Approximating lower-star persistence via 2D combinatorial map simplification

Filtration simplification consists of simplifying a given filtration while simultaneously controlling the perturbation in the associated persistence diagrams. In this paper, we propose a filtration simplification algorithm for orientable 2-dimensional (2D) manifolds with or without boundary ( meshes ) represented by2D combinatorial maps. Given a lower-star filtration of the mesh, faces are added into contiguous clusters according to a “height” function and a parameter . Faces in the same cluster are merged into a single face, resulting in a lower resolution mesh and a simpler filtration. We prove that the parameter bounds the perturbation in the original persistence diagrams, and we provide experiments demonstrating thecomputational advantages of the simplification process

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Conformational Ensembles and Sampled Energy Landscapes: Analysis and Comparison

We present novel algorithms and software addressing four core problemsin computational structural biology, namely analyzing a conformationalensemble, comparing two conformational ensembles, analyzing a sampledenergy landscape, and comparing two sampled energy landscapes. Usingrecent developments in computational topology, graph theory, andcombinatorial optimization, we make two notable contributions. First,we a present a generic algorithm analyzing height fields. We then usethis algorithm to perform density based clustering of conformations,and to analyze a sampled energy landscape in terms of basins andtransitions between them. In both cases, topological persistence isused to manage ruggedness. Second, we introduce two algorithms tocompare transition graphs. The first is the classical earth mover distance metric which depends only on local minimum energyconfigurations along with their statistical weights, while the secondincorporates topological constraints inherent to conformationaltransitions.Illustrations are provided on a simplified protein model (BLN69), whosefrustrated potential energy landscape has been thoroughly studied.The software implementing our tools is also made available, and shouldprove valuable wherever conformational ensembles and energy landscapesare used

Approximation algorithms for Vietoris-Rips and Čech filtrations

Persistent Homology is a tool to analyze and visualize the shape of data from a topological viewpoint. It computes persistence, which summarizes the evolution of topological and geometric information about metric spaces over multiple scales of distances. While computing persistence is quite efficient for low-dimensional topological features, it becomes overwhelmingly expensive for medium to high-dimensional features. In this thesis, we attack this computational problem from several different angles. We present efficient techniques to approximate the persistence of metric spaces. Three of our methods are tailored towards general point clouds in Euclidean spaces. We make use of high dimensional lattice geometry to reduce the cost of the approximations. In particular, we discover several properties of the Permutahedral lattice, whose Voronoi cell is well-known for its combinatorial properties. The last method is suitable for point clouds with low intrinsic dimension, where we exploit the structural properties of the point set to tame the complexity. In some cases, we achieve a reduction in size complexity by trading off the quality of the approximation. Two of our methods work particularly well in conjunction with dimension-reduction techniques: we arrive at the first approximation schemes whose complexities are only polynomial in the size of the point cloud, and independent of the ambient dimension. On the other hand, we provide a lower bound result: we construct a point cloud that requires super-polynomial complexity for a high-quality approximation of the persistence. Together with our approximation schemes, we show that polynomial complexity is achievable for rough approximations, but impossible for sufficiently fine approximations. For some metric spaces, the intrinsic dimension is low in small neighborhoods of the input points, but much higher for large scales of distances. We develop a concept of local intrinsic dimension to capture this property. We also present several applications of this concept, including an approximation method for persistence. This thesis is written in English.Persistent Homology ist eine Methode zur Analyse und Veranschaulichung von Daten aus topologischer Sicht. Sie berechnet eine topologische Zusammenfassung eines metrischen Raumes, die Persistence genannt wird, indem die topologischen Eigenschaften des Raumes über verschiedene Skalen von Abständen analysiert werden. Die Berechnung von Persistence ist für niederdimensionale topologische Eigenschaften effizient. Leider ist die Berechung für mittlere bis hohe Dimensionen sehr teuer. In dieser Dissertation greifen wir dieses Problem aus vielen verschiedenen Winkeln an. Wir stellen effiziente Techniken vor, um die Persistence für metrische Räume zu approximieren. Drei unserer Methoden eignen sich für Punktwolken im euklidischen Raum. Wir verwenden hochdimensionale Gittergeometrie, um die Kosten unserer Approximationen zu reduzieren. Insbesondere entdecken wir mehrere Eigenschaften des Permutahedral Gitters, dessen Voronoi-Zelle für ihre kombinatorischen Eigenschaften bekannt ist. Die vierte Methode eignet sich für Punktwolken mit geringer intrinsischer Dimension: wir verwenden die strukturellen Eigenschaften, um die Komplexität zu reduzieren. Für einige Methoden zeigen wir einen Trade-off zwischen Komplexität und Approximationsqualität auf. Zwei unserer Methoden funktionieren gut mit Dimensionsreduktionstechniken: wir präsentieren die erste Methode mit polynomieller Komplexität unabhängig von der Dimension. Wir zeigen auch eine untere Schranke. Wir konstruieren eine Punktwolke, für die die Berechnung der Persistence nicht in Polynomzeit möglich ist. Die bedeutet, dass in Polynomzeit nur eine grobe Approximation berechnet werden kann. Für gewisse metrische Räume ist die intrinsiche Dimension gering bei kleinen Skalen aber hoch bei großen Skalen. Wir führen das Konzept lokale intrinsische Dimension ein, um diesen Umstand zu fassen, und zeigen, dass es für eine gute Approximation von Persistenz benutzt werden kann. Diese Dissertation ist in englischer Sprache verfasst

Geometric and Topological Inference

Cambridge Texts in Applied MathematicsInternational audienceGeometric and topological inference deals with the retrieval of information about a geometric object that is only known through a finite set of possibly noisy sample points. Geometric and topological inference employs many tools from Computational Geometry and Applied Topology. It has connections to Manifold Learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of Topological Data Analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this book covers various aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. This first book on the subject can serve for teaching in a mathematical or computer science department, and will benefit to scientists and engineers interested in a geometric approach to Data Science