Coresets, complexity of shapes, and total sensitivity

Abstract

In this document, we consider coreset and total sensitivity for shape fitting problems. The shape fitting problems that are of considerable interest include: (1) (j, k) projective clustering problem, and (2) circle fitting problem on the plane. In (j, k) projective clustering, we are given a finite set of points P in d-dimensional Euclidean space, and the goal is to find a shape, which is a k-tuple j-flats (affine jsubspace), that best fits P . In circle fitting problem, given an input point set P ⊂ R, the goal is to find a circle that best fits P . In L1-fitting, the cost of fitting P to a shape F is defined as ∑ p∈P dist(p, F ), where dist(p, F ) is the cost of assigning p to F , while in L∞-fitting, maxp∈P dist(p, F ). We focus on L1-fitting. A coreset is a compact representation of the input point set. For a shape fitting problem, a coreset for a point set P is a weighted point set, with the property that the cost of fitting the coreset to a shape F approximates the cost of fitting P to F , for every shape in the family of shapes. Coreset of small (e.g., constant) cardinality is of interest, because one can afford to use off-shelf, perhaps computationally expensive algorithms to solve the geometric optimization problem for the coreset, and a good solution for the coreset is guaranteed to be also good for the original input. Depending on whether the fitting problem is L1 fitting or L∞ fitting, the coreset is L1 coreset or L∞ coreset, respectively. One way to obtain small coreset is via non-uniform sampling, using the framework by [30]. Given a point set P , the “importance” of each point p ∈ P is quantified by