Persistence landscapes of affine fractals-Reference-Cited by-同舟云学术

Persistence landscapes of affine fractals

Published:2022-01-01 Issue:1 Volume:55 Page:163-192
ISSN:2391-4661
Container-title:Demonstratio Mathematica
language:en
Short-container-title:

Author:

Catanzaro Michael J.¹,Przybylski Lee¹,Weber Eric S.¹

Affiliation:

1. Department of Mathematics, Iowa State University , 396 Carver Hall , Ames , IA 50011 , United States

Abstract

Abstract We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/dema-2022-0015/pdf

Reference40 articles.

1. P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Machine Learn. Res. 16 (2015), no. 1, 77–102.

2. V. Robins, Computational topology at multiple resolutions: foundations and applications to fractals and dynamics, Ph.D. thesis, University of Colorado, 2000.

3. G. Máté and D. W. Heermann, Persistence intervals of fractals, Phys. A Statist. Mech. Appl. 405 (2014), 252–259.

4. G. Carlsson, A. Zomorodian, A. Collins, and L. J. Guibas, Persistence barcodes for shapes, Int. J. Shape Model. 11 (2005), no. 2, 149–187.

5. E. Munch, K. Turner, P. Bendich, S. Mukherjee, J. Mattingly, and J. Harer, Probabilistic Fréchet means for time varying persistence diagrams, Electron. J. Statistic. 9 (2015), no. 1, 1173–1204.