Applied Mathematics

•  Kritika Singhal, Ohio State University
•  Sketching and Clustering Metric Measure Spaces
•  11/08/2019
•  4:10 PM - 5:00 PM
•  C304 Wells Hall

Two important optimization problems in the analysis of geometric data sets are clustering and simplification (sketching) of data. Clustering refers to partitioning a dataset, according to some rule, into sets of smaller size with the aim of extracting important information from the data. Sketching, or simplification of data, refers to approximating the input data with another dataset of much smaller size in such a way that properties of the input dataset are retained by the smaller dataset. In this sense, sketching facilitates understanding of the data. There are many clustering methods for metric spaces (mm spaces) already present in literature, such as k-center clustering, k-median clustering, k-means clustering, etc. A natural method for obtaining a k-sketch of a metric space (mm space) is by viewing the space of all metric spaces (mm space) as a metric under Gromov-Hausdorff (Gromov-Wasserstein) distance, and then determining, under this distance, the k point metric space (mm space) closest to the input metric space (mm space). These two problems of sketching and clustering, a priori, look completely unrelated. However, we establish a duality i.e. an equivalence between these notions of sketching and clustering. 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. 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 $\ell_p$-diameters of the clusters. We also identify procedures/maps that transform a solution of the sketching problem to a solution of the clustering problem, and vice-versa. These maps give rise to algorithms for performing these transformations and, by virtue of these algorithms, we are able to obtain an approximation to the k-sketch of a metric measure space (metric space) using known approximation algorithms for the respective clustering objectives. This is joint work with Facundo Memoli and Anastasios Sidiropoulos, and is available online at https://arxiv.org/abs/1801.00551.

Contact

Department of Mathematics
Michigan State University