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Ultra-low-dimensional embeddings of doubling metrics

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1 Ultra-low-dimensional embeddings of doubling metrics
T-H. Hubert Chan Max Planck Institute, Saarbrucken Anupam Gupta Carnegie Mellon University Kunal Talwar Microsoft Research SVC

2 Doubling Dimension Def: Every ball of radius 2R can be covered by 2k balls of radius R  Doubling dimension = k k has dimension (k) Abstract analog of Euclidean dimension 2R R

3 Doubling metric Def: Doubling metric = an n-point metric with doubling dimension constant (which is independent of n). Advantage: Robust definition, resistant to distorting points slightly. Points on a constant-dimensional manifold have constant doubling dimension! (Even with small noise.)

4 Low distortion Embedding
A map f: (X,d)m such that for all pairs x,y  X d(x,y) ≤ ║f(x)-f(y)║2 ≤ C d(x,y) Small C  f faithfully represents (X,d) Goal: Given an n-point metric space (X,d) with doubling dimension k, find an embedding into m with small distortion. “Distortion” of the embedding Ideally: dimension m and distortion C should be O(k), independent of n when (X,d) is Euclidean.

5 Our Results Dimension-Distortion Trade-off Theorem: Take any (not necessarily Euclidean) metric space (X,d) with doubling dimension k. Fix any integer T such that k loglog n ≤ T ≤ ln n. Then there exists a map f:X T into T-dimensional Euclidean space with distortion (dimD) O log n T

6 Comments on Our Results
A non-linear technique for dimensionality reduction. Interesting special cases of the tradeoff: Very low dimension: Dimension k loglog n for distortion ≈ O(log n) Balanced trade-off: Dimension log2/3 n for distortion O(k log2/3 n)

7 Tools Randomized low-diameter partitioning of doubling metrics
Co-ordinates at different scales combined using random +1/-1 linear combinations (reminiscent of random projections) Lovasz Local Lemma used to prove existence of an embedding with the desired bounds.

8 Future Work Our results apply to all doubling metrics
Thus cannot beat distortion O(log n) (there are known lower bounds) Future Work: better for Euclidean metrics ? Ideally: If (X,d) with doubling dimension k embeds into Euclidean space with distortion D, then want an embedding into O(k) dimensional Euclidean space with distortion O(D). Paper at


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