Related papers: Stability of persistence spaces of vector-valued c…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if…
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…
Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to…
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…
We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of…
The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…
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…
An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable…
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
Persistent homology and persistent entropy have recently become useful tools for patter recognition. In this paper, we find requirements under which persistent entropy is stable to small perturbations in the input data and scale invariant.…
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…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops,…
Over the past decade, the techniques of topological data analysis (TDA) have grown into prominence to describe the shape of data. In recent years, there has been increasing interest in developing statistical methods and in particular…
We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization…
In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistences module valued in any small abelian category.
The algebraic stability theorem for $\mathbb{R}$-persistence modules is a fundamental result in topological data analysis. We present a stability theorem for $n$-dimensional rectangle decomposable persistence modules up to a constant…