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.
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. 1. For metric spaces, we consider the case where the clustering objective is minimizing the maximum cluster diameter . We show that the ratio between the sketching and clustering objectives is constant over compact metric spaces . 2. We extend these results to the setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the clustering objective involves minimizing various notions of the l _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.
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We define the sketching/clustering gap to be the supremum over all mm-spaces of the ratio of the sketching and clustering objectives.
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the clustering objective is minimizing
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, and \rho is the Gromov-Hausdorff distance
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gap is constant for any compact metric space
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the above results to obtain constant gaps for the case of mm-spaces, where \rho is the p-Gromov-Wasserstein distance and
these results to
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setting of metric measure spaces where we prove that the ratio of sketching to clustering objectives is bounded both above and below by some universal constants. In this setting, the
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