One-dimensional reduction of multidimensional persistent homology

2010/04/09 by Francesca Cagliari, Barbara Di Fabio, Massimo Ferri · 52 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Persistent homology #Homology (biology) #Mathematics #Pure mathematics #Biology #Genetics #Algorithm #Gene

paper · pdf · doi:10.1090/s0002-9939-10-10312-8

published in Proceedings of the American Mathematical Society 138(8), 3003-3017 (American Mathematical Society)

openalex publication_date 2010/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23

Abstract

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i"> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding="application/x-tex">i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -essentiality of homological critical values conclude the paper.

Citations

• Multidimensional Size Functions for Shape Comparison
• Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
• Stability of Persistence Diagrams
• A refinement of multi-dimensional persistence
• A distance for similarity classes of submanifolds of a Euclidean space
• Barcodes: The persistent topology of data
• Topological Persistence and Simplification
• Persistent homology—a survey
• The Theory of Multidimensional Persistence
• Necessary Conditions for Discontinuities of Multidimensional Size Functions

Cited by

• Estimating Multidimensional Persistent Homology Through a Finite Sampling
• Topological data analysis [wikipedia]