Local Embeddings of Metric Spaces

Abstract

In many application areas, complex data sets are often represented by some metric space and metric embedding is used to provide a more structured representation of the data. In many of these applications much greater emphasis is put on preserving the local structure of the original space than on maintaining its complete structure. This is also the case in some networking applications where “small world” phenomena in communication patterns have been observed. Practical study of embedding has indeed involved with finding embeddings with this property. In this paper we initiate the study of local embeddings of metric spaces and provide embeddings with distortion depending solely on the local structure of the space.

Appendix: Proof of Lemma 5

The following decomposition lemma was shown in [2].

Lemma 49

(Probabilistic Decomposition)

For any metric space (X,d), a subset Z⊆X, a point v∈X, real parameters χ≥2,Δ>0, let r be a random variable sampled from a truncated exponential density function with parameter λ=8ln(χ)/Δ

$$f(r)=\left\{ \begin{array}{@{}l@{\quad}l} \frac{\chi^2}{1-\chi^{-2}}\lambda e^{-\lambda r} & r\in[\varDelta /4,\varDelta /2]\\ 0 & \mathrm{otherwise} \end{array} \right. $$

If S=B(v,r)∩Z and ${\bar{S}}= Z \setminus S$ then for any θ∈[χ ⁻¹,1) and any x∈Z:

$$\begin{aligned} \Pr \bigl[B(x, \eta \varDelta ) \bowtie(S,{\bar{S}}) \bigr] \leq (1-\theta) \biggl( \Pr \bigl[B(x, \eta \varDelta ) \nsubseteq{\bar{S}} \bigr] + \frac{2\theta}{\chi} \biggr), \end{aligned}$$

where η=2⁻⁴ln(1/θ)/lnχ.

We are now ready to prove Lemma 5. By sub-partition we mean a partition {C _i } _i lacking the requirement that ⋃ _i C _i =X. The intuition behind the construction is that we perform the partition of [2] as long as the local growth rate is small enough. Once the growth rate is large with respect to the decomposability parameter, we assign all the points who were not covered by the first partition a cluster generated by the probabilistic partition known to exists from Definition 10. This is done in two phases:

Phase 1 :

Define the sub-partition P ₁ of X into clusters by generating a sequence of clusters: C ₁,C ₂,…,C _s , for some s∈[n]. Notice that we are generating a distribution over sub-partitions and therefore the generated clusters are random variables. First we deterministically assign centers v ₁,v ₂,…,v _s and parameters χ ₁,χ ₂,…,χ _s . Let W ₁=X and j=1. Conduct the following iterative process:

• 1. Let v _j ∈W _j be the point minimizing $\hat{\chi_{j}}=\rho(x,2\varDelta ,\gamma)$ over all x∈W _j .

• 2. If $2^{6}\ln(\hat{\chi_{j}})>\tau^{-1}$ set s=j−1 and stop.

• 3. Set $\chi_{j} = \max\{2/\hat{\delta}^{1/4},\hat{\chi_{j}}\}$.

• 4. Let W _j+1=W _j ∖B(v _j ,Δ/4).

• 5. Set j=j+1. If W _j ≠∅ return to (1).

Now the algorithm for the partition and functions ξ,η is as follows: Let Z ₁=X. For j=1,2,3,…,s:

• 1. Let $(S_{v_{j}},{\bar{S}}_{v_{j}})$ be the partition created by invoking Lemma 49 on Z _j with center v=v _j and parameter χ=χ _j .

• 2. Set $C_{j}=S_{v_{j}}$, $Z_{j+1}={\bar{S}}_{v_{j}}$.

• 3. For all x∈C _j let $\eta_{P}(x)=2^{-7}/\max \{\ln\hat{\chi}_{j},\ln(1/\hat{\delta})\}$. If $\hat{\chi_{j}}\ge 1/\hat{\delta}$ set ξ _P (x)=1, otherwise set ξ _P (x)=0.

Fix some $\hat{\delta}\le\delta\le1$. Let θ=δ ^1/4. Note that $\theta\geq2\chi_{j}^{-1}$ for all j∈[s] as required. Recall that η _j =2⁻⁴ln(1/θ)/lnχ _j =2⁻⁶ln(1/δ)/lnχ _j (it is easy to verify that η _P (x)⋅ln(1/δ)≤η _j ). Observe that some clusters may be empty and that it is not necessarily the case that v _m ∈C _m .

Phase 2 :

In this phase we assign any points left unassigned from phase 1. Let $P'_{2}=\{D_{1},D_{2},\ldots,D_{t}\}$ be a Δ-bounded probabilistic partition of X, such that for all δ≤1 satisfying ln(1/δ)≤2⁶ τ ⁻¹, $P'_{2}$ is (τ⋅ln(1/δ),δ)-padded, this probabilistic partition exists by Definition 10. Let $Z=\bigcup_{i=1}^{s}C_{i}$ and $\bar{Z}=X\setminus Z$ (the unassigned points), then let $P_{2}=\{D_{1}\cap\bar{Z},D_{2}\cap\bar{Z},\ldots,D_{t}\cap\bar{Z}\}$. For all $x\in\bar{Z}$ let η _P (x)=τ/2 and ξ _P (x)=1. It can be checked that $\eta_{P}^{(\delta)}(x)\le\eta_{j}$ for all j∈[s]. Notice that by the stop condition of phase 1, $\tau\le2^{-6}/\ln\hat{\chi}_{j}$, since by definition $\tau\le 2^{-6}/\ln(1/\hat{\delta})$ as well follows that for all $x\in\bar{Z}$ and j∈[s], η _P (x)⋅ln(1/δ)≤η _j .

