Low Dimensional Embeddings of Doubling Metrics

Abstract

We study several embeddings of doubling metrics into low dimensional normed spaces, in particular into ℓ ₂ and ℓ _∞ . Doubling metrics are a robust class of metric spaces that have low intrinsic dimension, and often occur in applications. Understanding the dimension required for a concise representation of such metrics is a fundamental open problem in the area of metric embedding. Here we show that the n-vertex Laakso graph can be embedded into constant dimensional ℓ ₂ with the best possible distortion, which has implications for possible approaches to the above problem.

Since arbitrary doubling metrics require high distortion for embedding into ℓ ₂ and even into ℓ ₁, we turn to the ℓ _∞ space that enables us to obtain arbitrarily small distortion. We show embeddings of doubling metrics and their ”snowflakes” into low dimensional ℓ _∞ space that simplify and extend previous results.

A Lower Bound on the Dimension for 1+𝜖 Distortion Embedding to ℓ ∞

Here we sketch a proof that there exists a doubling metric such that a 1+𝜖 distortion embedding into ℓ _∞ requires dimension Ω(1/𝜖 ^c), for some constant c. We shall assume 𝜖 is sufficiently small (otherwise there is nothing to prove). The instance we take will be a subset of the unit circle in $\mathbb {R}^{2}$, let $S=\{(x,y)\in \mathbb {R}^{2}\mid x\ge \cos ^{2}(\pi /16), x^{2}+y^{2}=1\}$ (this is an arc of length π/8). Fix some 𝜖>0, and we shall take a suitable set X⊆S of size k=𝜖 ^−1/8 that will be determined later, with the usual Euclidean metric. Clearly the doubling constant of such an X will be at most 3. Consider a map $f:X\to \ell _{\infty }^{D}$ with distortion at most 1+𝜖. We may assume that f does not expand any distance (it is 1-Lipschitz). Note that in order to obey the distortion constraint, every pair x,y∈X must have a coordinate i∈[D] such that

$$ |f_{i}(x)-f_{i}(y)|\ge (1-\epsilon)\|x-y\|_{2}~. $$

(21)

For each i∈[D] define a graph G _i =(X,E _i ), where {x,y}∈E _i is an edge if indeed (21) holds.

Observation 4

If $D<\sqrt {k}/2$ , then there exists i∈[D] such that G _i contains a cycle of length 4.

Proof

Note that there are ${k\choose 2}$ pairs in X, each pair x,y has at least 1 coordinate i for which {x,y}∈E _i , thus if $D<\sqrt {k}/2$, there exists an i such that $|E_{i}|\ge {k\choose 2}/D>k^{3/2}-\sqrt {k}$, and it is known (see e.g. [5]) that such dense graphs have a cycle of length 4. □

Seeking contradiction, assume $D<\sqrt {k}/2$, let i∈[D] as in Observation 4, and let w,x,y,z∈X be the 4-cycle in G _i . Let a=f _i (w)−f _i (x), b=f _i (x)−f _i (y), c=f _i (y)−f _i (z), b=f _i (z)−f _i (w), then of course a+b+c+d=0. By definition of E _i , the embedding f _i must (approximately) preserve all the distances between the four pairs, we have that |a|∈[1−𝜖,1]∥w−x∥₂ and similarly for b,c,d. Thus it suffices to prove that there exists a choice of X, such that in any 4-cycle of X, the lengths of the four edges do not sum to 0 even if we allow a 1−𝜖 perturbation and multiplication of some of them by −1. Given the next Lemma, we conclude that $D\ge \sqrt {k}/2=\epsilon ^{-1/16}/2$.

Lemma 5

There exists X⊆S of size k=𝜖 ^−1/8 , such that for any 4-cycle in X with lengths a ^′ ,b ^′ ,c ^′ ,d ^′ the following holds: For each g∈{a,b,c,d} choose any g∈[1−𝜖,1]g′∪[−1,−1+𝜖]g ^′ , then a+b+c+d≠0.

Proof

We use the probabilistic method. Divide S into k consecutive pieces I ₁,…,I _k each of length π/(8k), and choose for s∈[k] each x _s uniformly and independently in the middle portion of I _s (which is of length π/(16k)). Consider a 4-cycle, whose elements w,x,y,z are chosen from pieces indexed 1≤i ₁<i ₂<i ₃<i ₄≤k (so that $w=x_{i_{1}}$ and so on). Fix any choice of w,x,y. For u∈{w,x,y} let 𝜃 _u be twice the angle between the vectors u,z, and note that ∥z−u∥₂=2 sin𝜃 _u . Observe that since each u∈{w,x,y} is chosen from the middle section of a piece, then

$$ |\theta_{w}-\theta_{x}|\ge \pi/(8k)~, $$

(22)

and similarly for w,y and x,y.

Let Φ=a+b+c+d be a random variable, we shall condition on choosing the first three points w,x,y and analyze the change to Φ as z goes over its possible values in its piece. There are two edges of the cycle touching z, w.l.o.g we assume that they are the edges to w,x of lengths a ^′,b ^′ respectively with a ^′>b ^′ (this does not matter for the lower bound on the change to Φ we will soon show). We assume z starts at the upmost position and moves downward (away from w,x,y). Changing z by an infinitesimally small δ increases a ^′ by

$$\begin{array}{@{}rcl@{}} 2(\sin(\theta_{w}+\delta)-\sin\theta_{w})&=&2\delta\cdot\frac{\sin(\theta_{w}+\delta)-\sin\theta_{w}}{\delta}\\ &\approx&2\delta\cos\theta_{w}\\ &\approx&2\delta(1-{\theta_{w}^{2}}/2)~, \end{array} $$

(23)

using Taylor expansion.² Similarly b ^′ increases by

$$ 2(\sin(\theta_{x}+\delta)-\sin\theta_{x})\approx 2\delta(1-{\theta_{x}^{2}}/2)~. $$

(24)

Since a and b may be multiples of (at least) 1−𝜖 of a ^′ and b ^′ respectively, and also pick different signs, we get that the total change to Φ is at least the absolute value of the difference of (23) and (24), adding the effect of multiplying by 1−𝜖, this is still at least

$$\begin{array}{@{}rcl@{}} \delta({\theta_{x}^{2}}-{\theta_{x}^{2}}) - 4\epsilon\delta &\ge& \delta(\theta_{w}+\theta_{x})(\theta_{w}-\theta_{x}) - 4\epsilon\delta\\ &\stackrel{(22)}{\ge}&\delta\cdot(\pi/(8k))^{2}-4\epsilon\delta\\ &\ge&\delta\epsilon^{1/4}/16~, \end{array} $$

where the last inequality holds for sufficiently small 𝜖. Observe that z is chosen from an interval of length π 𝜖 ^1/8/8, and that the change to Φ is monotonic once the signs of a,b have been determined, we obtain that for each choice of signs there is a ”bad” section of $I_{i_{4}}$ (from which z is chosen) of size O(𝜖 ^3/4) (because the total change from one end point of such bad interval to other end point is greater than 𝜖, thus Φ cannot be 0 at both end points). We conclude that the probability of choosing z from a bad interval (there at most 8 possible signs for three edges) is O(𝜖 ^3/4/𝜖 ^1/8)=𝜖 ^5/8. Taking a union bound over the k ⁴=𝜖 ^−1/2 possible 4-cycles, we get that there is a probability of at most O(𝜖 ^1/8) for having a bad 4-cycle, so for sufficiently small 𝜖 there exists a choice of x ₁,…,x _k that satisfy the assertion of the Lemma. □