Approximating Continuous Functions on Persistence Diagrams Using Template Functions

The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes th...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 23; no. 4; pp. 1215 - 1272
Main Authors Perea, Jose A., Munch, Elizabeth, Khasawneh, Firas A.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.08.2023
Springer
Springer Nature B.V
Subjects
Algorithms
Analysis
Applications of Mathematics
Approximation
Cognitive tasks
Computer Science
Continuity (mathematics)
Data analysis
Economics
Linear and Multilinear Algebras
Machine learning
Math Applications in Computer Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Supervised learning
Vector spaces
Germany
30L05
Persistent homology
68T05
Featurization
Bottleneck distance
Machine learning
55N31
Topological data analysis
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Summary:The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into vector spaces, in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization called template functions , and we show that it addresses the problem of approximating continuous functions on compact subsets of the space of persistence diagrams. Specifically, we begin by characterizing relative compactness with respect to the bottleneck distance, and then provide explicit theoretical methods for constructing compact-open dense subsets of continuous functions on persistence diagrams. These dense subsets—obtained via template functions—are leveraged for supervised learning tasks with persistence diagrams. Specifically, we test the method for classification and regression algorithms on several examples including shape data and dynamical systems.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-022-09567-7