Coresets for $(k, \ell)$-Median Clustering under the Fréchet Distance

We present an algorithm for computing $\epsilon$-coresets for $(k, \ell)$-median clustering of polygonal curves under the Fr\'echet distance. This type of clustering, which has recently drawn increasing popularity due to a growing number of applications, is an adaption of Euclidean $k$-median clustering: we are given a set of polygonal curves and want to compute $k$ center curves such that the sum of distances from the given curves to their nearest center curve is minimal. Additionally, we restrict the complexity, i.e., number of vertices, of the center curves to be at most $\ell$ each, to suppress overfitting. We achieve this result by applying the improved $\epsilon$-coreset framework by Braverman et al. to a generalized $k$-median problem over an arbitrary metric space. Later we combine this result with the recent result by Driemel et al. on the VC dimension of metric balls under the Fr\'echet distance. Finally, we show that $\epsilon$-coresets can be used to improve an existing approximation algorithm for $(1,\ell)$-median clustering under the Fr\'echet distance, taking a further step in the direction of practical $(k,\ell)$-median clustering algorithms.