Article

A PTAS for k-means clustering based on weak coresets

Authors:

Morteza Monemizadeh,

Christian SohlerAuthors Info & Claims

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

Pages 11 - 18

https://doi.org/10.1145/1247069.1247072

Published: 06 June 2007 Publication History

Abstract

Given a point set P ⊆ R^d the k-means clustering problem is to find a set C=(c₁,...,c_k) of k points and a partition of P into k clusters C₁,...,C_k such that the sum of squared errors ∑_i=1^k ∑_{p ∈ C_i} |p -c_i |₂² is minimized. For given centers this cost function is minimized byassigning points to the nearest center.The k-means cost function is probably the most widely used cost function in the area of clustering.In this paper we show that every unweighted point set P has a weak (ε, k)-coreset of size Poly(k,1/ε) for the k-means clustering problem, i.e. its size is independent of the cardinality |P| of the point set and the dimension d of the Euclidean space R^d. A weak coreset is a weighted set S ⊆ P together with a set T such that T contains a (1+ε)-approximation for the optimal cluster centers from P and for every set of kcenters from T the cost of the centers for S is a (1±ε)-approximation of the cost for P.We apply our weak coreset to obtain a PTAS for the k-means clustering problem with running time O(nkd + d · Poly(k/ε) + 2^Õ(k/ε)).

References

[1]

J.L. Bentley and J.B. Saxe. Decomposable searching problems I: Static--to--dynamic transformation. J. Algorithms, 1(4):301--358. 1980, pages 301--358, 1980.

[2]

M. Badoiu, S. Har--Peled, and P. Indyk. Approximate clustering via core-sets. Proc. 34th Annu. ACM Sympos. Theory Comput. (STOC), pages 396--407, 2002.

Digital Library

[3]

K. Chen. On k-Median clustering in high dimensions. Proc. 17th Annual ACM-SIAM Symposium of Discrete Algorithms (SODA), pages 1177--1185, 2006.

Digital Library

[4]

W. Fernandez de la Vega, M. Karpinski, C. Kenyon, and Y. Rabani. Approximation schemes for clustering problems. Proc. 35th Annu. ACM Sympos. Theory Comput. (STOC), pages 50--58, 2003.

Digital Library

[5]

M. Effros and L.J. Schulman. Deterministic clustering with data nets. Report TR04-085, Elec. Colloq. Comp. Complexity, http://www.eccc.uni--trier.de/eccc-reports/2004/TR04-085, 2003.

[6]

D. Feldman, A. Fiat, and M. Sharir. Coresets for weighted and their applications. Proc. 47th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), 2006.

Digital Library

[7]

G. Frahling and C. Sohler. Coresets in dynamic geometric data streams. Proc. 37th Annu. ACM Sympos. Theory Comput. (STOC), pages 209--217, 2005.

Digital Library

[8]

D. Haussler. Decision theoretic generalizations of the pac model for neural net and other learning applications. Information and Computation, 100(1):78--150, 1992.

Digital Library

[9]

S. Har-Peled and A. Kushal. Smaller coresets for k-median and k-means clustering. Proc. 21st Annu. ACM Sympos. Comput. Geom. (SOCG), pages 126--134, 2005.

Digital Library

[10]

S. Har-Peled and S. Mazumdar. Coresets for k-means and k-median clustering and their applications. Proc. 36th Annu. ACM Sympos. Theory Comput. (STOC), pages 291--300, 2004.

Digital Library

[11]

S. Har-Peled and K.R. Varadarajan. Approximation schemes for clustering problems. Proc. 18th Annu. ACM Sympos. Comput. Geom.(SoCG), pages 312--318, 2002.

[12]

M. Inaba, N. Katoh, and H. Imai. Applications of weighted voronoi diagrams and randomization to variance-based k-clustering. Proc. 10th Annu. ACM Sympos. Comput. Geom.(SoCG), pages 332--339, 1994.

Digital Library

[13]

A. Kumar, Y. Sabharwal, and S. Sen. A simple linear time (1+ε)-approximation algorithm for k-means clustering in any dimensions. Proc. 45th Annual Symposium on Foundations of Computer Science, pages 454--462, 2004.

Digital Library

[14]

A. Kumar, Y. Sabharwal, and S. Sen. Linear time algorithms for clustering problems in any dimensions. Proc. 32nd Annual Internat. Colloquium on Automata, Languages, and Programming (ICALP), pages 1374--1385, 2005.

Digital Library

[15]

S. Lloyd. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28:129--137, 1982.

Digital Library

[16]

N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215--245, 1995.

[17]

J. Matousek. On approximate geometric k-clustering. Discrete Comput. Geom., 24:61--84, 2000.

[18]

R.R. Mettu and C.G. Plaxton. Optimal time bounds for approximate clustering. Machine Learning, 56:35--60, 2004.

