Ultra-low-dimensional embeddings of doubling metrics

1. Ultra-low-dimensional embeddings of doubling metrics T-H. Hubert ChanMax Planck Institute, Saarbrucken Anupam GuptaCarnegie Mellon University KunalTalwarMicrosoft Research SVC

2. Doubling Dimension Def: Every ball of radius 2R can be covered by 2k balls of radius R Doubling dimension= k • khas 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)msuch 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 loglogn ≤ T ≤ lnn. Thenthere exists a map f:X Tinto T-dimensional Euclidean space with distortion • (dimD) • O • log n • T

6. Comments on Our Results A non-linear technique for dimensionality reduction. Interestingspecial cases of the tradeoff: • Verylow dimension: Dimension k loglog n for distortion≈ O(log n) • Balancedtrade-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) • LovaszLocal 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 http://www.cs.cmu.edu/~hubert