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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Podrobná bibliografie
Vydáno v:	Foundations of computational mathematics Ročník 23; číslo 4; s. 1215 - 1272
Hlavní autoři:	Perea, Jose A., Munch, Elizabeth, Khasawneh, Firas A.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.08.2023 Springer Nature B.V
Témata:	Algorithms 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 30L05 Persistent homology 68T05 Featurization Bottleneck distance Machine learning 55N31 Topological data analysis
ISSN:	1615-3375, 1615-3383
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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	1615-3375 1615-3383
DOI:	10.1007/s10208-022-09567-7