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On the Impossibility of Dimension Reduction for Doubling Subsets of L p

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On the Impossibility of Dimension Reduction for Doubling Subsetsof Lp

YairBartal

Lee-Ad Gottlieb

Ofer Neiman

- Lp spaces: Lpk is the metric space
- Let (X,d) be a finite metric space
- A map f:X→Lpk is called an embedding
- The embedding is non-expansive and has distortion D, if for all x,yϵX :

- Lemma: Any n points in L2 can be embedded into L2k, 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∞

- For general n-point sets in Lp, Ω(logDn) 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 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

- 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 L2k, k=O((log λ)/ε2) with 1+ε distortion

- 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 Lpkwith distortion at most D satisﬁes

- 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 Lpkwith 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

- 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)

G0

G1

- 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

G2

- 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 L2stretch is increased by some value, compared to the stretch of its parent edge
- When stretch(u,v) > k, we will have a contradiction, as

u

- 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

s

a

b

t

v

fj(a)

fj(b)

u

- 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)

s

a

b

t

v

fj(a)

fj(b)

Δj(u)

u

s

a

b

t

- The stretch of u,s will increase due to the j-th coordinate
- But may decrease due toother 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

fj(a)

fj(b)

Δj(u)

u

s

a

b

t

v

fh(a)

fh(b)

-Δh(u)

u

s

a

b

t

- Observe that in the j-thcoordinate:
- 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 L2stretch of {u,s} and {u,t} (in the j-th coordinate alone) is:

v

fj(a)

fj(b)

Δj(u)

- 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 mostkD2(otherwise the stretch will begreater than k)
- But # of iterations ≈ log n
- Finally,

u

s

a

t

b

v

- 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

u

b

a

s

t

ε

v

- 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

u

b

a

s

t

ε

v

- 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?