2019/09/21 by Vladyslav Oles, Nathan Lemons, Oles, Vladyslav +3
Computer Science · Engineering · #Topological and Geometric Data Analysis #Digital Image Processing Techniques #Automated Road and Building Extraction
paper · pdf · doi:10.48550/arxiv.1909.09772
Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we propose polynomial algorithm for estimating the so-called modified Gromov-Hausdorff (mGH) distance, whose topological equivalence with the standard Gromov-Hausdorff (GH) distance was established in Mémoli F, 2012. We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library \verb|scikit-tda|, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.