  Clustering of data points is a fundamental tool in data analysis. We consider
points $X$ in a relaxed metric space, where the triangle inequality holds
within 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 $Q$ such
that $|Q|=k$ and {\em clustering}, which returns an (approximate) minimizer of
$V(Q)$ of size $|Q|=k$. With very large data sets $X$, we seek efficient
constructions of small samples that act as surrogates to the full data for
performing these tasks. Existing constructions that provide quality guarantees
are 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 avoid
both these pitfalls using adaptive designs.
  At the core of our design is the {\em one2all} construction of
multi-objective probability-proportional-to-size (pps) samples: Given a set $M$
of centroids and $\alpha \geq 1$, one2all efficiently assigns probabilities to
points so that the clustering cost of {\em each} $Q$ with cost $V(Q) \geq
V(M)/\alpha$ 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 $M$, but
we adaptively balance $|M|$ and $\alpha$ to further reduce sample size. For
clustering, we design an adaptive wrapper that applies a base clustering
algorithm to a sample $S$. Our wrapper uses the smallest sample that provides
statistical guarantees that the quality of the clustering on the sample carries
over to the full data set. We demonstrate experimentally the huge gains of
using our adaptive instead of worst-case methods.

Machine Learning;Data Structures and Algorithms