Persistence diagrams are a main tool in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space is complicated. For that reason, summarizing and vectorizing these diagrams is an important topic currently researched in TDA. In this work, we provide a general framework of summarizing diagrams that we call Persistence Curves (PC). The main idea is so-called Fundamental Lemma of Persistent Homology, which is derived from the classic elder rule. Under this framework, certain  well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs. In particular, certain family of PCs admit the stabilityproperty under an additional assumption . Finally, we apply PCs to textures classification on four well-know texture datasets. The result outperforms several existing TDA methods.
Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space is complicated. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general framework of vectorizing diagrams that we call thePersistence Curves (PCs). We show that some well-known summaries, such as Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it. Second, we provide a theoretical foundation for the stability analysis of PCs. In addition, we propose several new summaries based on PC framework and investigate their stability . Finally, we demonstrate the practical uses of PCs on the texture classification on four public available texture datasets. We show the result of our proposed PCs outperforms several existing TDA methods.
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a main tool
one of the main tools
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summarizing and vectorizing these diagrams is
transforming these diagrams in a way that is compatible with machine learning is
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work, we provide
paper, our main contribution consists of three components. First, we develop
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summarizing
vectorizing
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Persistence Curves (PC). The main idea is so-called Fundamental Lemma of Persistent Homology, which is derived from the classic elder rule. Under this framework, certain
the
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Persistence Curves
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(PCs). We show that some
R
persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families
Betti number curves, the Euler Characteristic Curve, and Persistence Landscapes fall under the PC framework or are easily derived from it. Second, we provide a theoretical foundation for the stability analysis
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particular, certain family of PCs admit the stabilityproperty under an additional assumption
addition, we propose several new summaries based on PC framework and investigate their stability
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apply PCs to textures
demonstrate the practical uses of PCs on the texture
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well-know
public available
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The result
We show the result of our proposed PCs