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Graded Persistence Diagrams and Persistence Landscapes.

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Title: Graded Persistence Diagrams and Persistence Landscapes.
Authors: Betthauser, Leo1 (AUTHOR), Bubenik, Peter2 (AUTHOR) peter.bubenik@ufl.edu, Edwards, Parker B.3 (AUTHOR)
Source: Discrete & Computational Geometry. Jan2022, Vol. 67 Issue 1, p203-230. 28p.
Subjects: Cartesian plane, Maxima & minima, Number systems
Abstract: We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Möbius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or - 1 . The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the kth graded persistence diagram correspond to the local maxima and minima, respectively, of the kth persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between kth graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the kth graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams. [ABSTRACT FROM AUTHOR]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jan2022, Vol. 67 Issue 1, p203-230. 28p.
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  Data: We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Möbius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or - 1 . The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the kth graded persistence diagram correspond to the local maxima and minima, respectively, of the kth persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between kth graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the kth graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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              Text: Jan2022
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