Digital Library

[19]

R. Ostrovsky, Y. Rabani, L. Shulman, and C. Swamy. The effectiveness of lloyd-type methods for the k-means problem. Proc. 47th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), 2006.

Digital Library

Cited By

Fox EHuang HRaichel B(2024)Clustering with faulty centersComputational Geometry10.1016/j.comgeo.2023.102052117(102052)Online publication date: Feb-2024
https://doi.org/10.1016/j.comgeo.2023.102052
Jiang SKrauthgamer RLou JZhang Y(2024)Coresets for kernel clusteringMachine Learning10.1007/s10994-024-06540-z113:8(5891-5906)Online publication date: 22-Apr-2024
https://doi.org/10.1007/s10994-024-06540-z
Feng QFu B(2024)Speeding Up Constrained k-Means Through 2-MeansAlgorithmic Aspects in Information and Management10.1007/978-981-97-7801-0_5(52-63)Online publication date: 19-Sep-2024
https://doi.org/10.1007/978-981-97-7801-0_5
Show More Cited By

Index Terms

A PTAS for k-means clustering based on weak coresets
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

On coresets for k-means and k-median clustering
STOC '04: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing

In this paper, we show the existence of small coresets for the problems of computing k-median and k-means clustering for points in low dimension. In other words, we show that given a point set P in R^d, one can compute a weighted set S ⊆ P, of size O(k ε^-...
Smaller coresets for k-median and k-means clustering
SCG '05: Proceedings of the twenty-first annual symposium on Computational geometry

In this paper, we show that there exists a (k, ε)-coreset for k-median and k-means clustering of n points in R^d, which is of size independent of n. In particular, we construct a (k, ε)-coreset of size O(k²/ε^d) for k-median clustering, and of size O(k³/ε^...
Bi-criteria linear-time approximations for generalized k-mean/median/center
SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

We consider the problem of approximating a set P of n points in R^d by a collection of j-dimensional flats, andextensions thereof, under the standard median / mean / centermeasures, in which we wish to minimize, respectively, the sum of thedistances from ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SCG '07: Proceedings of the twenty-third annual symposium on Computational geometry

June 2007

404 pages

ISBN:9781595937056

DOI:10.1145/1247069

Program Chair:
Jeff Erickson
University of Illinois, Urbana-Champaign, USA

Copyright © 2007 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 June 2007

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

96
Total Citations
View Citations
765
Total Downloads

Downloads (Last 12 months)47
Downloads (Last 6 weeks)6

Reflects downloads up to 15 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Fox EHuang HRaichel B(2024)Clustering with faulty centersComputational Geometry10.1016/j.comgeo.2023.102052117(102052)Online publication date: Feb-2024
https://doi.org/10.1016/j.comgeo.2023.102052
Jiang SKrauthgamer RLou JZhang Y(2024)Coresets for kernel clusteringMachine Learning10.1007/s10994-024-06540-z113:8(5891-5906)Online publication date: 22-Apr-2024
https://doi.org/10.1007/s10994-024-06540-z
Feng QFu B(2024)Speeding Up Constrained k-Means Through 2-MeansAlgorithmic Aspects in Information and Management10.1007/978-981-97-7801-0_5(52-63)Online publication date: 19-Sep-2024
https://doi.org/10.1007/978-981-97-7801-0_5
Yang HPan XDeng JYin J(2024)An Effective RSP Data Sampling AlgorithmKnowledge Science, Engineering and Management10.1007/978-981-97-5501-1_25(331-342)Online publication date: 27-Jul-2024
https://doi.org/10.1007/978-981-97-5501-1_25
Dexter GWoodruff DDrineas PYasuda TOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Sketching algorithms for sparse dictionary learningProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668223(48431-48443)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668223
Esfandiari HMirrokni VZhong PAgrawal KShun J(2023)Brief Announcement: Streaming Balanced ClusteringProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591318(311-314)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591318
Abbasi FBanerjee SByrka JChalermsook PGadekar AKhodamoradi KMarx DSharma RSpoerhase J(2023)Parameterized Approximation Schemes for Clustering with General Norm Objectives2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00085(1377-1399)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00085
Zhao GLi GQin YYu Y(2023)Improved Distribution Matching for Dataset Condensation2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR52729.2023.00759(7856-7865)Online publication date: Jun-2023
https://doi.org/10.1109/CVPR52729.2023.00759
Lu JTang JXing BTang X(2022)Stochastic Approximate Algorithms for Uncertain Constrained K-Means ProblemMathematics10.3390/math1001014410:1(144)Online publication date: 4-Jan-2022
https://doi.org/10.3390/math10010144
Hoang NDang T(2022)Alpha Lightweight Coreset for k-Means Clustering2022 16th International Conference on Ubiquitous Information Management and Communication (IMCOM)10.1109/IMCOM53663.2022.9721770(1-8)Online publication date: 3-Jan-2022
https://doi.org/10.1109/IMCOM53663.2022.9721770
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents