Graded Persistence Diagrams and Persistence Landscapes

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.

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Stability of Persistence Diagrams

Discrete & Computational Geometry ◽

10.1007/s00454-006-1276-5 ◽

2006 ◽

Vol 37 (1) ◽

pp. 103-120 ◽

Cited By ~ 356

Author(s):

David Cohen-Steiner ◽

Herbert Edelsbrunner ◽

John Harer

Keyword(s):

Persistence Diagrams

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Optimization of Spectral Wavelets for Persistence-Based Graph Classification

Frontiers in Applied Mathematics and Statistics ◽

10.3389/fams.2021.651467 ◽

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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Digital image reduction for analysis of topological changes in the pore space of the rock matrix during chemical dissolution

Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie) ◽

10.26089/nummet.v21r327 ◽

2020 ◽

pp. 319-328

Author(s):

Д.И. Прохоров ◽

Я.В. Базайкин ◽

В.В. Лисица

Keyword(s):

Digital Image ◽

Linear Complexity ◽

Digital Images ◽

Pore Space ◽

Three Dimensional ◽

Chemical Dissolution ◽

Rock Matrix ◽

Persistence Diagrams ◽

Topological Changes

В работе предложен алгоритм редукции трехмерных цифровых изображений для ускорения вычисления персистентных диаграмм, характеризующих изменения в топологии порового пространства образцов горной породы. Воксели для удаления выбираются исходя из структуры своей окрестности, что позволяет редуцировать изображение за линейное время. Показано, что эффективность алгоритма существенно зависит от сложности устройства порового пространства и размеров шагов фильтрации. A new algorithm for the reduction of three-dimensional digital images is proposed to improve the performance of persistence diagrams computing. These diagrams represent changes in topology of the pore space in the rock matrix. The algorithm has a linear complexity, since the removal of the voxel is based on the structure of its neighborhood. It is shown that the efficiency of the algorithm depends heavily on the complexity of the pore space and the size of filtering steps.

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