Related papers: Some equivalence relation between persistent homol…
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…
The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex $K$. In case of $\dim K=1$ (i.e., a graph), by using the properties, we characterize strongly connectedness of critical…
In this paper we study functions on the interval that have the same persistent homology. By introducing an equivalence relation modeled after topological conjugacy, which we call graph-equivalence, a precise enumeration of functions with…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in…
We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of…
This note contributes to the point calculus of persistent homology by extending Alexander duality to real-valued functions. Given a perfect Morse function $f: S^{n+1} \to [0,1]$ and a decomposition $S^{n+1} = U \cup V$ such that $M = \U…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
Persistent Homology (PH) allows tracking homology features like loops, holes and their higher-dimensional analogs, along with a single-parameter family of nested spaces. Currently, computing descriptors for complex data characterized by…
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…
In this paper we describe a model based on persistent homology that describes interactions between mathematicians in terms of collaborations. Some ideas from classical data analysis are used.
We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…
Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…