Local Search Yields a PTAS for k-Means in Doubling Metrics

The most well known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly k-MEANS: given a set of data points and a parameter k, select k centres and partition the data points into k clusters around these centres so that the sum of squares of distances of the points to their cluster centre is minimized. Typically these data points lie in Euclidean space Rd for some d ≥ 2. k-MEANS and the first algorithms for it were introduced in the 1950's. Over the last six decades, hundreds of papers have studied this problem and different algorithms have been proposed for it. The most commonly used algorithm in practice is known as Lloyd-Forgy, which is also referred to as "the" k-MEANS algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [2004] analyzed a very simple local search heuristic to get a polynomial-time algorithm with approximation ratio 9 + ε for any fixed ε > 0 for k-Umeans in Euclidean space. Finding an algorithm with a better worst-case approximation guarantee has remained one of the biggest open questions in this area, in particular whether one can get a true PTAS for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for k-MEANS for Rd for any fixed d. More precisely, for any error parameter ε > 0, the local search algorithm that considers swaps of up to ρ = dO(d) · ε-O(d/ε) centres at a time will produce a solution using exactly k centres whose cost is at most a (1+ε)-factor greater than the optimum solution. Our analysis extends very easily to the more general settings where we want to minimize the sum of q'th powers of the distances between data points and their cluster centres (instead of sum of squares of distances as in k-MEANS) for any fixed q ≥ 1 and where the metric may not be Euclidean but still has fixed doubling dimension.

[1]  Sergei Vassilvitskii,et al.  How slow is the k-means method? , 2006, SCG '06.

[2]  Rafail Ostrovsky,et al.  Polynomial-time approximation schemes for geometric min-sum median clustering , 2002, JACM.

[3]  Shi Li,et al.  Approximating k-median via pseudo-approximation , 2012, STOC '13.

[4]  Alan M. Frieze,et al.  Clustering Large Graphs via the Singular Value Decomposition , 2004, Machine Learning.

[5]  Sariel Har-Peled,et al.  How Fast Is the k-Means Method? , 2005, SODA '05.

[6]  Anupam Gupta,et al.  Simpler Analyses of Local Search Algorithms for Facility Location , 2008, ArXiv.

[7]  Piotr Indyk,et al.  Approximate clustering via core-sets , 2002, STOC '02.

[8]  Meena Mahajan,et al.  The Planar k-means Problem is NP-hard I , 2009 .

[9]  Amit Kumar,et al.  A simple linear time (1 + /spl epsiv/)-approximation algorithm for k-means clustering in any dimensions , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[10]  Sanjoy Dasgupta.  How Fast Is k-Means? , 2003, COLT.

[11]  S. Dasgupta.  The hardness of k-means clustering , 2008 .

[12]  Christian Sohler,et al.  Theoretical Analysis of the k-Means Algorithm - A Survey , 2016, Algorithm Engineering.

[13]  Anil K. Jain.  Data clustering: 50 years beyond K-means , 2008, Pattern Recognit. Lett..

[14]  Claire Mathieu,et al.  The Unreasonable Success of Local Search: Geometric Optimization , 2014, ArXiv.

[15]  M. Inaba.  Application of weighted Voronoi diagrams and randomization to variance-based k-clustering , 1994, SoCG 1994.

[16]  Philip N. Klein,et al.  Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[17]  Patrik D'haeseleer,et al.  How does gene expression clustering work? , 2005, Nature Biotechnology.

[18]  L. Schulman,et al.  Universal ε-approximators for integrals , 2010, SODA '10.

[19]  Amit Kumar,et al.  Linear-time approximation schemes for clustering problems in any dimensions , 2010, JACM.

[20]  Amit Kumar,et al.  Clustering with Spectral Norm and the k-Means Algorithm , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[21]  Ke Chen,et al.  On Coresets for k-Median and k-Means Clustering in Metric and Euclidean Spaces and Their Applications , 2009, SIAM J. Comput..

[22]  Avrim Blum,et al.  Stability Yields a PTAS for k-Median and k-Means Clustering , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[23]  Dan Feldman,et al.  A PTAS for k-means clustering based on weak coresets , 2007, SCG '07.

[24]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[25]  Mary Inaba,et al.  Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract) , 1994, SCG '94.

[26]  Aravind Srinivasan,et al.  An Improved Approximation for k-Median and Positive Correlation in Budgeted Optimization , 2014, SODA.

[27]  Kamesh Munagala,et al.  Local search heuristic for k-median and facility location problems , 2001, STOC '01.

[28]  J. Matou.  On Approximate Geometric K-clustering , 1999 .

[29]  Ravishankar Krishnaswamy,et al.  The Hardness of Approximation of Euclidean k-Means , 2015, SoCG.

[30]  Shi Li,et al.  Approximating k-Median via Pseudo-Approximation , 2016, SIAM J. Comput..

[31]  Dan Feldman,et al.  Turning big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering , 2013, SODA.

[32]  Andrew M. Gross,et al.  Network-based stratification of tumor mutations , 2013, Nature Methods.

[33]  Shi Li,et al.  A Dependent LP-Rounding Approach for the k-Median Problem , 2012, ICALP.

[34]  Philip N. Klein,et al.  The power of local search for clustering , 2016, ArXiv.

[35]  Kamesh Munagala,et al.  Local Search Heuristics for k-Median and Facility Location Problems , 2004, SIAM J. Comput..

[36]  Marek Karpinski,et al.  Approximation schemes for clustering problems , 2003, STOC '03.

[37]  Kunal Talwar,et al.  Bypassing the embedding: algorithms for low dimensional metrics , 2004, STOC '04.

[38]  Satish Rao,et al.  A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem , 1999, ESA.

[39]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[40]  Andrea Vattani,et al.  k-means Requires Exponentially Many Iterations Even in the Plane , 2008, SCG '09.

[41]  Sayan Bandyapadhyay,et al.  On Variants of k-means Clustering , 2015, SoCG.

[42]  Bodo Manthey,et al.  Smoothed Analysis of the k-Means Method , 2011, JACM.

[43]  Sariel Har-Peled,et al.  On coresets for k-means and k-median clustering , 2004, STOC '04.

[44]  Claire Mathieu,et al.  Effectiveness of Local Search for Geometric Optimization , 2015, SoCG.

[45]  Sariel Har-Peled,et al.  Smaller Coresets for k-Median and k-Means Clustering , 2005, SCG.

[46]  Pierre Hansen,et al.  NP-hardness of Euclidean sum-of-squares clustering , 2008, Machine Learning.

[47]  Sergei Vassilvitskii,et al.  k-means++: the advantages of careful seeding , 2007, SODA '07.

[48]  Andrea Vattani.  The hardness of k-means clustering in the plane , 2010 .

[49]  David M. Mount,et al.  A local search approximation algorithm for k-means clustering , 2002, SCG '02.

[50]  Satish Rao,et al.  Approximation schemes for Euclidean k-medians and related problems , 1998, STOC '98.

[51]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.