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Statistical topological data analysis using persistence landscapes
 Journal of Machine Learning Research
"... Abstract. We define a new topological summary for data that we call the persistence landscape. In contrast to the standard topological summaries, the barcode and the persistence diagram, it is easy to combine with statistical analysis, and its associated computations are much faster. This summary o ..."
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Abstract. We define a new topological summary for data that we call the persistence landscape. In contrast to the standard topological summaries, the barcode and the persistence diagram, it is easy to combine with statistical analysis, and its associated computations are much faster. This summary obeys a Strong Law of Large Numbers and a Central Limit Theorem. Under certain finiteness conditions, this allows us to calculate approximate confidence intervals for the expected total squared persistence. With these results one can use ttests for statistical inference in topological data analysis. We apply these methods to numerous examples including random geometric complexes, random clique complexes, and Gaussian random fields. We also show that this summary is stable and gives lower bounds for the bottleneck distance and the Wasserstein distance. 1.
Stochastic convergence of persistence landscapes and silhouettes
, 2013
"... Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multiset of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize t ..."
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Cited by 5 (3 self)
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Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multiset of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landscape, introduced by Bubenik, which converts a diagram into a wellbehaved realvalued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory. 1 ar
Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces
, 2014
"... We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces, and illustrate their use in hypothesis testing and providing confidence intervals f ..."
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We study distributions of persistent homology barcodes associated to taking subsamples of a fixed size from metric measure spaces. We show that such distributions provide robust invariants of metric measure spaces, and illustrate their use in hypothesis testing and providing confidence intervals for topological data analysis.
Hypothesis Testing for Topological Data Analysis
, 2013
"... Persistence homology is a vital tool for topological data analysis. Previous work has developed some statistical estimators for characteristics of collections of persistence diagrams. However, tools that provide statistical inference for scenarios in which the observations are persistence diagrams ..."
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Persistence homology is a vital tool for topological data analysis. Previous work has developed some statistical estimators for characteristics of collections of persistence diagrams. However, tools that provide statistical inference for scenarios in which the observations are persistence diagrams are not developed. We propose the use of randomizationstyle null hypothesis significance tests (NHST) for these situations. We demonstrate this method to analyze a range of simulated and experimental data. 1
TOPOLOGICAL CONSISTENCY VIA KERNEL ESTIMATION
, 2014
"... We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology struc ..."
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We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of superlevel sets of both density and regression functions. Our method is based on kernel estimation. We apply this procedure to two problems: 1) inferring the homology structure of manifolds from noisy observations, 2) inferring the persistent homology (a multiscale extension of homology) of either density or regression functions. We prove consistency for both of these problems. In addition to the theoretical results we demonstrate these methods on simulated data for binary regression and clustering applications.
and
, 2014
"... In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in R3 and shapes in R2. This statistic is a collection of persistence diagrams – multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and e ..."
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In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in R3 and shapes in R2. This statistic is a collection of persistence diagrams – multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in R3 into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for probability densities on the space of piecewise linear shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data. persistence homology, surfaces, shape spaces, sufficient shape statistics Insert classification here 1
Journal of Computational Geometry jocg.org STOCHASTIC CONVERGENCE OF PERSISTENCE LANDSCAPES AND SILHOUETTES
"... Abstract. Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multiset of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we summar ..."
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Abstract. Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multiset of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we summarize persistent homology with a persistence landscape, introduced by Bubenik, which converts a diagram into a wellbehaved realvalued function. We investigate the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence of the bootstrap. In addition, we introduce an alternate functional summary of persistent homology, which we call the silhouette, and derive an analogous statistical theory. 1