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.
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 -