TY - GEN

T1 - A nonlinear approach to dimension reduction

AU - Gottlieb, Lee Ad

AU - Krauthgamer, Robert

PY - 2011

Y1 - 2011

N2 - 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 ⊂ C l2 can be embedded with arbitrarily low distortion into, l2D 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.

AB - 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 ⊂ C l2 can be embedded with arbitrarily low distortion into, l2D 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.

UR - http://www.scopus.com/inward/record.url?scp=79955709004&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973082.69

DO - 10.1137/1.9781611973082.69

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AN - SCOPUS:79955709004

SN - 9780898719932

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 888

EP - 899

BT - Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011

ER -