Sketching and Clustering Metric Measure Spaces.

Two important optimization problems in the analysis of geometric data sets are clustering and sketching. Here, clustering refers to the problem of partitioning some input metric measure space (mm-space) into $k$ clusters, minimizing some objective function $f$. Sketching, on the other hand, is the problem of approximating some mm-space by a smaller one supported on a set of $k$ points. Specifically, we define the $k$-sketch of some mm-space $M$ to be the nearest neighbor of $M$ in the set of $k$-point mm-spaces, under some distance function $rho$ on the set of mm-spaces. In this paper, we demonstrate a duality between general classes of clustering and sketching problems. We present a general method for efficiently transforming a solution for a clustering problem to a solution for a sketching problem, and vice versa, with approximately equal cost. More specifically, we obtain the following results. We define the sketching/clustering gap to be the supremum over all mm-spaces of the ratio of the sketching and clustering objectives. 1. For metric spaces, we consider the case where $f$ is the maximum cluster diameter, and $rho$ is the Gromov-Hausdorff distance. We show that the gap is constant for any compact metric space. 2. We extend the above results to obtain constant gaps for the case of mm-spaces, where $rho$ is the $p$-Gromov-Wasserstein distance and the clustering objective involves minimizing various notions of the $ell_p$-diameters of the clusters. 3. We consider two competing notions of sketching for mm-spaces, with one of them being more demanding than the other. These notions arise from two different definitions of $p$-Gromov-Wasserstein distance that have appeared in the literature. We then prove that whereas the gap between these can be arbitrarily large, in the case of doubling metric spaces the resulting sketching objectives are polynomially related.