Confidence sets for persistence diagrams

@article{Fasy2014ConfidenceSF,
  title={Confidence sets for persistence diagrams},
  author={Brittany Terese Fasy and Fabrizio Lecci and Alessandro Rinaldo and Larry A. Wasserman and Sivaraman Balakrishnan and Aarti Singh},
  journal={The Annals of Statistics},
  year={2014},
  volume={42}
}
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short lifetimes are informally considered to be "topological noise," and those with a long lifetime are considered to be "topological signal." In this paper, we bring some statistical ideas to persistent homology. In particular, we derive confidence sets that allow us to… 
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245 Citations
Highly Influential Citations
19
Background Citations
95
Methods Citations
63
Results Citations
4

245 Citations

On the Bootstrap for Persistence Diagrams and Landscapes

This paper uses a statistical technique, the empirical bootstrap, to separate topological signal from topological noise, and derives confidence sets for persistence diagrams and confidence bands for persistence landscapes.

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

It is shown that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties.

Convergence rates for persistence diagram estimation in topological data analysis

It is shown that the use of persistent homology can be naturally considered in general statistical frameworks and established convergence rates of persistence diagrams associated to data randomly sampled from any compact metric space to a well defined limit diagram encoding the topological features of the support of the measure from which the data have been sampled.

Stochastic Convergence of Persistence Landscapes and Silhouettes

An alternate functional summary of persistent homology is introduced, which is called the silhouette, and an analogous statistical theory is derived that investigates the statistical properties of landscapes, such as weak convergence of the average landscapes and convergence ofThe bootstrap.

Cycle Registration in Persistent Homology with Applications in Topological Bootstrap

A novel approach for comparing the persistent homology representations of two spaces (or filtrations) by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence.

Bayesian Estimation of Topological Features of Persistence Diagrams

The approach followed in this work makes use of features’ lifetimes and provides a full Bayesian clustering model, based on random partitions, in order to estimate Betti numbers.

Subsampling Methods for Persistent Homology

This work proposes to compute the persistent homology of several subsamples of the data and then combines the resulting estimates to prove that the subsampling approach carries stable topological information while achieving a great reduction in computational complexity.

A persistence landscapes toolbox for topological statistics

Confidence sets for persistent homology of the KDE filtration

When we observe a point cloud in the Euclidean space, the persistent homology of the upper level sets filtration of the density is one of the most important tools to understand topological features…

Persistence Terrace for Topological Inference of Point Cloud Data

A novel topological summary plot is proposed that incorporates a wide range of smoothing parameters and is robust, multi-scale, and parameter-free, and allows one to infer the size and point density of the topological features.
...

References

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This paper uses a statistical technique, the empirical bootstrap, to separate topological signal from topological noise, and derives confidence sets for persistence diagrams and confidence bands for persistence landscapes.

Optimal rates of convergence for persistence diagrams in Topological Data Analysis

It is shown that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties.

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