On the Sensitivity of Shape Fitting Problems

In this article, we study shape fitting problems, -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 -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 -coresets for several variants of the (j, k)-projective clustering problem. We also extend an earlier result on -coresets for the integer (j, k)-projective clustering problem in fixed dimension to the case of high dimension. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity