Related papers: Decomposition of Persistent Homology and Spectral …
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
We construct "barcodes" for the chain complexes over Novikov rings that arise in Novikov's Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these…
We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this…
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…
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…
Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective…
The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of the space of data, a persistence barcode tracks the evolution of its homological features. In this…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
We study the inverse problem for persistent homology: For a fixed simplicial complex $K$, we analyse the fiber of the continuous map $\mathrm{PH}$ on the space of filters that assigns to a filter $f: K \to \mathbb R$ the total barcode of…
The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees,…
In the world of chain complexes E_n-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic E_n-homology of an E_n-algebra computes the homology of an n-fold algebraic…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
As complex networks find applications in a growing range of disciplines, the diversity of naturally occurring and model networks being studied is exploding. The adoption of a well-developed collection of network taxonomies is a natural…
While there has been much interest in adapting conventional clustering procedures---and in higher dimensions, persistent homology methods---to directed networks, little is known about the convergence of such methods. In order to even…
This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of…
While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a…
We introduce a novel set of observables associated to the rapidly developing field of persistent homology for the quantitative characterization of nuclear collisions and their evolution. Persistent homology allows for the identification of…
Persistent homology, a method from topological data analysis, extracts robust, multi-scale features from data. It produces stable representations of time series by applying varying thresholds to their values (a process known as a…
We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential…