TY - CONF
AB - We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data.
AU - Kwitt, Roland
AU - Huber, Stefan
AU - Niethammer, Marc
AU - Lin, Weili
AU - Bauer, Ulrich
ID - 1424
TI - Statistical topological data analysis-A kernel perspective
VL - 28
ER -
TY - CONF
AB - Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.
AU - Reininghaus, Jan
AU - Huber, Stefan
AU - Bauer, Ulrich
AU - Kwitt, Roland
ID - 1483
TI - A stable multi-scale kernel for topological machine learning
ER -
TY - CONF
AB - Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.
AU - Edelsbrunner, Herbert
AU - Iglesias Ham, Mabel
AU - Kurlin, Vitaliy
ID - 1495
T2 - Proceedings of the 27th Canadian Conference on Computational Geometry
TI - Relaxed disk packing
VL - 2015-August
ER -
TY - CONF
AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1510
TI - On computability and triviality of well groups
VL - 34
ER -
TY - JOUR
AB - The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.
AU - Zobel, Valentin
AU - Jan Reininghaus
AU - Hotz, Ingrid
ID - 1531
JF - Mathematics and Visualization
TI - Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature
VL - 40
ER -
TY - JOUR
AB - We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.
AU - Knipl, Diána
AU - Pilarczyk, Pawel
AU - Röst, Gergely
ID - 1555
IS - 2
JF - SIAM Journal on Applied Dynamical Systems
TI - Rich bifurcation structure in a two patch vaccination model
VL - 14
ER -
TY - JOUR
AB - For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at {\tt http://www.pawelpilarczyk.com/combtop/}.
AU - Graff, Grzegorz
AU - Pilarczyk, Pawel
ID - 1563
IS - 1
JF - Topological Methods in Nonlinear Analysis
TI - An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds
VL - 45
ER -
TY - CONF
AB - My personal journey to the fascinating world of geometric forms started more than 30 years ago with the invention of alpha shapes in the plane. It took about 10 years before we generalized the concept to higher dimensions, we produced working software with a graphics interface for the three-dimensional case. At the same time, we added homology to the computations. Needless to say that this foreshadowed the inception of persistent homology, because it suggested the study of filtrations to capture the scale of a shape or data set. Importantly, this method has fast algorithms. The arguably most useful result on persistent homology is the stability of its diagrams under perturbations.
AU - Edelsbrunner, Herbert
ID - 1567
TI - Shape, homology, persistence, and stability
VL - 9411
ER -
TY - CONF
AB - Aiming at the automatic diagnosis of tumors from narrow band imaging (NBI) magnifying endoscopy (ME) images of the stomach, we combine methods from image processing, computational topology, and machine learning to classify patterns into normal, tubular, vessel. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.
AU - Dunaeva, Olga
AU - Edelsbrunner, Herbert
AU - Lukyanov, Anton
AU - Machin, Michael
AU - Malkova, Daria
ID - 1568
T2 - Proceedings - 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
TI - The classification of endoscopy images with persistent homology
ER -
TY - JOUR
AB - We prove that the dual of the digital Voronoi diagram constructed by flooding the plane from the data points gives a geometrically and topologically correct dual triangulation. This provides the proof of correctness for recently developed GPU algorithms that outperform traditional CPU algorithms for constructing two-dimensional Delaunay triangulations.
AU - Cao, Thanhtung
AU - Edelsbrunner, Herbert
AU - Tan, Tiowseng
ID - 1578
IS - 7
JF - Computational Geometry
TI - Triangulations from topologically correct digital Voronoi diagrams
VL - 48
ER -