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 into k clusters, minimizing some objective function f . Sketching, on the other hand, is the problem of approximating some metric measure space by a smaller one supported on a set of k points. Specifically, we define the k-sketch of some metric measure space M to be the nearest neighbor of M in the set of k-point metric measure spaces, under some distance function ρ on the set of metric measure 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 metric measure 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 ρ 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 metric measure spaces, where ρ is the p-Gromov-Wasserstein distance and the clustering objective involves minimizing various notions of the `p-diameters of the clusters. 3. We consider two competing notions of sketching for metric measure 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. ∗Department of Mathematics, The Ohio State University, [email protected],edu. †Department of Computer Science, University of Illinois at Chicago, [email protected]. ‡Department of Mathematics, The Ohio State University, [email protected]. ar X iv :1 80 1. 00 55 1v 1 [ cs .C G ] 2 J an 2 01 8