Related papers: The observable structure of persistence modules
One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the…
We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets,…
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…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a…
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…
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,…
Time-series of persistence diagrams, known as vineyards, have shown to be useful in diverse applications. A natural algebraic version of vineyards is a time series of persistence modules equipped with interleaving maps between the…
Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…
This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of…
Multidimensional persistence has been proposed to study the persistence of topological features in data indexed by multiple parameters. In this work, we further explore its algebraic complications from the point of view of higher…
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
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.…
In recent years, persistence modules have been viewed as graded modules with gradation over a preordered set serving as the indexing set. We provide sufficient criteria for a projective module over a PID to be free when the indexing set is…
In this article we study certain notions of `tameness' for the persistence modules studied in topological data analysis. In particular, we show that after adding infinitary points the so called finitely determined modules become finitely…
Commutative diagrams of vector spaces and linear maps over $\mathbb{Z}^2$ are objects of interest in topological data analysis (TDA) where this type of diagrams are called 2-parameter persistence modules. Given that quiver representation…
Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…