On the Sensitivity of Shape Fitting Problems

Abstract: In this article, we study shape fitting problems, $\epsilon$-coresets, and total sensitivity. We focus on the $(j,k)$-projective clustering problems, including $k$-median/$k$-means, $k$-line clustering, $j$-subspace approximation, and the integer $(j,k)$-projective clustering problem. We derive upper bounds of total sensitivities for these problems, and obtain $\epsilon$-coresets using these upper bounds. Using a dimension-reduction type argument, we are able to greatly simplify earlier results on total sensitivity for the $k$-median/$k$-means clustering problems, and obtain positively-weighted $\epsilon$-coresets for several variants of the $(j,k)$-projective clustering problem. We also extend an earlier result on $\epsilon$-coresets for the integer $(j,k)$-projective clustering problem in fixed dimension to the case of high dimension.