Metric Embeddings with Relaxed Guarantees

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Metric Embeddings with Relaxed Guarantees

• Hubert Chan

Joint work with Kedar Dhamdhere, Anupam
Gupta, Jon Kleinberg, Aleksandrs Slivkins
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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

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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.

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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!
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Roadmap

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

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Definition
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Problem
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Does such a result make sense?
Consider a uniform metric Un on n points.
Suppose f Un ? l2L is 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 (D0.5)L/(0.5)L
is at least n.
4. Hence, L ?(logD n)
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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/ ?.
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One of Our Results
Theorem For every ? gt 0, every finite metric
space can be embedded into Euclidean space with
O(log2 1/?) 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(log2 n) dimensions and O(log n) distortion.
With ? 1/2n2, our result reduces to Bourgains
theorem.
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Corresponding Lower Bounds
Theorem For every ? gt 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!
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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 ? gt 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(?)).
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Roadmap

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

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One mapping does it all
We have shown Given finite metric (V,d) and ? gt
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 ? gt 0, the mapping f is an ?-slack
embedding with good guarantees?
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Gracefully Degrading Embedding
Definition An embedding f (V,d) ? (V,d) has
gracefully degrading distortion D(?) if for every
? gt 0, f is an embedding with ?-slack distortion
D(?).
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Some results for GD Embeddings
Embedding into l2

• Abraham, Bartal, and Neiman obtained similar
  results independently.

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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 ?
?
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Questions?