On the Impossibility of Dimension Reduction for

Doubling Subsetsof Lp

Yair BartalLee-Ad Gottlieb

Ofer Neiman

Embedding and Distortion

• Lp spaces: Lpk is the metric space

• Let (X,d) be a finite metric space• A map f:X→ Lp

k is called an embedding• The embedding is non-expansive and has distortion

D, if for all x,yϵX :

JL Lemma• Lemma: Any n points in L2 can be embedded into L2

k, k=O((log n)/ε2) with 1+ε distortion

• Extremely useful for many applications:– Machine learning– Compressive sensing– Nearest Neighbor search– Many others…

• Limitations: specific to L2, dimension depends on n– There are lower bounds for dimension reduction in L1, L∞

Lower bounds on Dimension Reduction

• For general n-point sets in Lp, Ω(logD n) dimensions are required for distortion D (volume argument)

• BC’03 (and also LN’04, ACNN’11, R’12) showed strong impossibility results in L1

– The dimension must be for distortion D

Doubling Dimension

• Doubling constant: The minimal λ so that every ball of radius 2r can be covered by λ balls of radius r

• Doubling dimension: log2λ• A measure for dimensionality of a

metric space• Generalizes the dimension for normed

space: Lpk has doubling dimension Θ(k)

• The volume argument holds only for metrics with high doubling dimension

Overcoming the Lower Bounds?

• One could hope for an analogous version of the JL-Lemma for doubling subsets

• Question: Does every set of points in L2 of constant doubling dimension, embeds to constant dimensional space with constant distortion?

• More ambitiously: Any subset of L2 with doubling constant λ, can be embedded into L2

k, k=O((log λ)/ε2) with 1+ε distortion

Our Result

• Such a dimension reduction is impossible in the Lp spaces with p>2

• Thm: For any p>2 there is a constant c, such that for any n, there is a subset A of Lp of size n with doubling constant O(1), and any embedding of A into Lp

k with distortion at most D satisfies

Our Result• Thm: For any p>2 there is a constant c, such that for any n, there is a

subset A of Lp of size n with doubling constant O(1), and any embedding of A into Lp

k with distortion at most D satisfies

• Note: any sub-logarithmic dimension requires non-constant distortion

• We also show a similar bound for embedding from Lp into Lq, for all q≠2

• Lafforgue and Naor concurrently proved this using analytic tools, and their counterexample is based on the Heisenberg group

Implications

• Rules out a class of algorithms for NN-search, clustering, routing etc.

• The first non-trivial result on non-linear dimension reduction for Lp with p≠1,2,∞

• Comment: For p=1, there is a stronger lower bound for doubling subsets, the dimension of any embedding with distortion D (into L1) must be at least (LMN’05)

The Laakso Graph

• A recursive graph, Gi+1 is obtained from Gi by replacing every edge with a copy of G1

• A series-parallel graph• Has doubling constant 6

G0 G1 G2

Simple Case: p=∞

• The Laakso graph lies in high dimensional L∞

• Assume w.l.o.g that there is a non-expansive embedding f with distortion D into L∞

k

• Proof idea:– Follow the recursive construction– At each step, find an edge whose L2 stretch is increased by

some value, compared to the stretch of its parent edge

– When stretch(u,v) > k, we will have a contradiction, as•

v

Simple Case: p=∞

• Consider a single iteration• The pair a,b is an edge of the

previous iteration• Let fj be the j-th coordinate• There is a natural embedding

that does not increase stretch...• But then u,v may be distorted

a bs tu

fj(a) fj(b)

Simple Case: p=∞

• For simplicity (and w.l.o.g) assume – fj(s)=(fj(b)-fj(a))/4

– fj(t)=3(fj(b)-fj(a))/4

– fj(v)=(fj(b)-fj(a))/2

• Let Δj(u) be the difference between fj(u) and fj(v)

• The distortion D requirement imposes that for some j, Δj(u)>1/D (normalizing so that d(u,v)=1)

v

a bs tu

fj(a) fj(b)Δj(u)

Simple Case: p=∞

• The stretch of u,s will increase due to the j-th coordinate

• But may decrease due to other coordinates..

• Need to prove that for one of the pairs {u,s}, {u,t}, the total L2 stretch increases by at least – Compared to the stretch of a,b

v

a bs t

fj(a) fj(b)Δj(u)

u

v

a bs t

fh(a) fh(b)-Δh(u)

u

Simple Case: p=∞• Observe that in the j-th coordinate:

– If the distance between u,s increases by Δj(u),

– Then the distance between u,t decreases by Δj(u) (and vise versa)

• Denote by x the stretch of a,b in coordinate j

• The average of the L2 stretch of {u,s} and {u,t} (in the j-th coordinate alone) is:

v

a bs t

fj(a) fj(b)Δj(u)

u

Simple Case: p=∞

• For one of the pairs {u,s}, {u,t}, the total L2 stretch (over all coordinates) increases by

• Continue with this edge• The number of iterations must be at

most kD2 (otherwise the stretch will be greater than k)

• But # of iterations ≈ log n• Finally,

u

sa bt

v

Going Beyond Infinity

• For p<∞, we cannot use the Laakso graph– Requires high distortion to embed it into Lp

• Instead, we build an instance in Lp, inspired by the Laakso graph

• The new points u,v will use a new dimension

• Parameter ε determines the (scaled) u,v distance a s

u

v

t bε

Going Beyond Infinity

• Problem: the u,s distance is now larger than 1, roughly 1+εp

• Causes a loss of ≈ εp in the stretch of each level

• Since u,v are at distance ε, the increase to the stretch is now only (ε/D)2

• When p>2, there is a choice of ε for which the increase overcomes the loss

a su

v

t bε

Conclusion

• We show a strong lower bound against dimension reduction for doubling subsets of Lp, for any p>2

• Can our techniques be extended to 1<p<2 ?– The u,s distance when p<2 is quite large, ≈ 1+(p-1)ε2 , so a

different approach is required

• General doubling metrics embed to Lp with distortion O(log1/pn) (for p≥2)– Can this distortion bound be obtained in constant

dimension?