Metric Embeddings with Relaxed Guarantees

1. Metric Embeddings with Relaxed Guarantees Hubert Chan Joint work with Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, Aleksandrs Slivkins

2. Quality measured by distortion: The distortion of a mapping is D if for all pairs of nodes (x,y). Embedding & Distortion Central Idea: Given finite metric (V,d), embed into simpler metric (V’,d’), (e.g. Euclidean space l2) via mapping • Application in approximation algorithms, e.g. sparsest cut

3. Application in Networking • Suppose you want to: • find a “near” server in a network with replicated services • find a “near” copy of file in P2P system • Employ embedding techniques: • Point-to-point latencies treated as a metric (V,d) • Embed into Euclidean space f : (V,d) →l2 • Each node in the network gets virtual coordinates. • Latencies between points can be approximated.

4. Practical Issues Distortion may be too restrictive: • Some metrics embed into l2 with (log n) distortion. e.g. metrics induced by constant degree expanders • Some metrics embed into l2 with (log n) dimensions. e.g. uniform metrics It is sufficient in some applications to obtain good approximation for most pairs of nodes. For example, one might need a server that is among the nearest 1% of all nodes. We can do better in this case!

5. Roadmap • Motivation • Define Embeddings with Slack • Example & Results- Upper Bounds- Lower Bounds • Gracefully Degrading Embeddings • Open Questions

6. Recall: An embedding f : V→V’ has distortion D if Define: An embedding f : V→V’ has distortion D with -slack if for all pairs of nodes (x,y). for all but n2 pairs of nodes (x,y). Definition

7. Formally, A question posed by Kleinberg et al in FOCS ’04: Are there functions D() and L() such that for every  > 0, every finite metric space can be embedded into Euclidean space with L() dimensions and distortion D() with  -slack? Can every finite metric space can be embedded into Euclidean space with constant number of dimensions and constant distortion with constant slack? Problem

8. D Does such a result make sense? Consider a uniform metric Un on n points. Suppose f : Un→l2Lis an embedding into Euclidean space in L dimensions with (no slack) distortion D. Claim. # dimensions L = (logD n) 1. The images are contained in some ball of radius at most D. 2. Balls of radius 0.5 around the image of each point are pairwise disjoint. 3. Simple volume argument shows (D+0.5)L/(0.5)L is at least n. 4. Hence, L = (logD n)

9. 1/  1 2 3 4 … With slack comes more power… Consider the following embedding f : Un→ R. n Pairs within a cluster are ignored. How many pairs are ignored? • Each cluster has n points. • Each node ignores at most n other nodes. • At most n2 pairs are ignored. Distortion (with -slack) is 1/ .

10. One of Our Results Theorem For every  > 0, every finite metric space can be embedded into Euclidean space with O(log21/) dimensions and O(log 1/) distortion with  -slack. Theorem [Bourgain ’85] Every finite metric space of size n can be embedded into Euclidean space with O(log2n) dimensions and O(log n) distortion. With  = 1/2n2, our result reduces to Bourgain’s theorem.

11. Corresponding Lower Bounds Theorem For every  > 0, there exists a finite metric such that any -slack embedding into Euclidean space incurs distortion at least (log 1/). Theorem [Matousek ’97] There exists a finite metric space of size n such that any embedding into Euclidean space incurs distortion at least (log n). It seems we can just replace n with 1/ to get a lower bound for slack embeddings!

12. General Principle for Translating Lower Bounds Theorem Suppose there exists a family of metrics for which any embedding into l2 with at most L(n) dimensions incurs (no slack) distortion at least D(n). Then, for all  > 0, there exists a family of metrics for which any embedding into l2 with at most L(1/ 3 sqrt()) dimensions incurs (no slack) distortion at least D(1/ 3 sqrt()).

13. Roadmap • Motivation • Define Embeddings with Slack • Example & Results- Upper Bounds- Lower Bounds • Gracefully Degrading Embeddings • Open Questions

14. One mapping does it all… We have shown: Given finite metric (V,d) and  > 0, we can construct an -slack embedding with good guarantees. Question Given a finite metric (V,d), can we construct a single embedding f such that for every > 0, the mapping f is an -slack embedding with good guarantees?

15. Gracefully Degrading Embedding Definition An embedding f : (V,d) → (V’,d’) has gracefully degrading distortion D() if for every  > 0, f is an embedding with -slack distortion D().

16. Some results for GD Embeddings Embedding into l2 • Abraham, Bartal, and Neiman obtained similar results independently.

17. 1. Is there a function D() depending only on , such that every finite metric can be embedded into l2 with gracefully degrading distortion D() and polylog(n) dimensions? Theorem Any finite metric can be embedded into l1 with gracefully degrading distortion O(log 1/) in poly(n) dimensions. Open Questions 2. Given a metric with doubling dimension , is there a gracefully degrading embedding into l2 with the number of dimensions depending only on  ?

18. Questions?