Related papers: Worst-Case Examples for the Computation of Persist…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the…
We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital…
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…
Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we…
In this paper we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing…
A regularized version of Mixture Models is proposed to learn a principal graph from a distribution of $D$-dimensional data points. In the particular case of manifold learning for ridge detection, we assume that the underlying manifold can…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
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 diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…
The NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to $H$-free graphs, that is, graphs that do not contain some…
Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of networks flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and…
In this paper we focus on preprocessing for persistent homology computations. We adapt some techniques which were successfully used for standard homology computations. The main idea is to reduce the complex prior to generating its boundary…
This study introduces GCO-HPIF, a general machine-learning-based framework to predict and explain the computational hardness of combinatorial optimization problems that can be represented on graphs. The framework consists of two stages. In…
Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the…
We consider Holder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in `good continuation' of…
Shape graphs are complex geometrical structures commonly found in biological and anatomical systems. A shape graph is a collection of nodes, some connected by curvilinear edges with arbitrary shapes. Their high complexity stems from the…
We study the distributed message-passing model in which a communication network is represented by a graph G=(V,E). Usually, the measure of complexity that is considered in this model is the worst-case complexity, which is the largest number…
Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…