TY - GEN

T1 - Performance of Johnson–Lindenstrauss transform for k-means and k-medians clustering

AU - Makarychev, Konstantin

AU - Makarychev, Yury

AU - Razenshteyn, Ilya

N1 - Publisher Copyright:
© 2019 Association for Computing Machinery.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/6/23

Y1 - 2019/6/23

N2 - Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1 + ε) under a projection onto a random O(log(k/ε)/ε2)-dimensional subspace. Further, the cost of every clustering is preserved within (1 + ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p. For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan.

AB - Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1 + ε) under a projection onto a random O(log(k/ε)/ε2)-dimensional subspace. Further, the cost of every clustering is preserved within (1 + ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p. For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan.

KW - Clustering

KW - Dimension reduction

KW - Johnson-Lindenstrauss transform

KW - K-means

KW - K-medians

KW - Kirszbraun theorem

UR - http://www.scopus.com/inward/record.url?scp=85068734142&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068734142&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316350

DO - 10.1145/3313276.3316350

M3 - Conference contribution

AN - SCOPUS:85068734142

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1027

EP - 1038

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

T2 - 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019

Y2 - 23 June 2019 through 26 June 2019

ER -