rating” function of
event such that (Ai,Aj)єE then c(Ai)≥c(Aj), if

Pr  Ai

and
then

A j   p

jQ

Q  j Ai , Aj  E  c Ai   cAj 
epd  1  1


Pr   A j   0
 j[ n ]

Rating = radius
of scale.
Lower Bound – Net Pairs
►
►
►
►
►
A success event E(u,v) for a net pair u,v : there is
contribution from at least 1/16 of the coordinates.
Locality of partition – the net pair depend only on “nearby”
points, with distance < 8i.
Doubling constant λ, and ri≈8i/log n - there are at most
λloglog n such points, so d=λloglog n.
Taking D=O(log λ·loglog n) coordinates will give roughly
e-D= λ-loglog n failure probability.
By the local lemma, there is exists an embedding such that
E(u,v) holds for all net pairs.
Obtaining Lower Dimension
► To
use the LLL, probability to fail in more than
15/16 of the coordinates must be < λ-loglog n
► Instead of taking more coordinates, increase the
success probability in each coordinate.
► If probability to obtain contribution in each
coordinate >1-1/log n, it is enough to take O(log λ)
coordinates.
Similarly, if failure prob. in each
coordinate < log-θn, enough to
take O((log λ)/θ) coordinates
Using Several Scales
► Create
nets only every θloglog n scales.
► A pair (x,y) in scale i’ (i.e. d(x,y)≈8i’) will find a
close net pair in nearest smaller scale i.
► 8i’<logθn·8i, so lose a factor of logθn in the
i+θloglog n
distortion.
i’
θloglog n >
► Consider scales i-θloglog n,…,i.
i
i-θloglog n
Using Several Scales
► Take
u,v in the net with d(u,v)≈8i.
► A success in one of these scales will give
contribution >8i-θloglog n = 8i/logθn.
Lose a factor of logθn
in the distortion`
► The
success for u,v in each scale is :
 Unaffected by higher scales events
 Independent of events “far away” in the
same scale.
 Oblivious to events in lower scales.
► Probability that all scales failed<(7/8)θloglog n.
► Take only D=O((log λ)/θ) coordinates.
i+θloglog n
i
i-θloglog n
Constant Average Distortion
►
►
►
Scaling distortion – for every 0<ε<1 at most ε·n2 pairs with
distortion > polylog(1/ε).
Upper bound of log(1/ε), by standard techniques.
Lower bound:
 Define a net for any scale i>0 and ε=exp{-8j}.
 Every pair (x,y) needs contribution that depends on:
► d(x,y).
► The
ε-value of x,y.
 Sieve the nets to avoid dependencies between different scales and
different values of ε.
 Show that if a net pair succeeded, the points near it will also
succeed.
Constant Average Distortion
►
Lower bound cont…
 The local Lemma graph depends on ε, use the general case of local
Lemma.
 For a net pair (u,v) in scale 8i – consider scales:
8i-loglog(1/ε),…,8i-loglog(1/ε)/2.
 Requires dimension O(log λ·loglog λ).
The net
depends on λ.
Distortion-Dimension Tradeoff
D ≤ (log n)/log λ
: Olog n 1 p log n  D 11 p 
~
► Dimension : O D  log   log log n  D 
► Instead of assigning all scales to a single
coordinate:
► Distortion
 For each point x:
Divide the scales into D bunches of coordinates, in each
1

 i x   log n D
iBunch
 Create a hierarchical partition.
Upper bound needs the
x,y scales to be in the
same coordinates
Conclusion
► Main
result:
 Embedding metrics into their
intrinsic dimension.
► Open
problem:
For p>2 there is a doubling
metric space requiring
dimension at least Ω(log n)
for embedding into LP with
distortion O(log1/pn).
 Best distortion in dimension O(log λ).
 Dimension reduction in L2 :
►For
a doubling subset of L2 ,is there an embedding
into L2 with O(1) distortion and dimension O(dim(X))?