epsilon-Kernel Coresets for Stochastic Points

With the dramatic growth in the number of application domains that generateprobabilistic, noisy and uncertain data, there has been an increasing interest in designingalgorithms for geometric or combinatorial optimization problems over such data. Inthis paper, we initiate the study of constructing ε-kernel coresets for uncertain points.We consider uncertainty in the existential model where each point’s location is fixedbut only occurs with a certain probability, and the locational model where each pointhas a probability distribution describing its location. An ε-kernel coreset approximatesthe width of a point set in any direction. We consider approximating the expectedwidth (an ε-exp-kernel), as well as the probability distribution on the width (an(ε, τ)-quant-kernel) for any direction. We show that there exists a set of O(ε−(d−1)/2)deterministic points which approximate the expected width under the existential andlocational models, and we provide efficient algorithms for constructing such coresets.We show, however, it is not always possible to find a subset of the original uncertainpoints which provides such an approximation. However, if the existential probability ofeach point is lower bounded by a constant, an ε-exp-kernel is still possible. We alsoprovide efficient algorithms for construct an (ε, τ)-quant-kernel coreset in nearlylinear time. Our techniques utilize or connect to several important notions in probabilityand geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth,and may be useful in other geometric optimization problem in stochastic settings.Finally, combining with known techniques, we show a few applications to approximatingthe extent of uncertain functions, maintaining extent measures for stochastic movingpoints and some shape fitting problems under uncertainty.