Notes of Moses Charikar’s talk, jan. 17th

 

Dimension reduction in {\ell_1}

 

1. The problem

 

Problem: Given {n} points in Banach space {\ell_1}, how many dimensions are needed to represent pairwise distances to within {1\pm\epsilon} ?

In other words, given a finite subset {F\subset \ell_{1}^{d}}, we want a map {f:\ell_{1}^{d}\rightarrow\ell_{1}^{d'}} such that for all {x}, {y\in F},

\displaystyle  \begin{array}{rcl}  \frac{1}{1+\epsilon}\|x-y|_{1}\leq\|f(x)-f(y)|_{1}\leq \|x-y|_{1}. \end{array}

This works well in {\ell_{2}}, this is the Johnson-Lindenstrauss theorem: for an {n}-point set, {d'=O(\epsilon^{-2}\log n)} dimensions suffice. If {\epsilon=0}, {d'=n-1} are sufficient, and often necessary.

 

2. Old and new results

 

In {\ell_1}, exact reduction (i.e. {\epsilon=0}) requires {\Theta(n^2)} dimensions.

For {1+\epsilon} reduction, a result of Schechtman (later improved by Talagrand), states that {O(\epsilon^{-2}n\log n)} dimensions suffice. Newman and Rabinovich improved this to {O(\epsilon^{-2} n)} dimensions suffice. An unpublished result of Andoni, Naor and Newman says that {O(n/D)} dimensions suffice for large {D}.

Lower bounds. With Brinkman, I proved that {n^{\Omega(1/D^2)}} dimensions are necessary for distorsion {D}, [BC]. A simpler proof has been given since by James Lee and Assaf Naor, [LN].

Today, I focus on {D} close to {1}. Previously, I proved that {n^{\frac{1}{2}+\epsilon}} are needed. Alexandru Andoni, Ofer Neiman and Nguyen and I show

Theorem 1 For {1+\epsilon} reduction, {n^{1-O(1/\log(1/\epsilon))}} dimensions are needed.

 

3. The examples

 

The lower bound is based on a recursive {2k}-cycle graph. This means one starts with a cycle with two marked antipodal vertices {s} and {t}. Then we replace each edge with a copy of the cycle, and so on. For {k=2}, this is called the diamond graph (basis of the {n^{\Omega(1/D^2)}} lower bound). The diamond graph is isometrically embeddable in {\sqrt{n}} dimensions, so cannot give a sharp lower bound.

The {i}-th graph {G_i} has diameter {k^{i}}, {(2k)^{i}} edges, roughly {(2k)^{i}} vertices. It embeds isometrically in {k^i} dimensions. Note that {k^i \sim n^{1-1/\log k}}, fitting with the Andoni, Naor and Newman upper bound.

 

4. From distorsion to stretch

 

We first show that the {2k}-cycle needs {1/2\epsilon} dimensions for {1=\epsilon} distorsion., if {\epsilon\leq 1/k}. To prove this, think of an {\ell_1} embedding as a probabilistic line embedding. I.e.

\displaystyle  \begin{array}{rcl}  |x-y|=\sum_{i=1}^{n}|x_i -y_i|=\mathop{\mathbb E}(d|x_i -y_i|). \end{array}

Say a line embedding has stretch {s} if

\displaystyle  \begin{array}{rcl}  |f(x)-f(y)|\leq s.d(x,y). \end{array}

Say a distribution over line embeddings has stretch {s} if above inequality holds for all {f} in the support.

Our key estimate is the following: a (low) {d}-dimensional embedding has stretch {d}. So it suffices to estimate stretch of line embeddings from below.

 

5. Analysis of embeddings of the {2k}-cycle

 

The canonical embedding of the {2k}-cycle uses {k} dimensions. For instance, if {k=4}, 0000, 0001, 0011, 0111, 1111, 1110, 1100, 1000. When we view it as a distribution over line embedding, we rescale by {k}.

Let {L} be the sum of antipodals, {E} be the sum of edges. In the exact embedding case, {L=k^2}, {E=2k}. In the {(1+\epsilon)}-distorsion case, {L\geq k^2 /1+\epsilon} and {E\leq 2k}, so that

\displaystyle  \begin{array}{rcl}  2L-k(1-2\epsilon)E \geq 2\frac{k^2}{1+\epsilon}-k(1-2\epsilon)2k=2\epsilon k^2 . \end{array}

Now we show that low ({<1/2\epsilon}) stretch implies {2L-k(1-2\epsilon)E <2\epsilon k^2}. When the cycle is mapped to a line, every two antipodal pairs {(a,b)} and {(a',b')} are mapped to non nested intervals, so

\displaystyle  \begin{array}{rcl}  |f(a)-f(a')|+|f(b)-f(b')|\leq |f(a)-f(b)|+|f(a')-f(b')|. \end{array}

This implies that nearby antipodal pairs are mapped to comparable lengths. This propagates to not that nearby antipodal pairs, and so on… This leads to

\displaystyle  \begin{array}{rcl}  2L\leq kE_0 +(k-2)E_1 +(k-4)E_2 +\cdots+2E_{k/2 -1}, \end{array}

where {E_i} are edge lengths on the line. Think of {\epsilon} as {i/k}. Then

\displaystyle  \begin{array}{rcl}  2L-k(1-2\epsilon)E =2L-(k-2i)E \leq 2iE_0 +2(i-1)E_1 +2(i-2)E_2 +\cdots \end{array}

Assume that every edge has stretch at most {s}. Then {E_0 =2s} and other {E_j =4s}. This gives {2L-k(1-2\epsilon)E\leq 4si^2 \leq 2\epsilon k^2} if stretch is {\leq 1/2\epsilon}.

 

6. General case

 

Now we turn to graph {G_t}. It has roughly {(2k)^t} vertices. We show that for distorsion {1+1/k}, one needs stretch {\geq (k/16)^t \simeq n^{1- 5/\log k}}.

The canonical embedding of {G_t} is built recursively, adding {k} dimensions at each step. Edges have length {1}, antipodal pairs at level {i} are mapped to distance {k^{t-i+1}}. There are {2^{i-1}} such pairs.

Use a Poincaré inequality. Let {L_i} be the sums of antipodals at level {i} and {E} the sum of edges at level {t}. Now consider

\displaystyle  \begin{array}{rcl}  P=\sum_{i=1}^{t}2^{t+1-i}L_i -t(k-2)E. \end{array}

One shows that low ({1+1/k}) distorsion implies {P\geq t(2k)^t}, and low ({(k/16)^t}) stretch implies {P<t(2k)^t}. For this, one adds up estimates for {2L} for all cycles in the construction.

 

7. Comments and questions

 

This style of analysis can be used to give a simple proof of our previous result. It was discovered using LP methods, including experiments.

Regev: connection with functional analysis ? Answer: this does not go to {\ell_p} spaces. Dimension reduction in {\ell_p} is widely open.

Question: how tight is the result ? role of aspect ratio ? Answer: there is room for slight improvements.

Linial: do things generically embed well ? these examples look extreme. Answer: there are other graphs for which we care about embeddings. The recursive nature of examples suggests is there a parallel repetition phenomenon.

 

References

 

[BC] Brinkman, B; Charikar, Moses; On the Impossibility of Dimension Reduction in l1. FOCS (2003).

[LN] a, Embedding the Diamond Graph in Lp and Dimension Reduction in L1. GAFA 14 (2004) 745–747.

 

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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