Clear and Compress: Computing Persistent Homology in Chunks

We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by G\"unther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we improve the performance through parallelized computation.Comment: This result was presented at TopoInVis 2013 (http://www.sci.utah.edu/topoinvis13.html

Finding Pairwise Intersections Inside a Query Range

We study the following problem: preprocess a set O of objects into a data structure that allows us to efficiently report all pairs of objects from O that intersect inside an axis-aligned query range Q. We present data structures of size $O(n({\rm polylog} n))$ and with query time $O((k+1)({\rm polylog} n))$ time, where k is the number of reported pairs, for two classes of objects in the plane: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in R^3, we present a data structures of size $O(n\sqrt{n}({\rm polylog} n))$ and query time $O((\sqrt{n}+k)({\rm polylog} n))$. When the objects and query are fat, we obtain $O((k+1)({\rm polylog} n))$ query time using $O(n({\rm polylog} n))$ storage

Mind the Gap: A Study in Global Development through Persistent Homology

The Gapminder project set out to use statistics to dispel simplistic notions about global development. In the same spirit, we use persistent homology, a technique from computational algebraic topology, to explore the relationship between country development and geography. For each country, four indicators, gross domestic product per capita; average life expectancy; infant mortality; and gross national income per capita, were used to quantify the development. Two analyses were performed. The first considers clusters of the countries based on these indicators, and the second uncovers cycles in the data when combined with geographic border structure. Our analysis is a multi-scale approach that reveals similarities and connections among countries at a variety of levels. We discover localized development patterns that are invisible in standard statistical methods

Computing and reducing slope complexes

In this paper we provide a new characterization of cell de- composition (called slope complex) of a given 2-dimensional continuous surface. Each patch (cell) in the decomposition must satisfy that there exists a monotonic path for any two points in the cell. We prove that any triangulation of such surface is a slope complex and explain how to obtain new slope complexes with a smaller number of slope regions decomposing the surface. We give the minimal number of slope regions by counting certain bounding edges of a triangulation of the surface obtained from its critical points.Ministerio de Economía y Competitividad MTM2015-67072-

Hierarchical ordering of reticular networks

The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.Comment: 9 pages, 5 figures, During preparation of this manuscript the authors became aware of a related work by Katifori and Magnasco, concurrently submitted for publicatio

Topological characteristics of oil and gas reservoirs and their applications

We demonstrate applications of topological characteristics of oil and gas reservoirs considered as three-dimensional bodies to geological modeling.Comment: 12 page

Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational Geometr

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure

Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium on Intelligent Data Analysis (IDA 2017