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Register a new userUniversality of the Bottleneck Distance for Extended Persistence Diagrams
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Added by
Ulrich Bauer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The extended persistence diagram is an invariant of piecewise linear functions, introduced by Cohen-Steiner, Edelsbrunner, and Harer. The bottleneck distance has been introduced by the same authors as an extended pseudometric on the set of extended persistence diagrams, which is stable under perturbations of the function. We address the question whether the bottleneck distance is the largest possible stable distance, providing an affirmative answer.
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We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.
The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $mathbb{Z}^d$ is equally or perhaps more important in data science applications. However, in the multi-parameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the $1$-parameter case because there is no known extension of the persistence algorithm to multi-parameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module $M$ defined over the multiparameter $mathbb{Z}^d$.The algorithm first assumes that the module is presented with a set of $N$ generators and relations that are emph{distinctly graded}. Based on a generalized matrix reduction technique it runs in $O(N^{2omega+1})$ time where $omega<2.373$ is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in $tilde{O}(N^{6(d+1)})$ time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of $n$ simplices, our algorithm runs in $O(n^{2omega+1})$ time for $d=2$ and in $O(n^{d(2omega + 1)})$ time for $d>2$.
In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact spaces. Using recent developments in the theory of topological persistence we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Cech complexes built on top of compact spaces.
This paper presents an efficient algorithm for the progressive approximation of Wasserstein barycenters of persistence diagrams, with applications to the visual analysis of ensemble data. Given a set of scalar fields, our approach enables the computation of a persistence diagram which is representative of the set, and which visually conveys the number, data ranges and saliences of the main features of interest found in the set. Such representative diagrams are obtained by computing explicitly the discrete Wasserstein barycenter of the set of persistence diagrams, a notoriously computationally intensive task. In particular, we revisit efficient algorithms for Wasserstein distance approximation [12,51] to extend previous work on barycenter estimation [94]. We present a new fast algorithm, which progressively approximates the barycenter by iteratively increasing the computation accuracy as well as the number of persistent features in the output diagram. Such a progressivity drastically improves convergence in practice and allows to design an interruptible algorithm, capable of respecting computation time constraints. This enables the approximation of Wasserstein barycenters within interactive times. We present an application to ensemble clustering where we revisit the k-means algorithm to exploit our barycenters and compute, within execution time constraints, meaningful clusters of ensemble data along with their barycenter diagram. Extensive experiments on synthetic and real-life data sets report that our algorithm converges to barycenters that are qualitatively meaningful with regard to the applications, and quantitatively comparable to previous techniques, while offering an order of magnitude speedup when run until convergence (without time constraint). Our algorithm can be trivially parallelized to provide additional speedups in practice on standard workstations. [...]
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function $f : X to mathbb{R}$, where $X$ is a CW-complex. In the special case $X = [0,1]^N$, $N in mathbb{N}$ we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.
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