Persistence Paths and Signature Features in Topological Data Analysis

We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition o...

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Published in:	IEEE transactions on pattern analysis and machine intelligence Vol. 42; no. 1; pp. 192 - 202
Main Authors:	Chevyrev, Ilya, Nanda, Vidit, Oberhauser, Harald
Format:	Journal Article
Language:	English
Published:	United States IEEE 01.01.2020 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algebra Bar codes barcodes Bars Data analysis Extraterrestrial measurements Feature maps Homology Kernel kernel learning Mathematical analysis signature features Tensors Topological data analysis Vector space
ISSN:	0162-8828, 1939-3539, 2160-9292, 1939-3539
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Abstract	We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations-barcode to path, path to tensor series-results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.
AbstractList	We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations—barcode to path, path to tensor series—results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks. We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations-barcode to path, path to tensor series-results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations-barcode to path, path to tensor series-results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.
Author	Chevyrev, Ilya Oberhauser, Harald Nanda, Vidit
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SubjectTerms	Algebra Bar codes barcodes Bars Data analysis Extraterrestrial measurements Feature maps Homology Kernel kernel learning Mathematical analysis signature features Tensors Topological data analysis Vector space
Title	Persistence Paths and Signature Features in Topological Data Analysis
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