Clustering for edge-cost minimization (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
On the impossibility of dimension reduction in l1
Journal of the ACM (JACM)
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Metric structures in L1: dimension, snowflakes, and average distortion
European Journal of Combinatorics
Nearest-neighbor-preserving embeddings
ACM Transactions on Algorithms (TALG)
Embedding metric spaces in their intrinsic dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On low dimensional local embeddings
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Ultra-low-dimensional embeddings for doubling metrics
Journal of the ACM (JACM)
Geometry of Cuts and Metrics
Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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The l2 flattening lemma of Johnson and Lindenstrauss [JL84] is a powerful tool for dimension reduction. It has been conjectured that the target dimension bounds can be refined and bounded in terms of the intrinsic dimensionality of the data set (for example, the doubling dimension). One such problem was proposed by Lang and Plaut [LP01] (see also [GKL03, Mat02, ABN08, CGT10]), and is still open. We prove another result in this line of work: The snowflake metric d1/2 of a doubling set S ⊂ l2 can be embedded with arbitrarily low distortion into lD2, for dimension D that depends solely on the doubling constant of the metric. In fact, the target dimension is polylogarithmic in the doubling constant. Our techniques are robust and extend to the more difficult spaces l1 and l∞, although the dimension bounds here are quantitatively inferior than those for l2.