A persistence landscapes toolbox for topological statistics

@article{Bubenik2017APL,
  title={A persistence landscapes toolbox for topological statistics},
  author={Peter Bubenik and Pawel Dlotko},
  journal={J. Symb. Comput.},
  year={2017},
  volume={78},
  pages={91-114}
}
Multiparameter Persistence Landscapes
TLDR
It is shown that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance, and that the collection of multiparameters landscapes faithfully represents the rank invariant.
The Persistence Landscape and Some of Its Properties
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an
Persistent Homology: Functional Summaries of Persistence Diagrams for Time Series Analysis
TLDR
It is proved that the results reflected on the dependency relationship between persistence landscapes functional norms and variance-covariance for multivariate time series embedded via the sliding window method are valid for time series understanding them as a realization of a weakly stationary stochastic process.
Scalar Field Comparison with Topological Descriptors
TLDR
This work presents a state-of-the-art report on scalar field comparison using topological descriptors and provides a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles.
Kernel Method for Persistence Diagrams via Kernel Embedding and Weight Factor
TLDR
A kernel method on persistence diagrams is proposed that allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis.
Topological Data Analysis for Object Data
TLDR
A framework for using persistence landscapes to vectorize object data and perform statistical analysis is presented and the most persistent features are shown to be “topological noise” and the statistical analysis depends on the less persistent features which are referred to as the “geometric signal”.
Persistent homology detects curvature
TLDR
It is proved that persistent homology detects the curvature of disks from which points have been sampled, and a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space is described.
Topological spaces of persistence modules and their properties
TLDR
This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.
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References

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A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.
Stochastic Convergence of Persistence Landscapes and Silhouettes
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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.
On the Bootstrap for Persistence Diagrams and Landscapes
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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.
Statistical topology using persistence landscapes
TLDR
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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This paper derives confidence sets that allow us to separate topological signal from topological noise, and brings some statistical ideas to persistent homology.
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Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
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It is shown that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension.
A stable multi-scale kernel for topological machine learning
TLDR
This work designs a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data that is positive definite and proves its stability with respect to the 1-Wasserstein distance.
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