Persistence Curves: A canonical framework for summarizing persistence diagrams

Chung, Yu-Min; Lawson, Austin

doi:10.1007/s10444-021-09893-4

Cited by 33 publications

(40 citation statements)

References 35 publications

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“…Persistence Curves, first defined in [6], draws inspiration from the Fundamental Lemma of Persistent Homology, which states that the Betti number of an element in the filtration β k (f −1 ((−∞, x]) is given by i≤x j>x µ k ai,aj . The statement here says that the k-th Betti number of the subspace f −1 ((−∞, x]) is given exactly by the number of points in the box formed by the points up and to the left of the diagonal at x.…”

Section: Persistence Curvesmentioning

confidence: 99%

“…[6] provides an explicit form of C. Since we will not use the its explicit form in this work, we refer interested readers to [6] for more details about the constant C.…”

Section: Persistence Curvesmentioning

confidence: 99%

“…One may choose P based on their prior knowledge about the dataset. [6] provides a list of persistence curves. The other parameter, α, is a weighting of the distance between the original time series and the distance between their topological transformations.…”

Section: New Metricsmentioning

confidence: 99%

“…Summaries such as the persistence images [1], the persistence landscapes [4], and the persistent entropy [2]. The particular interest in this paper is the persistence curve framework [6]. In that work, they showed this framework capable of generating new summaries as well as several well-known summaries such as persistence landscapes and persistent entropy.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Persistent Homology Approach to Time Series Classification

Chung¹,

Cruse²,

Lawson³

2020

Preprint

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Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that tracks changes in topological information. These changes can be recorded in multi-sets known as persistence diagrams. Converting information stored in persistence diagrams into a form compatible with modern machine learning algorithms is a major vein of research in TDA. Persistence curves, a recently developed framework, provides a canonical and flexible way to encode the information presented in persistence diagrams into vectors. In this work, we propose a new set of metrics based on persistence curves. We prove the stability of the proposed metrics. Finally, we apply these metrics to the UCR Time Series Classification Archive. These empirical results show that our metrics perform better than the relevant benchmark in most cases and warrant further study.

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Section: Persistence Curvesmentioning

confidence: 99%

“…[6] provides an explicit form of C. Since we will not use the its explicit form in this work, we refer interested readers to [6] for more details about the constant C.…”

Section: Persistence Curvesmentioning

confidence: 99%

Section: New Metricsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Persistent Homology Approach to Time Series Classification

Chung¹,

Cruse²,

Lawson³

2020

Preprint

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“…In [6] a new class of one-dimensional smooth functional summaries was introduced called Gaussian persistence curves (GPC's). These functional summaries were built by combining (a slight variation of) the persistence curve framework from [7] with the persistence surfaces construction from [1], and they were used to study the texture classification of grey-scale images [6].…”

Section: Introductionmentioning

confidence: 99%

Gaussian Persistence Curves

Chung¹,

Hüll²,

Lawson³

et al. 2022

Preprint

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Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a persistence diagram. To utilize machine and deep learning methods on persistence diagrams, These diagrams are further summarized by transforming them into functions. In this paper we investigate the stability and injectivity of a class of smooth, one-dimensional functional summaries called Gaussian persistence curves.

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Classification of stochastic processes with topological data analysis

Güzel

Kaygun

2023

Concurrency and Computation

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SummaryIn this study, we demonstrate that engineered topological features can distinguish time series sampled from different stochastic processes with different noise characteristics, in both balanced and unbalanced sampling schemes. We compare our classification results against the results of the same classification on features coming from descriptive statistics and the wavelet transform. We conclude that machine learning models built on engineered topological features alone perform consistently better than those built on the standard statistical and wavelet features for time series classification tasks. We also apply dimension reduction techniques to our engineered features and compare the result of our classification models before and after dimensionality reduction. Finally, we also show that in our calculations of the engineered topological features, employing parallel computing methods does yield significant improvements in run time and memory footprint.

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Persistence Curves: A canonical framework for summarizing persistence diagrams

Cited by 33 publications

References 35 publications

A Persistent Homology Approach to Time Series Classification

A Persistent Homology Approach to Time Series Classification

Gaussian Persistence Curves

Classification of stochastic processes with topological data analysis

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