scholarly journals Strong topology on the set of persistence diagrams

2019 ◽  
Author(s):  
M. Zarichnyi ◽  
A. Savchenko ◽  
V. Kiosak
Keyword(s):  
Strong Topology ◽  
Persistence Diagrams

scholarly journals Limit theorems for persistence diagrams

2018 ◽  
Vol 28 (5) ◽  
pp. 2740-2780 ◽  
Author(s):  
Yasuaki Hiraoka ◽  
Tomoyuki Shirai ◽  
Khanh Duy Trinh
Keyword(s):  
Limit Theorems ◽  
Persistence Diagrams

scholarly journals Contractibility of a persistence map preimage

2020 ◽  
Vol 4 (4) ◽  
pp. 509-523
Author(s):  
Jacek Cyranka ◽  
Konstantin Mischaikow ◽  
Charles Weibel
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Time Series ◽  
Data Driven ◽  
Dimensional System ◽  
Multiple Time ◽  
Multiple Time Series ◽  
Topological Features ◽  
Persistence Diagram ◽  
Persistence Diagrams

Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

scholarly journals Stability of Persistence Diagrams

2006 ◽  
Vol 37 (1) ◽  
pp. 103-120 ◽  
Author(s):  
David Cohen-Steiner ◽  
Herbert Edelsbrunner ◽  
John Harer
Keyword(s):  
Persistence Diagrams

scholarly journals Optimization of Spectral Wavelets for Persistence-Based Graph Classification

2021 ◽  
Vol 7 ◽  
Author(s):  
Ka Man Yim ◽  
Jacob Leygonie
Keyword(s):  
Data Science ◽  
Persistent Homology ◽  
A Priori ◽  
Graph Classification ◽  
Classification Problems ◽  
Geometric Properties ◽  
Node Attributes ◽  
Science Problem ◽  
Theoretical Foundations ◽  
Persistence Diagrams

A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimizes the choice of wavelet for a dataset of graphs, such that their associated persistence diagrams capture features of the graphs that are best suited to a given data science problem. Since the spectral wavelet signature of a graph is derived from its Laplacian, our framework encodes geometric properties of graphs in their associated persistence diagrams and can be applied to graphs without a priori node attributes. We apply our framework to graph classification problems and obtain performances competitive with other persistence-based architectures. To provide the underlying theoretical foundations, we extend the differentiability result for ordinary persistent homology to extended persistent homology.

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