Witnessed k-Distance

Leonidas Guibas, Quentin Mérigot, Dmitriy Morozov.
DCG: Discrete and Computational Geometry, vol. 49, pages 22-45, 2013.
SoCG'11: Proceedings of the Annual Symposium on Computational Geometry, pages 57-64, 2011.
DOI: 10.1007/s00454-012-9465-x
arXiv: 1102.4972
Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.
k-Distance sublevelsets
Frédéric Chazal, David Cohen-Steiner, Quentin Mérigot. Geometric inference for probability measures. Foundations of Computational Mathematics, 11:733–751, 2011.