Related papers: Persistence modules induced by inner functions
The stability of topological persistence is one of the fundamental issues in topological data analysis. Numerous methods have been proposed to address the stability of persistent modules or persistence diagrams. Recently, the concept of…
This paper complements the work done on simultaneous approximation results in classical Banach spaces, by focusing on approximation by finite Blaschke products. We prove the existence of a finite Blaschke product that approximates a…
Consider the space $\mathcal{F}$ of all inner functions on the unit open disk under the uniform topology, which is a metric topology induced by the $H^{\infty}$-norm. In the present paper, a class of Blaschke products, denoted by…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
In a nutshell, submodular functions encode an intuitive notion of diminishing returns. As a result, submodularity appears in many important machine learning tasks such as feature selection and data summarization. Although there has been a…
In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of…
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance…
Persistence is an important characteristic of many complex systems in nature, related to how long the system remains at a certain state before changing to a different one. The study of complex systems' persistence involves different…
In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in…
Interleaving distances are used widely in Topological Data Analysis (TDA) as a tool for comparing topological signatures of datasets. The theory of interleaving distances has been extended through various category-theoretic constructions,…
Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the…
The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a…
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects…
We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…
The purpose of this paper is to make a comprehensive connection between the basic results and properties derived from the two kinds of topologies (namely the $(\epsilon,\lambda)-$topology introduced by the author and the stronger locally…
Interspike intervals describe the output of neurons. Signal transmission in a neuronal network implies that the output of some neurons becomes the input of others. The output should reproduce the main features of the input to avoid a…
In this article we continue the study of holonomic modules over sheaves of Cherednik algebras, initiated by the third author in [Tho18]. Working with arbitrary parameters, we first develop a theory of $b$-functions to prove that…
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…
We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their…