Related papers: Interleaving distances from height-difference func…
In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset…
Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is…
Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance $d_{\mathrm{I}}$ between poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between…
In order to better understand and to compare interleavings between persistence modules, we elaborate on the algebraic structure of interleavings in general settings. In particular, we provide a representation-theoretic framework for…
Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving…
In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that…
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck…
The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence (AI) in general, since they can serve as an organizing principle by which individuals classify objects, form concepts…
We demonstrate that an equivalence of categories using $\varepsilon$-interleavings as a fundamental component exists between the model of persistence modules as graded modules over a polynomial ring and the model of persistence modules as…
Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the…
Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…
We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric…
A lower bound for the interleaving distance on persistence vector spaces is given in terms of rank invariants. This offers an alternative proof of the stability of rank invariants.
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…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
We give formulas for calculating the interleaving distance between rectangle persistence modules that depend solely on the geometry of the underlying rectangles. Moreover, we extend our results to calculate the bottleneck distance for…
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval…