# Probability measures on the space of persistence diagrams

@article{Mileyko2011ProbabilityMO, title={Probability measures on the space of persistence diagrams}, author={Yuriy Mileyko and Sayan Mukherjee and John Harer}, journal={Inverse Problems}, year={2011}, volume={27}, pages={124007} }

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional probabilities. This provides a theoretical basis for a statistical treatment of persistence diagrams, for example computing sample averages and sample variances of persistence diagrams. We first prove that the space of persistence diagrams with the Wasserstein metric is complete and separable. We then…

## 210 Citations

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A persistence diagram characterizes robust geometric and topological features in data. Data, which will be treated here, are assumed to be drawn from a probability distribution and then the…

Means and medians of sets of persistence diagrams

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The space of persistence diagrams is looked at under a variety of different metrics which are analogous to L p metrics on the space of functions, which gives the natural definitions of both the mean and median of a finite number of persistence diagram.

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This work provides a modification and formalization of the persistence intensity function, which can be used to visualize multiple diagrams, perform clustering and conduct two-sample tests.

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- 2012

A new descriptor for persistent homology is defined, which is thought of as an embedding of the usual descriptors, barcodes and persistence diagrams, into a space of functions, which inherits an L norm, and it is shown that this metric space is complete and separable.

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- MathematicsProceedings of the American Mathematical Society
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We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an…

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