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Clustering of data points is a fundamental tool in data analysis. We considerpoints $X$ in a relaxed metric space, where the triangle inequality holdswithin a constant factor. The {\em cost} of clustering $X$ by $Q$ is$V(Q)=\sum_{x\in X} d_{xQ}$. Two basic tasks, parametrized by $k \geq 1$, are{\em cost estimation}, which returns (approximate) $V(Q)$ for queries $Q$ suchthat $|Q|=k$ and {\em clustering}, which returns an (approximate) minimizer of$V(Q)$ of size $|Q|=k$. With very large data sets $X$, we seek efficientconstructions of small samples that act as surrogates to the full data forperforming these tasks. Existing constructions that provide quality guaranteesare either worst-case, and unable to benefit from structure of real data sets,or make explicit strong assumptions on the structure. We show here how to avoidboth these pitfalls using adaptive designs.  At the core of our design is the {\em one2all} construction ofmulti-objective probability-proportional-to-size (pps) samples: Given a set $M$of centroids and $\alpha \geq 1$, one2all efficiently assigns probabilities topoints so that the clustering cost of {\em each} $Q$ with cost $V(Q) \geqV(M)/\alpha$ can be estimated well from a sample of size $O(\alpha|M|\epsilon^{-2})$. For cost queries, we can obtain worst-case sample size$O(k\epsilon^{-2})$ by applying one2all to a bicriteria approximation $M$, butwe adaptively balance $|M|$ and $\alpha$ to further reduce sample size. Forclustering, we design an adaptive wrapper that applies a base clusteringalgorithm to a sample $S$. Our wrapper uses the smallest sample that providesstatistical guarantees that the quality of the clustering on the sample carriesover to the full data set. We demonstrate experimentally the huge gains ofusing our adaptive instead of worst-case methods.

Clustering Small Samples with Quality Guarantees: Adaptivity with One2all pps