@phdthesis{6287,
abstract = {The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the Čech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's.},
author = {Nikitenko, Anton},
pages = {86},
publisher = {IST Austria},
title = {{Discrete Morse theory for random complexes }},
doi = {10.15479/AT:ISTA:th_873},
year = {2017},
}
@article{481,
abstract = {We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.},
author = {Biedl, Therese and Huber, Stefan and Palfrader, Peter},
journal = {International Journal of Computational Geometry and Applications},
number = {3-4},
pages = {211 -- 229},
publisher = {World Scientific Publishing},
title = {{Planar matchings for weighted straight skeletons}},
doi = {10.1142/S0218195916600050},
volume = {26},
year = {2017},
}
@article{568,
abstract = {We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).},
author = {Franek, Peter and Krcál, Marek},
issn = {15320073},
journal = {Homology, Homotopy and Applications},
number = {2},
pages = {313 -- 342},
publisher = {International Press},
title = {{Persistence of zero sets}},
doi = {10.4310/HHA.2017.v19.n2.a16},
volume = {19},
year = {2017},
}
@unpublished{6288,
abstract = {The order-k Voronoi tessellation of a locally finite set X⊆ℝn decomposes ℝn into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold. },
author = {Edelsbrunner, Herbert and Nikitenko, Anton},
booktitle = {arXiv:1709.09380},
pages = {11},
title = {{Poisson-Delaunay mosaics of order k}},
year = {2017},
}
@inproceedings{688,
abstract = {We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. },
author = {Edelsbrunner, Herbert and Wagner, Hubert},
issn = {18688969},
location = {Brisbane, Australia},
pages = {391--3916},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Topological data analysis with Bregman divergences}},
doi = {10.4230/LIPIcs.SoCG.2017.39},
volume = {77},
year = {2017},
}