Define P=P ₁∪P ₂. We now prove the properties in the lemma for some x∈X, first consider the sub-partition P ₁, and the distribution over the clusters C ₁,C ₂,…,C _s as defined above. For 1≤m≤s, define the events:

$$\begin{aligned} \mathcal{Z}_m = & \bigl\{ \forall j, 1 \leq j < m,\ B(x, \eta_j \varDelta ) \subseteq Z_{j+1}\bigr\} , \\ \mathcal{E}_m = & \bigl\{ \exists j, \, m \leq j < s\ \mathrm{s.t.}\ B(x,\eta_j \varDelta ) \bowtie(S_{v_j},{\bar{S}}_{v_j}) \wedge \mathcal{Z}_m \bigr\} . \end{aligned}$$

Also let T=T _x =B(x,Δ). We prove the following inductive claim: For every 1≤m≤s:

$$\begin{aligned} \Pr[ \mathcal{E}_m \mid\mathcal{Z}_m] \leq(1-\theta) \biggl(1 + \theta\sum_{j \geq m, v_j \in T} \chi_j^{-1} \biggr). \end{aligned}$$

(23)

Note that $\Pr[ \mathcal{E}_{s} ]=0$. Assume the claim holds for m+1 and we will prove for m. Define the events:

$$\begin{aligned} \mathcal{F}_m = & \bigl\{ B(x,\eta_m \varDelta ) \bowtie (S_{v_m},{\bar{S}}_{v_m}) \wedge\mathcal{Z}_m\bigr\} , \\ \mathcal{G}_m = & \bigl\{ B(x,\eta_m \varDelta ) \subseteq{ \bar{S}}_{v_m} \wedge \mathcal{Z}_m \bigr\} = \mathcal{Z}_{m+1} , \\ \bar{\mathcal{G}}_m = & \bigl\{ B(x,\eta_m \varDelta ) \nsubseteq{\bar {S}}_{v_m} \wedge \mathcal{Z}_m \bigr\} =\{ { \bar{\mathcal{Z}}}_{m+1} \wedge\mathcal{Z}_m\}. \end{aligned}$$

First we bound $\Pr[ \mathcal{F}_{m} ]$. Recall that the center v _m of C _m and the value of χ _m are determined deterministically. The radius r _m is chosen from the interval [Δ/4,Δ/2]. Since η _m ≤1/2, if $B(x,\eta_{m} \varDelta ) \bowtie (S_{v_{m}},{\bar{S}}_{v_{m}})$ then d(v _m ,x)≤Δ, and thus v _m ∈T. Therefore if v _m ∉T then $\Pr[ \mathcal{F}_{m} ] = 0$. Otherwise, we shall use Lemma 49 (note that conditioning on $\mathcal{Z}_{m}$ only determines Z _m , all the other parameters are determined deterministically),

$$\begin{aligned} \Pr[\mathcal{F}_m\mid\mathcal{Z}_m ] =& \Pr\bigl[ B(x,\eta_m \varDelta ) \bowtie (S_{v_m},{ \bar{S}}_{v_m}) | \mathcal{Z}_m \bigr] \\ \le& (1-\theta) \bigl(\Pr\bigl[ B(x,\eta_m \varDelta ) \nsubseteq { \bar{S}}_{v_m} |\mathcal{Z}_m \bigr] + \theta \chi_m^{-1}\bigr) \\ =& (1-\theta) \bigl(\Pr[ \bar{\mathcal{G}}_m \mid \mathcal{Z}_m] +\theta \chi_m^{-1}\bigr). \end{aligned}$$

(24)

Since the choice of radius is the only randomness in the process of creating P ₁, the event of padding for z∈Z, and the event B(z,η _P (z)Δ)∩Z=∅ for $z\in\bar{Z}$ are independent of all choices of radii for centers v _j ∉T _z . That is, for any assignment to clusters of points outside B(z,2Δ) (which may determine radius choices for points in X∖B(x,Δ)), the padding probability will not be affected. Using the induction hypothesis we prove the inductive claim:

$$\begin{aligned} \Pr[\mathcal{E}_m \mid\mathcal{Z}_m] \leq& \Pr[ \mathcal{F}_m \mid \mathcal{Z}_m] + \Pr[ \mathcal{G}_m \mid\mathcal{Z}_m] \Pr[ \mathcal{E}_{m+1} \mid\mathcal{Z}_m\wedge\mathcal{G}_m] \\ \leq& (1-\theta) \bigl(\Pr[ \bar{\mathcal{G}}_m \mid \mathcal{Z}_m] + \theta\mathbf{1}_{\{v_m \in T\}} \chi_m^{-1} \bigr) \\ &{}+ \Pr[ \mathcal{G}_m \mid\mathcal{Z}_m] \cdot(1- \theta) \biggl(1 + \theta \sum_{j \geq m+1, v_j \in T} \chi_j^{-1}\biggr) \\ \leq& (1-\theta) \biggl(1 + \theta\sum_{j \geq m, v_j \in T} \chi_j^{-1}\biggr). \end{aligned}$$

