Related papers: Stability of Bipath Persistence Diagrams
The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their…
This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a…
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistences module valued in any small abelian category.
Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the…
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a filtered space, shown via persistence diagrams. Additionally,…
Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques.…
We apply persistent homology, the dominant tool from the field of topological data analysis, to study electoral redistricting. Our method combines the geographic information from a political districting plan with election data to produce a…
Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb].…
In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least $k$, then it contains a cycle of…
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability…
Persistent homology (PH) is a recently developed theory in the field of algebraic topology to study shapes of datasets. It is an effective data analysis tool that is robust to noise and has been widely applied. We demonstrate a general…
The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of…
These notes are a self-contained short proof of the stability of persistence diagrams.
In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…
Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for…
In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as…
Persistent homology is a tool that can be employed to summarize the shape of data by quantifying homological features. When the data is an object in $\mathbb{R}^d$, the (augmented) persistent homology transform ((A)PHT) is a family of…