Related papers: Persistent Mayer homology and persistent Mayer Lap…
We extend recent results on discrete approximations of the Laplacian in $\mathbf{R}^d$ with norm resolvent convergence to the corresponding results for Dirichlet and Neumann Laplacians on a half-space. The resolvents of the discrete…
Consider $n$ points independently sampled from a density $p$ of class $\mathcal{C}^2$ on a smooth compact $d$-dimensional sub-manifold $\mathcal{M}$ of $\mathbb{R}^m$, and consider the generator of a random walk visiting these points…
Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the…
Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…
The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key…
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a…
The theory of persistence modules on the commutative ladders $CL_n(\tau)$ provides an extension of persistent homology. However, an efficient algorithm to compute the generalized persistence diagrams is still lacking. In this work, we view…
For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often…
In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the…
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…
For an increasing monotone graph property $\mP$ the \emph{local resilience} of a graph $G$ with respect to $\mP$ is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of…
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…
We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they…
Capturing complex high-order interactions among data is an important task in many scenarios. A common way to model high-order interactions is to use hypergraphs whose topology can be mathematically represented by tensors. Existing methods…
This work introduces a number of algebraic topology approaches, such as multicomponent persistent homology, multi-level persistent homology and electrostatic persistence for the representation, characterization, and description of small…
A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the…
Given topological spaces X and Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X -> Y . We consider a computational version, where X, Y are given as finite simplicial complexes, and the…
We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families (visibility (natural and…