TY - JOUR
AB - We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.
AU - Akopyan, Arseniy
AU - Pirogov, Sergey
AU - Rybko, Aleksandr
ID - 1828
IS - 1
JF - Journal of Statistical Physics
TI - Invariant measures of genetic recombination process
VL - 160
ER -
TY - JOUR
AB - We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid Z2.
AU - Pausinger, Florian
AU - Steinerberger, Stefan
ID - 1938
IS - 6
JF - Physics Letters, Section A
TI - On the distribution of local extrema in quantum chaos
VL - 379
ER -
TY - JOUR
AB - Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility.
AU - Edelsbrunner, Herbert
AU - Jablonski, Grzegorz
AU - Mrozek, Marian
ID - 2035
IS - 5
JF - Foundations of Computational Mathematics
TI - The persistent homology of a self-map
VL - 15
ER -
TY - THES
AB - This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold.
AU - Pausinger, Florian
ID - 1399
TI - On the approximation of intrinsic volumes
ER -
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 -