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Proximity of persistence modules and their diagrams
TLDR
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
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The Structure and Stability of Persistence Modules
This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such…
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Persistence stability for geometric complexes
TLDR
The properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces are studied.
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Towards persistence-based reconstruction in euclidean spaces
TLDR
A novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data is introduced.
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Provably good sampling and meshing of surfaces
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Sliced Wasserstein Kernel for Persistence Diagrams
TLDR
A new kernel for PDs is defined, which is not only provably stable but also discriminative (with a bound depending on the number of points in the PDs) w.r.t. the first diagram distance between PDs.
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Persistence-Based Clustering in Riemannian Manifolds
TLDR
A clustering scheme that combines a mode-seeking phase with a cluster merging phase in the corresponding density map, and whose output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density.
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Persistence Theory - From Quiver Representations to Data Analysis
  • S. Oudot
  • Mathematics
    Mathematical surveys and monographs
  • 2 December 2015
TLDR
Theoretical foundations: Algebraic persistence, Stability Applications: Topological inference, and Perspectives: New trends in topological data analysis.
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Gromov‐Hausdorff Stable Signatures for Shapes using Persistence
We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We…
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Provably Good Surface Sampling and Approximation
TLDR
It is proved that the algorithm can generate good samples on S as well as triangulated surfaces that approximate S, and it is shown that the restricted Delaunay triangulation Del\S of the points has the same topology type as S, that the Hausdorff distance between Del \S and S can be made arbitrarily small, and that the authors can bound the aspect ratio of the facets ofDel\S.
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