TY - THES
AB - 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.
AU - Nikitenko, Anton
ID - 6287
TI - Discrete Morse theory for random complexes
ER -
TY - CONF
AB - 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.
AU - Edelsbrunner, Herbert
AU - Wagner, Hubert
ID - 688
SN - 18688969
TI - Topological data analysis with Bregman divergences
VL - 77
ER -
TY - JOUR
AB - We answer a question of M. Gromov on the waist of the unit ball.
AU - Akopyan, Arseniy
AU - Karasev, Roman
ID - 707
IS - 4
JF - Bulletin of the London Mathematical Society
SN - 00246093
TI - A tight estimate for the waist of the ball
VL - 49
ER -
TY - CONF
AB - We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.
AU - Heiss, Teresa
AU - Wagner, Hubert
ED - Felsberg, Michael
ED - Heyden, Anders
ED - Krüger, Norbert
ID - 833
SN - 03029743
TI - Streaming algorithm for Euler characteristic curves of multidimensional images
VL - 10424
ER -
TY - CONF
AB - Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.
AU - Ethier, Marc
AU - Jablonski, Grzegorz
AU - Mrozek, Marian
ID - 836
SN - 978-331956930-7
T2 - Special Sessions in Applications of Computer Algebra
TI - Finding eigenvalues of self-maps with the Kronecker canonical form
VL - 198
ER -