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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Published in:	Foundations of computational mathematics Vol. 23; no. 4; pp. 1215 - 1272
Main Authors:	Perea, Jose A., Munch, Elizabeth, Khasawneh, Firas A.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2023 Springer Nature B.V
Subjects:	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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Abstract	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.
AbstractList	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. 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.
Author	Perea, Jose A. Khasawneh, Firas A. Munch, Elizabeth
Author_xml	– sequence: 1 givenname: Jose A. surname: Perea fullname: Perea, Jose A. email: j.pereabenitez@northeastern.edu organization: Department of Mathematics and Khoury College of Computer Sciences, Northeastern University – sequence: 2 givenname: Elizabeth surname: Munch fullname: Munch, Elizabeth organization: Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University – sequence: 3 givenname: Firas A. surname: Khasawneh fullname: Khasawneh, Firas A. organization: Department of Mechanical Engineering, Michigan State University
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Snippet	The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires...
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SubjectTerms	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
Title	Approximating Continuous Functions on Persistence Diagrams Using Template Functions
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