The second inequality follows from (24) and the induction hypothesis. Fix some x∈X, T=T _x . Observe that for all v _j ∈T, d(v _j ,x)≤Δ, and so we get B(v _j ,2Δ/γ)⊆B(x,2Δ). On the other hand B(v _j ,2γΔ)⊇B(x,2Δ). Note that the definition of W _j suggests that if v _j is a center then all the other points in B(v _j ,Δ/4) cannot be a center as well, therefore for any j≠j′, d(v _j ,v _j′)>Δ/4≥4Δ/γ, so that B(v _j ,2Δ/γ)∩B(v _j′,2Δ/γ)=∅. Hence, we get:

$$\begin{aligned} \sum_{j \geq1, v_j \in T} \chi_j^{-1} \le& \sum_{j\ge1,v_j \in T} \hat{\chi_j}^{-1}\\ \le& \sum_{j\ge1, v_j \in T} \frac{|B(v_j,2\varDelta /\gamma)|}{|B(v_j,2\gamma \varDelta )|}\\ \leq& \sum_{j\ge1, v_j \in T} \frac{|B(v_j,2\varDelta /\gamma)|}{|B(x, 2\varDelta )|} \leq1. \end{aligned}$$

We conclude from (23) with m=1 that

$$\begin{aligned} \Pr[\mathcal{E}_1\mid\mathcal{Z}_1] \leq(1-\theta) \biggl(1 + \theta\cdot\sum_{j \geq1, v_j \in T} \chi_j^{-1} \biggr) \leq(1-\theta)(1+\theta)\le1-\delta^{1/2}. \end{aligned}$$

Hence there is probability at least δ ^1/2 that event $\mathcal{E}_{1}$ does not occurs. Given that this happens, we will show that there is probability at least δ ^1/2 that x is padded. If x∈Z, then let j∈[s] such that P(x)=C _j , then η _P (x)⋅ln(1/δ)≤η _j and so B(x,η _P (x)⋅ln(1/δ)Δ)⊆B(x,η _j Δ). Note that if x∈Z is padded in P ₁ it will be padded in P. If x∉Z: since for any j∈[s], η _P (x)⋅ln(1/δ)≤η _j we have that $\neg\mathcal{E}_{1}$ suggests that B(x,η _P (x)⋅ln(1/δ)Δ)∩Z=∅. As P ₂ is performed independently of P ₁ we have Pr[B(x,(τ/2)ln(1/δ))⊆P ₂(x)]≥δ ^1/2, hence

$$\begin{aligned} \Pr\bigl[B\bigl(x,(\tau/2)\ln(1/\delta)\bigr)\subseteq P(x)\bigr] \ge&\Pr\bigl[B\bigl(x,(\tau/2)\ln(1/\delta)\bigr)\subseteq P(x)\mid\neg\mathcal{E}_1\bigr]\cdot\Pr[\neg\mathcal{E}_1]\\ \ge&\delta ^{1/2}\cdot \delta^{1/2}=\delta. \end{aligned}$$

It follows that $\hat{\mathcal{P}}$ is uniformly padded. Finally, we show the properties stated in the lemma. The first property follows from the stop condition in phase 1 and from the definition of η _P (x). The second property holds: first take x∈Z and let j be such that x∈C _j , then ξ _P (x)=1 suggests that $\hat{\chi}_{j}\ge 1/\hat{\delta}$ hence $\eta_{P}(x)=2^{-7}/\ln\hat{\chi_{j}}= 2^{-7}/\ln\rho(v_{j},2\varDelta ,\gamma)$ and by the minimality of v _j , η _P (x)≥2⁻⁷/lnρ(x,2Δ,γ). By definition $\eta_{P}(x)\le2^{-7}/\ln(1/\hat{\delta})$. If x∉Z then η _P (x)=τ/2, by the stop condition of phase 1 $\tau/2\ge2^{-7}/\ln\hat{\chi}_{j}$. Again by definition of $\hat{\delta}$ it follows that $\tau/2\le2^{-7}/\ln(1/\hat{\delta})$. As for the third property, which is meaningful only for x∈Z, let j be such that x∈C _j , then ξ _P (x)=0 suggests that $\hat{\chi}_{j}< 1/\hat{\delta}$ hence $\eta_{P}(x)=2^{-7}/\ln(1/\hat{\delta})$ and since d(x,v _j )≤Δ also $\bar{\rho}(x,2\varDelta ,\gamma)\le \rho(v_{j},2\varDelta ,\gamma) < 1/\hat{\delta}$.